lastfm_and_wiki_parsing.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import requests\n",
    "from bs4 import BeautifulSoup"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Target: all h2 and h3 HTML headings, [link](https://en.wikipedia.org/wiki/Bias%E2%80%93variance_tradeoff)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "r_1 = requests.get('https://en.wikipedia.org/wiki/Bias-variance_tradeoff')\n",
    "soup_1 = BeautifulSoup(r_1.text, 'html.parser')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Motivation\n",
      "Bias–variance decomposition of squared error\n",
      "Derivation\n",
      "Application to regression\n",
      "Application to classification\n",
      "Application to reinforcement learning\n",
      "Approaches\n",
      "k-nearest neighbors\n",
      "Application to human learning\n",
      "See also\n",
      "References\n"
     ]
    }
   ],
   "source": [
    "for heads in soup_1.findAll('span', attrs={'class':'mw-headline'}):\n",
    "    print(heads.text)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Target: titles of all articles in the category 'Machine Learning Algorithms', [link](https://en.wikipedia.org/wiki/Category:Machine_learning_algorithms)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "r_2 = requests.get('https://en.wikipedia.org/wiki/Category:Machine_learning_algorithms')\n",
    "soup_2 = BeautifulSoup(r_2.text, 'html.parser')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Adagrad\n",
      "Algorithms of Oppression\n",
      "Almeida–Pineda recurrent backpropagation\n",
      "Augmented Analytics\n",
      "Backpropagation\n",
      "Bioz\n",
      "Bootstrap aggregating\n",
      "Cancer Likelihood in Plasma\n",
      "CN2 algorithm\n",
      "Constructing skill trees\n",
      "Deep reinforcement learning\n",
      "Dehaene–Changeux model\n",
      "Diffusion map\n",
      "Dominance-based rough set approach\n",
      "Dynamic time warping\n",
      "Elastic net regularization\n",
      "Error-driven learning\n",
      "Evolutionary multimodal optimization\n",
      "Expectation–maximization algorithm\n",
      "Extremal Ensemble Learning\n",
      "FastICA\n",
      "Forward–backward algorithm\n",
      "GeneRec\n",
      "Genetic Algorithm for Rule Set Production\n",
      "Growing self-organizing map\n",
      "Hyper basis function network\n",
      "IDistance\n",
      "Incremental learning\n",
      "K-nearest neighbors algorithm\n",
      "Kernel methods for vector output\n",
      "Kernel principal component analysis\n",
      "Label propagation algorithm\n",
      "Lasso (statistics)\n",
      "Leabra\n",
      "Linde–Buzo–Gray algorithm\n",
      "Local outlier factor\n",
      "Logic learning machine\n",
      "LogitBoost\n",
      "Loss functions for classification\n",
      "Manifold alignment\n",
      "Minimum redundancy feature selection\n",
      "Mixture of experts\n",
      "Multiple kernel learning\n",
      "Non-negative matrix factorization\n",
      "Online machine learning\n",
      "Out-of-bag error\n",
      "Prefrontal cortex basal ganglia working memory\n",
      "Prototype methods\n",
      "PVLV\n",
      "Q-learning\n",
      "Quadratic unconstrained binary optimization\n",
      "Query-level feature\n",
      "Quickprop\n",
      "Radial basis function network\n",
      "Randomized weighted majority algorithm\n",
      "Repeated incremental pruning to produce error reduction (RIPPER)\n",
      "Rprop\n",
      "Rule-based machine learning\n",
      "Skill chaining\n",
      "Sparse PCA\n",
      "State–action–reward–state–action\n",
      "Stochastic gradient descent\n",
      "Structured kNN\n",
      "T-distributed stochastic neighbor embedding\n",
      "Triplet loss\n",
      "Wake-sleep algorithm\n",
      "Weighted majority algorithm (machine learning)\n",
      "Zero-shot learning\n"
     ]
    }
   ],
   "source": [
    "for i in soup_2.findAll('div', attrs={'class':'mw-category-group'}):\n",
    "    for elem in i.findAll('li'):\n",
    "        print(elem.text)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Target: names of all black metal bands from the list, [link](https://www.last.fm/tag/black+metal/artists)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Page https://www.last.fm/tag/black+metal/artists?page=1 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=2 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=3 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=4 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=5 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=6 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=7 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=8 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=9 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=10 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=11 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=12 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=13 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=14 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=15 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=16 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=17 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=18 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=19 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=20 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=21 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=22 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=23 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=24 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=25 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=26 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=27 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=28 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=29 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=30 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=31 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=32 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=33 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=34 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=35 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=36 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=37 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=38 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=39 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=40 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=41 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=42 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=43 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=44 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=45 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=46 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=47 is loaded.\n",
      "Page https://www.last.fm/tag/black+metal/artists?page=48 is loaded.\n",
      "\n",
      "CPU times: user 5.65 s, sys: 199 ms, total: 5.85 s\n",
      "Wall time: 1min 2s\n"
     ]
    }
   ],
   "source": [
    "%%time\n",
    "\n",
    "URLs = []\n",
    "soup_3 = []\n",
    "\n",
    "# getting all links\n",
    "for page_num in range(1,49):\n",
    "    URLs.append('https://www.last.fm/tag/black+metal/artists?page=' + str(page_num))\n",
    "\n",
    "# extracting\n",
    "for i in URLs:\n",
    "    soup_3.append(BeautifulSoup((requests.get(i)).text, 'html.parser'))\n",
    "    print('Page', i, 'is loaded.')\n",
    "print('')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "48\n"
     ]
    }
   ],
   "source": [
    "# list of HTML texts\n",
    "print(len(soup_3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# getting band names\n",
    "\n",
    "bands = []\n",
    "\n",
    "for p in range(0, 48):\n",
    "    for heads in soup_3[p].findAll('h3'):\n",
    "        bands.append(heads.text)\n",
    "    bands.remove('Scrobble from Spotify?')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Burzum\n",
      "Immortal\n",
      "Dimmu Borgir\n",
      "Darkthrone\n",
      "Satyricon\n",
      "Mayhem\n",
      "Emperor\n",
      "Bathory\n",
      "Behemoth\n",
      "Cradle of Filth\n",
      "Marduk\n",
      "Gorgoroth\n",
      "Ulver\n",
      "1349\n",
      "Enslaved\n",
      "Dark Funeral\n",
      "Carpathian Forest\n",
      "Venom\n",
      "Anaal Nathrakh\n",
      "Summoning\n",
      "Dissection\n",
      "Celtic Frost\n",
      "Samael\n",
      "Drudkh\n",
      "Deathspell Omega\n",
      "Taake\n",
      "Finntroll\n",
      "Rotting Christ\n",
      "Agalloch\n",
      "Watain\n",
      "Wolves in the Throne Room\n",
      "Nargaroth\n",
      "Shining\n",
      "Windir\n",
      "Alcest\n",
      "Arcturus\n",
      "Borknagar\n",
      "Belphegor\n",
      "Blut aus Nord\n",
      "Naglfar\n",
      "Xasthur\n",
      "Impaled Nazarene\n",
      "Nokturnal Mortum\n",
      "Graveworm\n",
      "Old Man's Child\n",
      "Moonspell\n",
      "Satanic Warmaster\n",
      "Peste Noire\n",
      "Primordial\n",
      "Beherit\n",
      "Keep of Kalessin\n",
      "Lifelover\n",
      "Judas Iscariot\n",
      "Amesoeurs\n",
      "Limbonic Art\n",
      "Tsjuder\n",
      "Leviathan\n",
      "Behexen\n",
      "Sigh\n",
      "Anorexia Nervosa\n",
      "Horna\n",
      "Negură Bunget\n",
      "Melechesh\n",
      "Nachtmystium\n",
      "Abigor\n",
      "Absu\n",
      "Moonsorrow\n",
      "Falkenbach\n",
      "Graveland\n",
      "Darkspace\n",
      "Ajattara\n",
      "Enthroned\n",
      "Funeral Mist\n",
      "Dark Fortress\n",
      "Dornenreich\n",
      "Arckanum\n",
      "Forgotten Tomb\n",
      "Endstille\n",
      "Inquisition\n",
      "Hellhammer\n",
      "Deströyer 666\n",
      "Silencer\n",
      "Lunar Aurora\n",
      "Kampfar\n",
      "Isengard\n",
      "Mütiilation\n",
      "Sargeist\n",
      "Catamenia\n",
      "Ragnarok\n",
      "Eisregen\n",
      "Nattefrost\n",
      "Urgehal\n",
      "Vesania\n",
      "Mgła\n",
      "Absurd\n",
      "ColdWorld\n",
      "Hate Forest\n",
      "Lord Belial\n",
      "Inferius Torment\n",
      "Gehenna\n",
      "Vreid\n",
      "Kvelertak\n",
      "Ihsahn\n",
      "Nortt\n",
      "Sarcófago\n",
      "Dødheimsgard\n",
      "Agathodaimon\n",
      "Craft\n",
      "Diabolical Masquerade\n",
      "Sodom\n",
      "Ad Hominem\n",
      "Altar of Plagues\n",
      "Mercyful Fate\n",
      "Antaeus\n",
      "Katharsis\n",
      "Setherial\n",
      "Urfaust\n",
      "Nocte Obducta\n",
      "Azaghal\n",
      "Akercocke\n",
      "Sólstafir\n",
      "Aura Noir\n",
      "Vlad Tepes\n",
      "Justin Bieber\n",
      "The Ruins of Beverast\n",
      "Bethlehem\n",
      "Deafheaven\n",
      "Темнозорь\n",
      "Armagedda\n",
      "Aborym\n",
      "Ondskapt\n",
      "Thyrfing\n",
      "Secrets of the Moon\n",
      "Paysage d'Hiver\n",
      "The Kovenant\n",
      "Barathrum\n",
      "Arkona\n",
      "I\n",
      "Månegarm\n",
      "Aaskereia\n",
      "Infernal War\n",
      "Fen\n",
      "Moonblood\n",
      "Gallhammer\n",
      "Lantlôs\n",
      "Abigail Williams\n",
      "Nocturnal Depression\n",
      "Archgoat\n",
      "Bal-Sagoth\n",
      "Darkened Nocturn Slaughtercult\n",
      "Astarte\n",
      "Helrunar\n",
      "Ildjarn\n",
      "Wyrd\n",
      "Glorior Belli\n",
      "Kroda\n",
      "Nagelfar\n",
      "Furia\n",
      "Ancient\n",
      "Theatres Des Vampires\n",
      "Woods of Ypres\n",
      "Abyssic Hate\n",
      "Austere\n",
      "Khold\n",
      "Blasphemy\n",
      "Thorns\n",
      "Solefald\n",
      "A Forest of Stars\n",
      "In the Woods...\n",
      "Velvet Cacoon\n",
      "Krieg\n",
      "Aosoth\n",
      "Nifelheim\n",
      "Lurker of Chalice\n",
      "...and Oceans\n",
      "Forgotten Woods\n",
      "Chthonic\n",
      "Panopticon\n",
      "Nyktalgia\n",
      "Carach Angren\n",
      "Children of Bodom\n",
      "Imperium Dekadenz\n",
      "Clandestine Blaze\n",
      "Opera IX\n",
      "Koldbrann\n",
      "Ancient Rites\n",
      "Massemord\n",
      "Krallice\n",
      "Walknut\n",
      "Gris\n",
      "Stormlord\n",
      "Besatt\n",
      "Mortifera\n",
      "Make a Change... Kill Yourself\n",
      "Celeste\n",
      "Woods of Desolation\n",
      "Helheim\n",
      "Kriegsmaschine\n",
      "Oranssi Pazuzu\n",
      "Angantyr\n",
      "Vintersorg\n",
      "Grand Belial's Key\n",
      "Black Witchery\n",
      "Triptykon\n",
      "Goatwhore\n",
      "Baptism\n",
      "Enochian Crescent\n",
      "Von\n",
      "Alghazanth\n",
      "Abruptum\n",
      "Drowning the Light\n",
      "Astrofaes\n",
      "Sear Bliss\n",
      "Hypothermia\n",
      "Forefather\n",
      "Desaster\n",
      "Pest\n",
      "Siebenbürgen\n",
      "Vinterriket\n",
      "Master's Hammer\n",
      "Zyklon\n",
      "Einherjer\n",
      "Thy Light\n",
      "Krohm\n",
      "Mirrorthrone\n",
      "Goatmoon\n",
      "King Diamond\n",
      "Lux Occulta\n",
      "Agrypnie\n",
      "Weakling\n",
      "Woods of Infinity\n",
      "Totalselfhatred\n",
      "I Shalt Become\n",
      "Myrkgrav\n",
      "Cobalt\n",
      "Hecate Enthroned\n",
      "Tormentor\n",
      "Cadaveria\n",
      "Sacramentum\n",
      "Trelldom\n",
      "Trist\n",
      "Mysticum\n",
      "Sterbend\n",
      "Corpus Christii\n",
      "An Autumn for Crippled Children\n",
      "Ljå\n",
      "Belketre\n",
      "October Falls\n",
      "Svartsyn\n",
      "Ofermod\n",
      "Liturgy\n",
      "Shade Empire\n",
      "Verdunkeln\n",
      "Heretoir\n",
      "Twilight\n",
      "Black Funeral\n",
      "Gnaw Their Tongues\n",
      "Celestia\n",
      "Akitsa\n",
      "Nocternity\n",
      "Minas Morgul\n",
      "Darkestrah\n",
      "Kvist\n",
      "Ved Buens Ende\n",
      "Ov Hell\n",
      "Batushka\n",
      "Lucifugum\n",
      "Dub Buk\n",
      "Manes\n",
      "Toxic Holocaust\n",
      "Teitanblood\n",
      "Aeternus\n",
      "Cor Scorpii\n",
      "Bilskirnir\n",
      "Root\n",
      "Necromantia\n",
      "Troll\n",
      "Thyrane\n",
      "Belenos\n",
      "Antestor\n",
      "Zyklon-B\n",
      "Arkhon Infaustus\n",
      "Wedard\n",
      "Morgul\n",
      "Necrophobic\n",
      "Forest Silence\n",
      "Winterfylleth\n",
      "Black Messiah\n",
      "Der Weg einer Freiheit\n",
      "Anti\n",
      "Merrimack\n",
      "Skyforger\n",
      "Temple of Baal\n",
      "The Meads of Asphodel\n",
      "Nae'blis\n",
      "Portal\n",
      "Blood of Kingu\n",
      "Kat\n",
      "Mörk Gryning\n",
      "Elysian Blaze\n",
      "Varathron\n",
      "Den Saakaldte\n",
      "Luror\n",
      "Darzamat\n",
      "Menace Ruine\n",
      "Nightbringer\n",
      "Skeletonwitch\n",
      "Khors\n",
      "Farsot\n",
      "Thunderbolt\n",
      "Morbid\n",
      "Striborg\n",
      "Sabbat\n",
      "Deicide\n",
      "Countess\n",
      "Crionics\n",
      "Hell Militia\n",
      "Beastcraft\n",
      "Iperyt\n",
      "Circle of Ouroborus\n",
      "Negator\n",
      "Nehëmah\n",
      "Negură Bunget\n",
      "Code\n",
      "Todtgelichter\n",
      "Happy Days\n",
      "Branikald\n",
      "Abigail\n",
      "Forest\n",
      "Averse Sefira\n",
      "Iskald\n",
      "Forteresse\n",
      "Skagos\n",
      "Brocken Moon\n",
      "Thy Catafalque\n",
      "Aes Dana\n",
      "Ash Borer\n",
      "Krypt\n",
      "Vinterland\n",
      "IXXI\n",
      "Blaze of Perdition\n",
      "Revenge\n",
      "Nazgul\n",
      "Profanatica\n",
      "The Axis of Perdition\n",
      "Hellveto\n",
      "Blutklinge\n",
      "M8L8TH\n",
      "Lustre\n",
      "Sombres Forêts\n",
      "Pensées Nocturnes\n",
      "Kataxu\n",
      "Bestial Mockery\n",
      "Peccatum\n",
      "Ne Obliviscaris\n",
      "Cirith Gorgor\n",
      "Thy Serpent\n",
      "Torgeist\n",
      "Forest of Fog\n",
      "Blodsrit\n",
      "Wodensthrone\n",
      "Acherontas\n",
      "Dødsengel\n",
      "Illnath\n",
      "Mortuary Drape\n",
      "Old Wainds\n",
      "Christ Agony\n",
      "Elffor\n",
      "Mithotyn\n",
      "Ascension\n",
      "Bone Awl\n",
      "Valkyrja\n",
      "Ashen Light\n",
      "Graupel\n",
      "The Abyss\n",
      "Falls of Rauros\n",
      "Monarque\n",
      "Dawn\n",
      "Profanum\n",
      "Wigrid\n",
      "Mystic Circle\n",
      "Svartidauði\n",
      "Raventale\n",
      "Horn\n",
      "Tvangeste\n",
      "Dodsferd\n",
      "Mystifier\n",
      "Caïna\n",
      "Proclamation\n",
      "Cultes des Ghoules\n",
      "Aäkon Këëtrëh\n",
      "Ancient Wisdom\n",
      "Woe\n",
      "Kristallnacht\n",
      "Gontyna Kry\n",
      "Beatrik\n",
      "Nachtfalke\n",
      "Ram-Zet\n",
      "Zemial\n",
      "Morrigan\n",
      "Al-Namrood\n",
      "Menhir\n",
      "Bestial Warlust\n",
      "Nessie & Her Beard\n",
      "Hades\n",
      "Ophthalamia\n",
      "Cryptic Wintermoon\n",
      "Celestial Bloodshed\n",
      "Hermh\n",
      "Negative Plane\n",
      "Tulus\n",
      "Bishop Of Hexen\n",
      "Epheles\n",
      "Evilfeast\n",
      "Pseudogod\n",
      "Equilibrium\n",
      "Strid\n",
      "Angmar\n",
      "Brenoritvrezorkre\n",
      "Inferno\n",
      "Szron\n",
      "Lyrinx\n",
      "Musta Surma\n",
      "Sun of the Blind\n",
      "Stielas Storhett\n",
      "Dragonlord\n",
      "Varg\n",
      "Spektr\n",
      "Demonaz\n",
      "Nazxul\n",
      "Finsterforst\n",
      "Sun of the Sleepless\n",
      "Sombre Chemin\n",
      "Odal\n",
      "Klabautamann\n",
      "Ravencult\n",
      "Thou Art Lord\n",
      "Lik\n",
      "Amon Amarth\n",
      "Fäulnis\n",
      "Veil\n",
      "Lutomysl\n",
      "Iskra\n",
      "Svarttjern\n",
      "Vargsang\n",
      "Ludicra\n",
      "Kawir\n",
      "Fleurety\n",
      "Thränenkind\n",
      "Cold Body Radiation\n",
      "The Legion\n",
      "Arkha Sva\n",
      "War\n",
      "Rivendell\n",
      "Necros Christos\n",
      "Gestapo 666\n",
      "Totenburg\n",
      "XIV Dark Centuries\n",
      "Covenant\n",
      "Mord\n",
      "Demoncy\n",
      "Kult ov Azazel\n",
      "Hellsaw\n",
      "Disiplin\n",
      "Elite\n",
      "Obtained Enslavement\n",
      "Havohej\n",
      "Mortuus\n",
      "Aeba\n",
      "Slavland\n",
      "S.V.E.S.T.\n",
      "Otargos\n",
      "Impiety\n",
      "Gloomy Grim\n",
      "Moëvöt\n",
      "Heresi\n",
      "Witchmaster\n",
      "Conqueror\n",
      "Midnight Odyssey\n",
      "Myrkur\n",
      "Zarathustra\n",
      "Maniac Butcher\n",
      "Morbosidad\n",
      "God Seed\n",
      "Witchery\n",
      "Gaahlskagg\n",
      "Haemoth\n",
      "Nuit Noire\n",
      "Mustan Kuun Lapset\n",
      "Riger\n",
      "Blackdeath\n",
      "Blacklodge\n",
      "North\n",
      "Non Opus Dei\n",
      "Unexpect\n",
      "Arthemesia\n",
      "Avsky\n",
      "Dordeduh\n",
      "Plaga\n",
      "Clair Cassis\n",
      "Spite Extreme Wing\n",
      "Ásmegin\n",
      "Fjoergyn\n",
      "Bulldozer\n",
      "Wolfsmond\n",
      "Loits\n",
      "Hypomanie\n",
      "Young and in the Way\n",
      "Drautran\n",
      "Melencolia Estatica\n",
      "Black Murder\n",
      "Camulos\n",
      "Blasphemophagher\n",
      "Sarke\n",
      "Funeral Winds\n",
      "Morowe\n",
      "Opeth\n",
      "Semargl\n",
      "Seth\n",
      "Samsas Traum\n",
      "Alkonost\n",
      "Nachtmahr\n",
      "Fluisterwoud\n",
      "Mgła\n",
      "Csejthe\n",
      "Life is Pain\n",
      "Throne of Katarsis\n",
      "Diocletian\n",
      "Sorhin\n",
      "Gospel of the Horns\n",
      "Heimdalls Wacht\n",
      "Geist\n",
      "Abazagorath\n",
      "Fear of Eternity\n",
      "Nåstrond\n",
      "Alda\n",
      "Naer Mataron\n",
      "Skogen\n",
      "Gernotshagen\n",
      "Bergthron\n",
      "Isvind\n",
      "Dragged Into Sunlight\n",
      "Leichenzug\n",
      "Darkwoods My Betrothed\n",
      "Susperia\n",
      "Nahvalr\n",
      "Sigrblot\n",
      "Orcivus\n",
      "Ceremonial Castings\n",
      "Veles\n",
      "Dismal Euphony\n",
      "Furze\n",
      "Lugubrum\n",
      "Thulcandra\n",
      "Daemonarch\n",
      "Njiqahdda\n",
      "God Dethroned\n",
      "Urn\n",
      "Funeris Nocturnum\n",
      "Gallowbraid\n",
      "Der Stürmer\n",
      "TrollfesT\n",
      "Marblebog\n",
      "Grabnebelfürsten\n",
      "Zuriaake\n",
      "Mistur\n",
      "Miserere Luminis\n",
      "Bornholm\n",
      "Cult of Fire\n",
      "Truppensturm\n",
      "Jotunspor\n",
      "Be Persecuted\n",
      "Frost\n",
      "Midnight\n",
      "Butterfly Temple\n",
      "Seigneur Voland\n",
      "Crimson Moonlight\n",
      "Inferi\n",
      "Faustcoven\n",
      "Tenebrae in Perpetuum\n",
      "Evil\n",
      "Psychonaut 4\n",
      "Apati\n",
      "Bosse-de-Nage\n",
      "Svart\n",
      "Joyless\n",
      "Ash Pool\n",
      "Saltus\n",
      "Rossomahaar\n",
      "Misþyrming\n",
      "Skitliv\n",
      "Asmodeus\n",
      "Mystic Forest\n",
      "Vrolok\n",
      "Enslavement of Beauty\n",
      "Aluk Todolo\n",
      "Slechtvalk\n",
      "Dark Fury\n",
      "Denial of God\n",
      "Sun Devoured Earth\n",
      "Black Autumn\n",
      "Triumphator\n",
      "Halgadom\n",
      "Devian\n",
      "Dark Tribe\n",
      "Ruins\n",
      "Infernal\n",
      "Thor's Hammer\n",
      "Adorned Brood\n",
      "Sarkom\n",
      "Murmuüre\n",
      "Vordr\n",
      "Wolfnacht\n",
      "Horde\n",
      "Dimhymn\n",
      "Moon\n",
      "Storm\n",
      "Bloodthorn\n",
      "Magog\n",
      "Sortsind\n",
      "Caladan Brood\n",
      "Thurisaz\n",
      "Fuck Off And Die!\n",
      "Bewitched\n",
      "Obtest\n",
      "Trollech\n",
      "Fukpig\n",
      "Isolation\n",
      "Dark Forest\n",
      "Malign\n",
      "Sadistik Exekution\n",
      "Nachtblut\n",
      "Imperial Crystalline Entombment\n",
      "Xibalba\n",
      "The Black\n",
      "Acrimonious\n",
      "Lyfthrasyr\n",
      "Impaled Northern Moonforest\n",
      "Aryan Blood\n",
      "Cosmic Church\n",
      "Kadenzza\n",
      "Darvulia\n",
      "Sorgsvart\n",
      "Dragobrath\n",
      "Endless Dismal Moan\n",
      "Abbath\n",
      "Ewiges Reich\n",
      "Deinonychus\n",
      "Warloghe\n",
      "Raate\n",
      "Reverence\n",
      "Ikuinen Kaamos\n",
      "Volahn\n",
      "Necrodeath\n",
      "Trollheim's Grott\n",
      "Líam\n",
      "Slagmaur\n",
      "Zorn\n",
      "Arafel\n",
      "Blood Red Fog\n",
      "Borgne\n",
      "Förgjord\n",
      "Dodecahedron\n",
      "Svartahrid\n",
      "Deathcult\n",
      "Sui Caedere\n",
      "Eternity\n",
      "Goatpenis\n",
      "Abusiveness\n",
      "Bekhira\n",
      "Radigost\n",
      "Luna Ad Noctum\n",
      "Saor\n",
      "Dråpsnatt\n",
      "Runenblut\n",
      "Evol\n",
      "Stutthof\n",
      "Martyrdöd\n",
      "Dolorian\n",
      "Funeral Mourning\n",
      "Djevel\n",
      "Tartaros\n",
      "Benighted in Sodom\n",
      "Ekove Efrits\n",
      "Pestilential Shadows\n",
      "Dantalion\n",
      "Die Apokalyptischen Reiter\n",
      "Hexis\n",
      "All The Cold\n",
      "Himinbjørg\n",
      "Diabolicum\n",
      "Nocturnal\n",
      "Kromlek\n",
      "Moloch\n",
      "Armaggedon\n",
      "Ygg\n",
      "...aaaarrghh...\n",
      "Thantifaxath\n",
      "Bergraven\n",
      "Horned Almighty\n",
      "Legion of Doom\n",
      "Uruk-Hai\n",
      "Infernum\n",
      "Ohtar\n",
      "IC Rex\n",
      "The Great Old Ones\n",
      "Sworn\n",
      "Exordium\n",
      "Heol Telwen\n",
      "Nihil Nocturne\n",
      "Draugnim\n",
      "Black Altar\n",
      "Todeskult\n",
      "Nightfall\n",
      "Sorcier Des Glaces\n",
      "Dusk\n",
      "Decayed\n",
      "Hellfucked\n",
      "Vital Remains\n",
      "Svarti Loghin\n",
      "Spectral Lore\n",
      "Mare\n",
      "Istapp\n",
      "Velnias\n",
      "Osculum Infame\n",
      "Desolation Triumphalis\n",
      "Askival\n",
      "Unanimated\n",
      "Pantheon I\n",
      "Wolvhammer\n",
      "Ithdabquth Qliphoth\n",
      "Sort Vokter\n",
      "Vitsaus\n",
      "Capricornus\n",
      "Vestiges\n",
      "Fanisk\n",
      "Ensiferum\n",
      "Häive\n",
      "Thy Primordial\n",
      "Kekal\n",
      "Galar\n",
      "Teratism\n",
      "Aryan Terrorism\n",
      "Bloodhammer\n",
      "Alrakis\n",
      "Ephel Duath\n",
      "Ador Dorath\n",
      "Transcending Bizarre?\n",
      "The Secret\n",
      "Fell Voices\n",
      "Gromm\n",
      "Weltmacht\n",
      "Ride For Revenge\n",
      "Neige et Noirceur\n",
      "Petrychor\n",
      "Valfunde\n",
      "Castevet\n",
      "Kadotus\n",
      "Nordglanz\n",
      "Cryfemal\n",
      "Hypocrisy\n",
      "(V.E.G.A.)\n",
      "Frostmoon Eclipse\n",
      "Drottnar\n",
      "Sacrilegious Impalement\n",
      "Arch Enemy\n",
      "Moontower\n",
      "Avathar\n",
      "Malfeitor\n",
      "Pagan Reign\n",
      "In Vain\n",
      "Orcustus\n",
      "Hordagaard\n",
      "Janvs\n",
      "Bann\n",
      "Tombs\n",
      "Agatus\n",
      "Botanist\n",
      "Funeral Procession\n",
      "Spear of Longinus\n",
      "Wolven Ancestry\n",
      "Luctus\n",
      "Hetroertzen\n",
      "Karg\n",
      "Thaw\n",
      "Mactätus\n",
      "Seviss\n",
      "Nefandus\n",
      "Fauna\n",
      "Nitberg\n",
      "Hades Almighty\n",
      "I'm in a Coffin\n",
      "Заводь\n",
      "Trancelike Void\n",
      "Dopamine\n",
      "Sacrificia Mortuorum\n",
      "Ashdautas\n",
      "Arizmenda\n",
      "Harakiri for the Sky\n",
      "In Flames\n",
      "Tyrant\n",
      "Addaura\n",
      "Order Of The Ebon Hand\n",
      "Anubi\n",
      "Godkiller\n",
      "Antediluvian\n",
      "Die Toten kehren wieder mit dem Wind\n",
      "Aetherius Obscuritas\n",
      "Abyssos\n",
      "666\n",
      "Absentia Lunae\n",
      "Selbstmord\n",
      "Nidingr\n",
      "Diaboli\n",
      "Bloodrain\n",
      "Niflheim\n",
      "Gorath\n",
      "Vhernen\n",
      "Verge\n",
      "Graven\n",
      "Dernier Martyr\n",
      "Empyrium\n",
      "Dorn\n",
      "Leech\n",
      "Locrian\n",
      "Devilish Impressions\n",
      "Angst Skvadron\n",
      "Coffinworm\n",
      "Waldgeflüster\n",
      "Chaos Moon\n",
      "Forest Stream\n",
      "Hail Spirit Noir\n",
      "Unholy Matrimony\n",
      "Andras\n",
      "Balrog\n",
      "Mondstille\n",
      "Angra Mainyu\n",
      "Azhubham Haani\n",
      "Exiled From Light\n",
      "Bloodline\n",
      "Berserk\n",
      "Abused Majesty\n",
      "Wrath of the Weak\n",
      "Enbilulugugal\n",
      "Nachtreich\n",
      "Árstíðir Lífsins\n",
      "Obscurity\n",
      "Weapon\n",
      "Myrkr\n",
      "Old\n",
      "Eternal Majesty\n",
      "Fimbulwinter\n",
      "Lönndom\n",
      "Black Anvil\n",
      "Nekrokrist SS\n",
      "Sad\n",
      "Dark Tranquillity\n",
      "Unlight\n",
      "Abhorer\n",
      "Nebelhorn\n",
      "Obscurus Advocam\n",
      "Sapthuran\n",
      "Avichi\n",
      "Germ\n",
      "Ghost Bath\n",
      "The Austrasian Goat\n",
      "Nocturnal Breed\n",
      "Sleeping Peonies\n",
      "De Magia Veterum\n",
      "Wold\n",
      "Eikenskaden\n",
      "Tymah\n",
      "Morte Incandescente\n",
      "Fullmoon\n",
      "Flagellant\n",
      "Anal Blasphemy\n",
      "Munruthel\n",
      "Sturmkaiser\n",
      "Shyy\n",
      "Throne of Ahaz\n",
      "Black Dawn\n",
      "Megiddo\n",
      "Orlog\n",
      "Woodtemple\n",
      "Apostasy\n",
      "Foscor\n",
      "Pantheon\n",
      "Nazgûl\n",
      "Sektemtum\n",
      "Мор\n",
      "Kalmah\n",
      "One Tail, One Head\n",
      "Eisblut\n",
      "Iuvenes\n",
      "Lutomysl\n",
      "Aryan Art\n",
      "Treblinka\n",
      "Old Funeral\n",
      "Svartkraft\n",
      "Nasheim\n",
      "Thornium\n",
      "Abyssmal Sorrow\n",
      "Pagan Hellfire\n",
      "Bölzer\n",
      "Malveillance\n",
      "Wolfshade\n",
      "Vader\n",
      "Wormlust\n",
      "Kathaarsys\n",
      "Raven Dark\n",
      "Crebain\n",
      "Bestial Summoning\n",
      "Nordreich\n",
      "Soliness\n",
      "Carpe Noctem\n",
      "Curse\n",
      "Mörker\n",
      "Axamenta\n",
      "Yoga\n",
      "Macabre Omen\n",
      "Cry of Silence\n",
      "Amestigon\n",
      "Vulcano\n",
      "Myrkskog\n",
      "Sauron\n",
      "Folkvang\n",
      "Drastus\n",
      "In Lingua Mortua\n",
      "Sinmara\n",
      "Warhead\n",
      "Numen\n",
      "Kältetod\n",
      "Black Crucifixion\n",
      "Airs\n",
      "Azazel\n",
      "Black Countess\n",
      "Anthelion\n",
      "De Silence et d'Ombre\n",
      "Sturmgeist\n",
      "Vi\n",
      "Bael\n",
      "Demoniac\n",
      "Frostkrieg\n",
      "Torturium\n",
      "Procer Veneficus\n",
      "Slidhr\n",
      "Throes of Dawn\n",
      "Lord Mantis\n",
      "Elderwind\n",
      "Rimfrost\n",
      "Barbatos\n",
      "Dekadent\n",
      "Infestus\n",
      "Mortualia\n",
      "Noenum\n",
      "Korgonthurus\n",
      "Satans Elite Kommando\n",
      "Manierisme\n",
      "Katatonia\n",
      "Norther\n",
      "Abgott\n",
      "Ungod\n",
      "Lugubre\n",
      "Dead Reptile Shrine\n",
      "Handful of Hate\n",
      "Hromovlad\n",
      "Schammasch\n",
      "Christicide\n",
      "Permafrost\n",
      "Black Shadow\n",
      "Sportlov\n",
      "Dishammer\n",
      "Calvarium\n",
      "Hateful Abandon\n",
      "Sad Legend\n",
      "Mutilation Rites\n",
      "Ayat\n",
      "Todesstoß\n",
      "Wurdulak\n",
      "Yellow Eyes\n",
      "Ancestral\n",
      "Svartediket\n",
      "Anhedonia\n",
      "Viking Crown\n",
      "Mourning Dawn\n",
      "Grimlair\n",
      "Acheron\n",
      "Taran\n",
      "Asgaard\n",
      "Bestial Raids\n",
      "Mord'A'Stigmata\n",
      "Uada\n",
      "Taunusheim\n",
      "Dies Ater\n",
      "Vesperian Sorrow\n",
      "Geimhre\n",
      "Acid Enema\n",
      "Malicious Secrets\n",
      "Unlord\n",
      "The Black Dahlia Murder\n"
     ]
    }
   ],
   "source": [
    "for band in bands:\n",
    "    print(band)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1000\n"
     ]
    }
   ],
   "source": [
    "print(len(bands))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "<h1 class=\"firstHeading\" id=\"firstHeading\" lang=\"en\">Bias–variance tradeoff</h1>\n",
      "<div class=\"mw-body-content\" id=\"bodyContent\">\n",
      "<div class=\"noprint\" id=\"siteSub\">From Wikipedia, the free encyclopedia</div>\n",
      "<div id=\"contentSub\"><span class=\"mw-redirectedfrom\">  (Redirected from <a class=\"mw-redirect\" href=\"/w/index.php?title=Bias-variance_tradeoff&amp;redirect=no\" title=\"Bias-variance tradeoff\">Bias-variance tradeoff</a>)</span></div>\n",
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      "<div class=\"mw-content-ltr\" dir=\"ltr\" id=\"mw-content-text\" lang=\"en\"><div class=\"mw-parser-output\"><table class=\"vertical-navbox nowraplinks\" style=\"float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;background:#f8f9fa;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%\"><tbody><tr><td style=\"padding-top:0.4em;line-height:1.2em\">Part of a series on</td></tr><tr><th style=\"padding:0.2em 0.4em 0.2em;padding-top:0;font-size:145%;line-height:1.2em\"><a href=\"/wiki/Machine_learning\" title=\"Machine learning\">Machine learning</a><br/>and<br/><a href=\"/wiki/Data_mining\" title=\"Data mining\">data mining</a></th></tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\">Problems</div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Statistical_classification\" title=\"Statistical classification\">Classification</a></li>\n",
      "<li><a href=\"/wiki/Cluster_analysis\" title=\"Cluster analysis\">Clustering</a></li>\n",
      "<li><a href=\"/wiki/Regression_analysis\" title=\"Regression analysis\">Regression</a></li>\n",
      "<li><a href=\"/wiki/Anomaly_detection\" title=\"Anomaly detection\">Anomaly detection</a></li>\n",
      "<li><a href=\"/wiki/Automated_machine_learning\" title=\"Automated machine learning\">AutoML</a></li>\n",
      "<li><a href=\"/wiki/Association_rule_learning\" title=\"Association rule learning\">Association rules</a></li>\n",
      "<li><a href=\"/wiki/Reinforcement_learning\" title=\"Reinforcement learning\">Reinforcement learning</a></li>\n",
      "<li><a href=\"/wiki/Structured_prediction\" title=\"Structured prediction\">Structured prediction</a></li>\n",
      "<li><a href=\"/wiki/Feature_engineering\" title=\"Feature engineering\">Feature engineering</a></li>\n",
      "<li><a href=\"/wiki/Feature_learning\" title=\"Feature learning\">Feature learning</a></li>\n",
      "<li><a href=\"/wiki/Online_machine_learning\" title=\"Online machine learning\">Online learning</a></li>\n",
      "<li><a href=\"/wiki/Semi-supervised_learning\" title=\"Semi-supervised learning\">Semi-supervised learning</a></li>\n",
      "<li><a href=\"/wiki/Unsupervised_learning\" title=\"Unsupervised learning\">Unsupervised learning</a></li>\n",
      "<li><a href=\"/wiki/Learning_to_rank\" title=\"Learning to rank\">Learning to rank</a></li>\n",
      "<li><a href=\"/wiki/Grammar_induction\" title=\"Grammar induction\">Grammar induction</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><div style=\"display:inline-block; padding:0.1em 0;line-height:1.2em;\"><a href=\"/wiki/Supervised_learning\" title=\"Supervised learning\">Supervised learning</a><br/><style data-mw-deduplicate=\"TemplateStyles:r886047488\">.mw-parser-output .nobold{font-weight:normal}</style><span class=\"nobold\"><span style=\"font-size:85%;\">(<b><a href=\"/wiki/Statistical_classification\" title=\"Statistical classification\">classification</a></b> • <b><a href=\"/wiki/Regression_analysis\" title=\"Regression analysis\">regression</a></b>)</span></span> </div></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Decision_tree_learning\" title=\"Decision tree learning\">Decision trees</a></li>\n",
      "<li><a href=\"/wiki/Ensemble_learning\" title=\"Ensemble learning\">Ensembles</a>\n",
      "<ul><li><a href=\"/wiki/Bootstrap_aggregating\" title=\"Bootstrap aggregating\">Bagging</a></li>\n",
      "<li><a href=\"/wiki/Boosting_(machine_learning)\" title=\"Boosting (machine learning)\">Boosting</a></li>\n",
      "<li><a href=\"/wiki/Random_forest\" title=\"Random forest\">Random forest</a></li></ul></li>\n",
      "<li><a href=\"/wiki/K-nearest_neighbors_algorithm\" title=\"K-nearest neighbors algorithm\"><i>k</i>-NN</a></li>\n",
      "<li><a href=\"/wiki/Linear_regression\" title=\"Linear regression\">Linear regression</a></li>\n",
      "<li><a href=\"/wiki/Naive_Bayes_classifier\" title=\"Naive Bayes classifier\">Naive Bayes</a></li>\n",
      "<li><a href=\"/wiki/Artificial_neural_network\" title=\"Artificial neural network\">Artificial neural networks</a></li>\n",
      "<li><a href=\"/wiki/Logistic_regression\" title=\"Logistic regression\">Logistic regression</a></li>\n",
      "<li><a href=\"/wiki/Perceptron\" title=\"Perceptron\">Perceptron</a></li>\n",
      "<li><a href=\"/wiki/Relevance_vector_machine\" title=\"Relevance vector machine\">Relevance vector machine (RVM)</a></li>\n",
      "<li><a class=\"mw-redirect\" href=\"/wiki/Support-vector_machine\" title=\"Support-vector machine\">Support vector machine (SVM)</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><a href=\"/wiki/Cluster_analysis\" title=\"Cluster analysis\">Clustering</a></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/BIRCH\" title=\"BIRCH\">BIRCH</a></li>\n",
      "<li><a class=\"mw-redirect\" href=\"/wiki/CURE_data_clustering_algorithm\" title=\"CURE data clustering algorithm\">CURE</a></li>\n",
      "<li><a href=\"/wiki/Hierarchical_clustering\" title=\"Hierarchical clustering\">Hierarchical</a></li>\n",
      "<li><a href=\"/wiki/K-means_clustering\" title=\"K-means clustering\"><i>k</i>-means</a></li>\n",
      "<li><a href=\"/wiki/Expectation%E2%80%93maximization_algorithm\" title=\"Expectation–maximization algorithm\">Expectation–maximization (EM)</a></li>\n",
      "<li><br/><a href=\"/wiki/DBSCAN\" title=\"DBSCAN\">DBSCAN</a></li>\n",
      "<li><a href=\"/wiki/OPTICS_algorithm\" title=\"OPTICS algorithm\">OPTICS</a></li>\n",
      "<li><a class=\"mw-redirect\" href=\"/wiki/Mean-shift\" title=\"Mean-shift\">Mean-shift</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><a href=\"/wiki/Dimensionality_reduction\" title=\"Dimensionality reduction\">Dimensionality reduction</a></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Factor_analysis\" title=\"Factor analysis\">Factor analysis</a></li>\n",
      "<li><a href=\"/wiki/Canonical_correlation\" title=\"Canonical correlation\">CCA</a></li>\n",
      "<li><a href=\"/wiki/Independent_component_analysis\" title=\"Independent component analysis\">ICA</a></li>\n",
      "<li><a href=\"/wiki/Linear_discriminant_analysis\" title=\"Linear discriminant analysis\">LDA</a></li>\n",
      "<li><a href=\"/wiki/Non-negative_matrix_factorization\" title=\"Non-negative matrix factorization\">NMF</a></li>\n",
      "<li><a href=\"/wiki/Principal_component_analysis\" title=\"Principal component analysis\">PCA</a></li>\n",
      "<li><a href=\"/wiki/Proper_generalized_decomposition\" title=\"Proper generalized decomposition\">PGD</a></li>\n",
      "<li><a href=\"/wiki/T-distributed_stochastic_neighbor_embedding\" title=\"T-distributed stochastic neighbor embedding\">t-SNE</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><a href=\"/wiki/Structured_prediction\" title=\"Structured prediction\">Structured prediction</a></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Graphical_model\" title=\"Graphical model\">Graphical models</a>\n",
      "<ul><li><a href=\"/wiki/Bayesian_network\" title=\"Bayesian network\">Bayes net</a></li>\n",
      "<li><a href=\"/wiki/Conditional_random_field\" title=\"Conditional random field\">Conditional random field</a></li>\n",
      "<li><a href=\"/wiki/Hidden_Markov_model\" title=\"Hidden Markov model\">Hidden Markov</a></li></ul></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><a href=\"/wiki/Anomaly_detection\" title=\"Anomaly detection\">Anomaly detection</a></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a class=\"mw-redirect\" href=\"/wiki/K-nearest_neighbors_classification\" title=\"K-nearest neighbors classification\"><i>k</i>-NN</a></li>\n",
      "<li><a href=\"/wiki/Local_outlier_factor\" title=\"Local outlier factor\">Local outlier factor</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><a href=\"/wiki/Artificial_neural_network\" title=\"Artificial neural network\">Artificial neural network</a></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Autoencoder\" title=\"Autoencoder\">Autoencoder</a></li>\n",
      "<li><a href=\"/wiki/Deep_learning\" title=\"Deep learning\">Deep learning</a></li>\n",
      "<li><a href=\"/wiki/DeepDream\" title=\"DeepDream\">DeepDream</a></li>\n",
      "<li><a href=\"/wiki/Multilayer_perceptron\" title=\"Multilayer perceptron\">Multilayer perceptron</a></li>\n",
      "<li><a href=\"/wiki/Recurrent_neural_network\" title=\"Recurrent neural network\">RNN</a>\n",
      "<ul><li><a href=\"/wiki/Long_short-term_memory\" title=\"Long short-term memory\">LSTM</a></li>\n",
      "<li><a href=\"/wiki/Gated_recurrent_unit\" title=\"Gated recurrent unit\">GRU</a></li>\n",
      "<li><a href=\"/wiki/Echo_state_network\" title=\"Echo state network\">ESN</a></li></ul></li>\n",
      "<li><a href=\"/wiki/Restricted_Boltzmann_machine\" title=\"Restricted Boltzmann machine\">Restricted Boltzmann machine</a></li>\n",
      "<li><a href=\"/wiki/Generative_adversarial_network\" title=\"Generative adversarial network\">GAN</a></li>\n",
      "<li><a href=\"/wiki/Self-organizing_map\" title=\"Self-organizing map\">SOM</a></li>\n",
      "<li><a href=\"/wiki/Convolutional_neural_network\" title=\"Convolutional neural network\">Convolutional neural network</a>\n",
      "<ul><li><a href=\"/wiki/U-Net\" title=\"U-Net\">U-Net</a></li></ul></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><a href=\"/wiki/Reinforcement_learning\" title=\"Reinforcement learning\">Reinforcement learning</a></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Q-learning\" title=\"Q-learning\">Q-learning</a></li>\n",
      "<li><a href=\"/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action\" title=\"State–action–reward–state–action\">SARSA</a></li>\n",
      "<li><a href=\"/wiki/Temporal_difference_learning\" title=\"Temporal difference learning\">Temporal difference (TD)</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\">Theory</div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a class=\"mw-redirect\" href=\"/wiki/Bias%E2%80%93variance_dilemma\" title=\"Bias–variance dilemma\">Bias–variance dilemma</a></li>\n",
      "<li><a href=\"/wiki/Computational_learning_theory\" title=\"Computational learning theory\">Computational learning theory</a></li>\n",
      "<li><a href=\"/wiki/Empirical_risk_minimization\" title=\"Empirical risk minimization\">Empirical risk minimization</a></li>\n",
      "<li><a href=\"/wiki/Occam_learning\" title=\"Occam learning\">Occam learning</a></li>\n",
      "<li><a href=\"/wiki/Probably_approximately_correct_learning\" title=\"Probably approximately correct learning\">PAC learning</a></li>\n",
      "<li><a href=\"/wiki/Statistical_learning_theory\" title=\"Statistical learning theory\">Statistical learning</a></li>\n",
      "<li><a href=\"/wiki/Vapnik%E2%80%93Chervonenkis_theory\" title=\"Vapnik–Chervonenkis theory\">VC theory</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\">Machine-learning venues</div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Conference_on_Neural_Information_Processing_Systems\" title=\"Conference on Neural Information Processing Systems\">NeurIPS</a></li>\n",
      "<li><a href=\"/wiki/International_Conference_on_Machine_Learning\" title=\"International Conference on Machine Learning\">ICML</a></li>\n",
      "<li><a href=\"/wiki/Machine_Learning_(journal)\" title=\"Machine Learning (journal)\">ML</a></li>\n",
      "<li><a href=\"/wiki/Journal_of_Machine_Learning_Research\" title=\"Journal of Machine Learning Research\">JMLR</a></li>\n",
      "<li><a class=\"external text\" href=\"https://arxiv.org/list/cs.LG/recent\" rel=\"nofollow\">ArXiv:cs.LG</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\"><a href=\"/wiki/Glossary_of_artificial_intelligence\" title=\"Glossary of artificial intelligence\">Glossary of artificial intelligence</a></div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/Glossary_of_artificial_intelligence\" title=\"Glossary of artificial intelligence\">Glossary of artificial intelligence</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"padding:0 0.1em 0.4em\">\n",
      "<div class=\"NavFrame collapsed\" style=\"border:none;padding:0\"><div class=\"NavHead\" style=\"font-size:105%;background:transparent;text-align:left\">Related articles</div><div class=\"NavContent\" style=\"font-size:105%;padding:0.2em 0 0.4em;text-align:center\"><div class=\"hlist\">\n",
      "<ul><li><a href=\"/wiki/List_of_datasets_for_machine-learning_research\" title=\"List of datasets for machine-learning research\">List of datasets for machine-learning research</a></li>\n",
      "<li><a href=\"/wiki/Outline_of_machine_learning\" title=\"Outline of machine learning\">Outline of machine learning</a></li></ul>\n",
      "</div></div></div></td>\n",
      "</tr><tr><td style=\"text-align:right;font-size:115%;padding-top: 0.6em;\"><div class=\"plainlinks hlist navbar mini\"><ul><li class=\"nv-view\"><a href=\"/wiki/Template:Machine_learning_bar\" title=\"Template:Machine learning bar\"><abbr title=\"View this template\">v</abbr></a></li><li class=\"nv-talk\"><a href=\"/wiki/Template_talk:Machine_learning_bar\" title=\"Template talk:Machine learning bar\"><abbr title=\"Discuss this template\">t</abbr></a></li><li class=\"nv-edit\"><a class=\"external text\" href=\"https://en.wikipedia.org/w/index.php?title=Template:Machine_learning_bar&amp;action=edit\"><abbr title=\"Edit this template\">e</abbr></a></li></ul></div></td></tr></tbody></table>\n",
      "<style data-mw-deduplicate=\"TemplateStyles:r923042769/mw-parser-output/.tmulti\">.mw-parser-output .tmulti .thumbinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{text-align:left;background-color:transparent}.mw-parser-output .tmulti .thumbcaption-center{text-align:center;background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .thumbcaption{text-align:center}}</style><div class=\"thumb tmulti tright\"><div class=\"thumbinner\" style=\"width:204px;max-width:204px\"><div class=\"trow\"><div class=\"tsingle\" style=\"width:202px;max-width:202px\"><div class=\"thumbimage\"><a class=\"image\" href=\"/wiki/File:Test_function_and_noisy_data.png\"><img alt=\"\" data-file-height=\"901\" data-file-width=\"1201\" decoding=\"async\" height=\"150\" src=\"//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Test_function_and_noisy_data.png/200px-Test_function_and_noisy_data.png\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/6/64/Test_function_and_noisy_data.png/300px-Test_function_and_noisy_data.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/64/Test_function_and_noisy_data.png/400px-Test_function_and_noisy_data.png 2x\" width=\"200\"/></a></div><div class=\"thumbcaption\">Function and noisy data.</div></div></div><div class=\"trow\"><div class=\"tsingle\" style=\"width:202px;max-width:202px\"><div class=\"thumbimage\"><a class=\"image\" href=\"/wiki/File:Radial_basis_function_fit,_spread%3D5.png\"><img alt=\"\" data-file-height=\"901\" data-file-width=\"1201\" decoding=\"async\" height=\"150\" src=\"//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Radial_basis_function_fit%2C_spread%3D5.png/200px-Radial_basis_function_fit%2C_spread%3D5.png\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Radial_basis_function_fit%2C_spread%3D5.png/300px-Radial_basis_function_fit%2C_spread%3D5.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Radial_basis_function_fit%2C_spread%3D5.png/400px-Radial_basis_function_fit%2C_spread%3D5.png 2x\" width=\"200\"/></a></div><div class=\"thumbcaption\">spread=5</div></div></div><div class=\"trow\"><div class=\"tsingle\" style=\"width:202px;max-width:202px\"><div class=\"thumbimage\"><a class=\"image\" href=\"/wiki/File:Radial_basis_function_fit,_spread%3D1.png\"><img alt=\"\" data-file-height=\"901\" data-file-width=\"1201\" decoding=\"async\" height=\"150\" src=\"//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Radial_basis_function_fit%2C_spread%3D1.png/200px-Radial_basis_function_fit%2C_spread%3D1.png\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Radial_basis_function_fit%2C_spread%3D1.png/300px-Radial_basis_function_fit%2C_spread%3D1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Radial_basis_function_fit%2C_spread%3D1.png/400px-Radial_basis_function_fit%2C_spread%3D1.png 2x\" width=\"200\"/></a></div><div class=\"thumbcaption\">spread=1</div></div></div><div class=\"trow\"><div class=\"tsingle\" style=\"width:202px;max-width:202px\"><div class=\"thumbimage\"><a class=\"image\" href=\"/wiki/File:Radial_basis_function_fit,_spread%3D0.1.png\"><img alt=\"\" data-file-height=\"901\" data-file-width=\"1201\" decoding=\"async\" height=\"150\" src=\"//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Radial_basis_function_fit%2C_spread%3D0.1.png/200px-Radial_basis_function_fit%2C_spread%3D0.1.png\" srcset=\"//upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Radial_basis_function_fit%2C_spread%3D0.1.png/300px-Radial_basis_function_fit%2C_spread%3D0.1.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3f/Radial_basis_function_fit%2C_spread%3D0.1.png/400px-Radial_basis_function_fit%2C_spread%3D0.1.png 2x\" width=\"200\"/></a></div><div class=\"thumbcaption\">spread=0.1</div></div></div><div class=\"trow\" style=\"display:flex\"><div class=\"thumbcaption\" style=\"background-color:transparent\">A function (red) is approximated using <a class=\"mw-redirect\" href=\"/wiki/Radial_basis_functions\" title=\"Radial basis functions\">radial basis functions</a> (blue). Several trials are shown in each graph. For each trial, a few noisy data points are provided as a training set (top). For a wide spread (image 2) the bias is high: the RBFs cannot fully approximate the function (especially the central dip), but the variance between different trials is low. As spread decreases (image 3 and 4) the bias decreases: the blue curves more closely approximate the red. However, depending on the noise in different trials the variance between trials increases. In the lowermost image the approximated values for x=0 varies wildly depending on where the data points were located.</div></div></div></div>\n",
      "<p>In <a href=\"/wiki/Statistics\" title=\"Statistics\">statistics</a> and <a href=\"/wiki/Machine_learning\" title=\"Machine learning\">machine learning</a>, the <b>bias–variance tradeoff</b> is the property of a set of predictive models whereby models with a lower <a href=\"/wiki/Bias_of_an_estimator\" title=\"Bias of an estimator\">bias</a> in <a href=\"/wiki/Statistical_parameter\" title=\"Statistical parameter\">parameter</a> <a href=\"/wiki/Estimation_theory\" title=\"Estimation theory\">estimation</a> have a higher <a href=\"/wiki/Variance\" title=\"Variance\">variance</a> of the parameter estimates across <a href=\"/wiki/Sample_(statistics)\" title=\"Sample (statistics)\">samples</a>, and vice versa. The <b>bias–variance dilemma</b> or <b>bias–variance problem</b> is the conflict in trying to simultaneously minimize these two sources of <a class=\"mw-redirect\" href=\"/wiki/Errors_and_residuals_in_statistics\" title=\"Errors and residuals in statistics\">error</a> that prevent <a href=\"/wiki/Supervised_learning\" title=\"Supervised learning\">supervised learning</a> algorithms from generalizing beyond their <a class=\"mw-redirect\" href=\"/wiki/Training_set\" title=\"Training set\">training set</a><sup class=\"reference\" id=\"cite_ref-1\"><a href=\"#cite_note-1\">[1]</a></sup><sup class=\"reference\" id=\"cite_ref-2\"><a href=\"#cite_note-2\">[2]</a></sup>:\n",
      "</p>\n",
      "<ul><li>The <a href=\"/wiki/Bias_of_an_estimator\" title=\"Bias of an estimator\"><i>bias error</i></a> is an error from erroneous assumptions in the learning <a href=\"/wiki/Algorithm\" title=\"Algorithm\">algorithm</a>. High bias can cause an algorithm to miss the relevant relations between features and target outputs (underfitting).</li>\n",
      "<li>The <i><a href=\"/wiki/Variance\" title=\"Variance\">variance</a></i> is an error from sensitivity to small fluctuations in the training set. High variance can cause an algorithm to model the random <a href=\"/wiki/Noise_(signal_processing)\" title=\"Noise (signal processing)\">noise</a> in the training data, rather than the intended outputs (<a href=\"/wiki/Overfitting\" title=\"Overfitting\">overfitting</a>).</li></ul>\n",
      "<p>The <b>bias–variance decomposition</b> is a way of analyzing a learning algorithm's <a href=\"/wiki/Expected_value\" title=\"Expected value\">expected</a> <a href=\"/wiki/Generalization_error\" title=\"Generalization error\">generalization error</a> with respect to a particular problem as a sum of three terms, the bias, variance, and a quantity called the <i>irreducible error</i>, resulting from noise in the problem itself.\n",
      "</p><p>This tradeoff applies to all forms of <a href=\"/wiki/Supervised_learning\" title=\"Supervised learning\">supervised learning</a>: <a href=\"/wiki/Statistical_classification\" title=\"Statistical classification\">classification</a>, <a href=\"/wiki/Regression_analysis\" title=\"Regression analysis\">regression</a> (function fitting),<sup class=\"reference\" id=\"cite_ref-geman_3-0\"><a href=\"#cite_note-geman-3\">[3]</a></sup><sup class=\"reference\" id=\"cite_ref-4\"><a href=\"#cite_note-4\">[4]</a></sup> and <a href=\"/wiki/Structured_prediction\" title=\"Structured prediction\">structured output learning</a>, though, it does not apply in all learning algorithms.<sup class=\"reference\" id=\"cite_ref-nealThesis2019_5-0\"><a href=\"#cite_note-nealThesis2019-5\">[5]</a></sup><sup class=\"reference\" id=\"cite_ref-nealBlog2020_6-0\"><a href=\"#cite_note-nealBlog2020-6\">[6]</a></sup> It has also been invoked to explain the effectiveness of heuristics in human learning.<sup class=\"reference\" id=\"cite_ref-ReferenceA_7-0\"><a href=\"#cite_note-ReferenceA-7\">[7]</a></sup>\n",
      "</p><p>It is important to note that the bias-variance tradeoff is not universal.<sup class=\"reference\" id=\"cite_ref-nealThesis2019_5-1\"><a href=\"#cite_note-nealThesis2019-5\">[5]</a></sup><sup class=\"reference\" id=\"cite_ref-nealBlog2020_6-1\"><a href=\"#cite_note-nealBlog2020-6\">[6]</a></sup><sup class=\"reference\" id=\"cite_ref-neal2018_8-0\"><a href=\"#cite_note-neal2018-8\">[8]</a></sup> For example, <i>both</i> bias <i>and</i> variance decrease when increasing the width of a neural network.<sup class=\"reference\" id=\"cite_ref-neal2018_8-1\"><a href=\"#cite_note-neal2018-8\">[8]</a></sup> This means that it is not necessary to control the size of a neural network to control variance. This does not contradict the bias-variance decomposition because the bias-variance decomposition does not imply a bias-variance tradeoff.<sup class=\"reference\" id=\"cite_ref-nealBlog2020_6-2\"><a href=\"#cite_note-nealBlog2020-6\">[6]</a></sup><sup class=\"reference\" id=\"cite_ref-nealThesis2019_5-2\"><a href=\"#cite_note-nealThesis2019-5\">[5]</a></sup>\n",
      "</p>\n",
      "<div aria-labelledby=\"mw-toc-heading\" class=\"toc\" id=\"toc\" role=\"navigation\"><input class=\"toctogglecheckbox\" id=\"toctogglecheckbox\" role=\"button\" style=\"display:none\" type=\"checkbox\"/><div class=\"toctitle\" dir=\"ltr\" lang=\"en\"><h2 id=\"mw-toc-heading\">Contents</h2><span class=\"toctogglespan\"><label class=\"toctogglelabel\" for=\"toctogglecheckbox\"></label></span></div>\n",
      "<ul>\n",
      "<li class=\"toclevel-1 tocsection-1\"><a href=\"#Motivation\"><span class=\"tocnumber\">1</span> <span class=\"toctext\">Motivation</span></a></li>\n",
      "<li class=\"toclevel-1 tocsection-2\"><a href=\"#Bias–variance_decomposition_of_squared_error\"><span class=\"tocnumber\">2</span> <span class=\"toctext\">Bias–variance decomposition of squared error</span></a>\n",
      "<ul>\n",
      "<li class=\"toclevel-2 tocsection-3\"><a href=\"#Derivation\"><span class=\"tocnumber\">2.1</span> <span class=\"toctext\">Derivation</span></a></li>\n",
      "</ul>\n",
      "</li>\n",
      "<li class=\"toclevel-1 tocsection-4\"><a href=\"#Application_to_regression\"><span class=\"tocnumber\">3</span> <span class=\"toctext\">Application to regression</span></a></li>\n",
      "<li class=\"toclevel-1 tocsection-5\"><a href=\"#Application_to_classification\"><span class=\"tocnumber\">4</span> <span class=\"toctext\">Application to classification</span></a></li>\n",
      "<li class=\"toclevel-1 tocsection-6\"><a href=\"#Application_to_reinforcement_learning\"><span class=\"tocnumber\">5</span> <span class=\"toctext\">Application to reinforcement learning</span></a></li>\n",
      "<li class=\"toclevel-1 tocsection-7\"><a href=\"#Approaches\"><span class=\"tocnumber\">6</span> <span class=\"toctext\">Approaches</span></a>\n",
      "<ul>\n",
      "<li class=\"toclevel-2 tocsection-8\"><a href=\"#k-nearest_neighbors\"><span class=\"tocnumber\">6.1</span> <span class=\"toctext\"><i>k</i>-nearest neighbors</span></a></li>\n",
      "</ul>\n",
      "</li>\n",
      "<li class=\"toclevel-1 tocsection-9\"><a href=\"#Application_to_human_learning\"><span class=\"tocnumber\">7</span> <span class=\"toctext\">Application to human learning</span></a></li>\n",
      "<li class=\"toclevel-1 tocsection-10\"><a href=\"#See_also\"><span class=\"tocnumber\">8</span> <span class=\"toctext\">See also</span></a></li>\n",
      "<li class=\"toclevel-1 tocsection-11\"><a href=\"#References\"><span class=\"tocnumber\">9</span> <span class=\"toctext\">References</span></a></li>\n",
      "</ul>\n",
      "</div>\n",
      "<h2><span class=\"mw-headline\" id=\"Motivation\">Motivation</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=1\" title=\"Edit section: Motivation\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<p>The bias-variance tradeoff is a central problem in supervised learning. Ideally, one wants to <a href=\"/wiki/Model_selection\" title=\"Model selection\">choose a model</a> that both accurately captures the regularities in its training data, but also <a href=\"/wiki/Generalization\" title=\"Generalization\">generalizes</a> well to unseen data. Unfortunately, it is typically impossible to do both simultaneously. High-variance learning methods may be able to represent their training set well but are at risk of overfitting to noisy or unrepresentative training data. In contrast, algorithms with high bias typically produce simpler models that don't tend to overfit but may <i>underfit</i> their training data, failing to capture important regularities.\n",
      "</p><p>Models with high variance are usually more complex (e.g. higher-order regression polynomials), enabling them to represent the training set more accurately. In the process, however, they may also represent a large <a href=\"/wiki/Noise_(signal_processing)\" title=\"Noise (signal processing)\">noise</a> component in the training set, making their predictions less accurate – despite their added complexity. In contrast, models with higher bias tend to be relatively simple (low-order or even linear regression polynomials) but may produce lower variance predictions when applied beyond the training set.\n",
      "</p>\n",
      "<h2><span id=\"Bias.E2.80.93variance_decomposition_of_squared_error\"></span><span class=\"mw-headline\" id=\"Bias–variance_decomposition_of_squared_error\">Bias–variance decomposition of squared error</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=2\" title=\"Edit section: Bias–variance decomposition of squared error\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<p>Suppose that we have a training set consisting of a set of points <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle x_{1},\\dots ,x_{n}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>1</mn>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>,</mo>\n",
      "<mo>…<!-- … --></mo>\n",
      "<mo>,</mo>\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>n</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle x_{1},\\dots ,x_{n}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"x_{1},\\dots ,x_{n}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753\" style=\"vertical-align: -0.671ex; width:10.11ex; height:2.009ex;\"/></span> and real values <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle y_{i}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>y</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>i</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle y_{i}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"y_{i}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f\" style=\"vertical-align: -0.671ex; width:1.939ex; height:2.009ex;\"/></span> associated with each point <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle x_{i}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>i</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle x_{i}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"x_{i}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158\" style=\"vertical-align: -0.671ex; width:2.129ex; height:2.009ex;\"/></span>. We assume that there is a function with noise <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle y=f(x)+\\varepsilon }\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>y</mi>\n",
      "<mo>=</mo>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mo>+</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle y=f(x)+\\varepsilon }</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle y=f(x)+\\varepsilon }\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/93f385c9e8b0b5372dd00cfcaa57b1fe20bb812b\" style=\"vertical-align: -0.838ex; width:12.595ex; height:2.843ex;\"/></span>, where the noise, <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\varepsilon }\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>ε<!-- ε --></mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\varepsilon }</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\varepsilon \" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173\" style=\"vertical-align: -0.338ex; width:1.083ex; height:1.676ex;\"/></span>, has zero mean and variance <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\sigma ^{2}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msup>\n",
      "<mi>σ<!-- σ --></mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\sigma ^{2}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\sigma ^{2}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5\" style=\"vertical-align: -0.338ex; width:2.385ex; height:2.676ex;\"/></span>.\n",
      "</p><p>We want to find a function <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}(x;D)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}(x;D)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle {\\hat {f}}(x;D)}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/104cc1ab5f9c97a82bdf05c6e5999d69b9db08a7\" style=\"vertical-align: -0.838ex; width:7.796ex; height:3.343ex;\"/></span>, that approximates the true function <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle f(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle f(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"f(x)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074\" style=\"vertical-align: -0.838ex; width:4.418ex; height:2.843ex;\"/></span> as well as possible, by means of some learning algorithm based on a training dataset (sample) <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle D=\\{(x_{1},y_{1})\\dots ,(x_{n},y_{n})\\}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>D</mi>\n",
      "<mo>=</mo>\n",
      "<mo fence=\"false\" stretchy=\"false\">{</mo>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>1</mn>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>,</mo>\n",
      "<msub>\n",
      "<mi>y</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>1</mn>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mo>…<!-- … --></mo>\n",
      "<mo>,</mo>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>n</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>,</mo>\n",
      "<msub>\n",
      "<mi>y</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>n</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mo fence=\"false\" stretchy=\"false\">}</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle D=\\{(x_{1},y_{1})\\dots ,(x_{n},y_{n})\\}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle D=\\{(x_{1},y_{1})\\dots ,(x_{n},y_{n})\\}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/757190ec5d200f1ca40a0d23e32f090a12352b6d\" style=\"vertical-align: -0.838ex; width:27.049ex; height:2.843ex;\"/></span>. We make \"as well as possible\" precise by measuring the <a href=\"/wiki/Mean_squared_error\" title=\"Mean squared error\">mean squared error</a> between <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle y}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>y</mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle y}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"y\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d\" style=\"vertical-align: -0.671ex; width:1.155ex; height:2.009ex;\"/></span> and <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}(x;D)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}(x;D)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle {\\hat {f}}(x;D)}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/104cc1ab5f9c97a82bdf05c6e5999d69b9db08a7\" style=\"vertical-align: -0.838ex; width:7.796ex; height:3.343ex;\"/></span>: we want <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle (y-{\\hat {f}}(x;D))^{2}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>y</mi>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<msup>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle (y-{\\hat {f}}(x;D))^{2}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle (y-{\\hat {f}}(x;D))^{2}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/fcff55548f0c80eb86405dd0e2e894b6d3734f1a\" style=\"vertical-align: -0.838ex; width:14.656ex; height:3.343ex;\"/></span> to be minimal, both for <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle x_{1},\\dots ,x_{n}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>1</mn>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>,</mo>\n",
      "<mo>…<!-- … --></mo>\n",
      "<mo>,</mo>\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>n</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle x_{1},\\dots ,x_{n}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"x_{1},\\dots ,x_{n}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/e5afdbc2d248d8fa9ba2c4f5188d946a0537e753\" style=\"vertical-align: -0.671ex; width:10.11ex; height:2.009ex;\"/></span> <i>and for points outside of our sample</i>. Of course, we cannot hope to do so perfectly, since the <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle y_{i}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>y</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>i</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle y_{i}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"y_{i}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/67d30d30b6c2dbe4d6f150d699de040937ecc95f\" style=\"vertical-align: -0.671ex; width:1.939ex; height:2.009ex;\"/></span> contain noise <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\varepsilon }\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>ε<!-- ε --></mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\varepsilon }</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\varepsilon \" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173\" style=\"vertical-align: -0.338ex; width:1.083ex; height:1.676ex;\"/></span>; this means we must be prepared to accept an <i>irreducible error</i> in any function we come up with.\n",
      "</p><p>Finding an <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\hat {f}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb\" style=\"vertical-align: -0.671ex; width:1.699ex; height:3.176ex;\"/></span> that generalizes to points outside of the training set can be done with any of the countless algorithms used for supervised learning. It turns out that whichever function <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\hat {f}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb\" style=\"vertical-align: -0.671ex; width:1.699ex; height:3.176ex;\"/></span> we select, we can decompose its <a href=\"/wiki/Expected_value\" title=\"Expected value\">expected</a> error on an unseen sample <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle x}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>x</mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle x}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"x\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4\" style=\"vertical-align: -0.338ex; width:1.33ex; height:1.676ex;\"/></span> as follows:<sup class=\"reference\" id=\"cite_ref-islr_9-0\"><a href=\"#cite_note-islr-9\">[9]</a></sup><sup class=\"reference\" style=\"white-space:nowrap;\">:<span>34</span></sup><sup class=\"reference\" id=\"cite_ref-ESL_10-0\"><a href=\"#cite_note-ESL-10\">[10]</a></sup><sup class=\"reference\" style=\"white-space:nowrap;\">:<span>223</span></sup>\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {E} _{D}{\\Big [}{\\big (}y-{\\hat {f}}(x;D){\\big )}^{2}{\\Big ]}={\\Big (}\\operatorname {Bias} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}{\\Big )}^{2}+\\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}+\\sigma ^{2}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">[</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mi>y</mi>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">]</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo>=</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<msub>\n",
      "<mi>Bias</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">[</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">]</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo>+</mo>\n",
      "<msub>\n",
      "<mi>Var</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">[</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">]</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo>+</mo>\n",
      "<msup>\n",
      "<mi>σ<!-- σ --></mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {E} _{D}{\\Big [}{\\big (}y-{\\hat {f}}(x;D){\\big )}^{2}{\\Big ]}={\\Big (}\\operatorname {Bias} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}{\\Big )}^{2}+\\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}+\\sigma ^{2}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {E} _{D}{\\Big [}{\\big (}y-{\\hat {f}}(x;D){\\big )}^{2}{\\Big ]}={\\Big (}\\operatorname {Bias} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}{\\Big )}^{2}+\\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}+\\sigma ^{2}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/d5a2f3d7e452720a1f105dff963ad490221a4a80\" style=\"vertical-align: -1.838ex; width:67.734ex; height:5.176ex;\"/></span></dd></dl>\n",
      "<p>where\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {Bias} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}=\\operatorname {E} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}-f(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>Bias</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
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      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">]</mo>\n",
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      "</mrow>\n",
      "<mo>=</mo>\n",
      "<msub>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
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      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">[</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">]</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {Bias} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}=\\operatorname {E} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}-f(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {Bias} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}=\\operatorname {E} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}-f(x)}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/a0013efa4f5587aa74b8c5511bf4b5864c3f5c56\" style=\"vertical-align: -1.005ex; width:39.742ex; height:3.509ex;\"/></span></dd></dl>\n",
      "<p>and\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}=\\operatorname {E} _{D}[{\\big (}\\operatorname {E} _{D}[{\\hat {f}}(x;D)]-{\\hat {f}}(x;D){\\big )}^{2}].}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>Var</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
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      "</mrow>\n",
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      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">]</mo>\n",
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      "<mo>=</mo>\n",
      "<msub>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">(</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<msub>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>D</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">)</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>.</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}=\\operatorname {E} _{D}[{\\big (}\\operatorname {E} _{D}[{\\hat {f}}(x;D)]-{\\hat {f}}(x;D){\\big )}^{2}].}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}=\\operatorname {E} _{D}[{\\big (}\\operatorname {E} _{D}[{\\hat {f}}(x;D)]-{\\hat {f}}(x;D){\\big )}^{2}].}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/00217394951130843b79dfb8bbd6e2374515bbbf\" style=\"vertical-align: -1.005ex; width:50.221ex; height:3.676ex;\"/></span></dd></dl>\n",
      "<p>The expectation ranges over different choices of the training set <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle D=\\{(x_{1},y_{1})\\dots ,(x_{n},y_{n})\\}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>D</mi>\n",
      "<mo>=</mo>\n",
      "<mo fence=\"false\" stretchy=\"false\">{</mo>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>1</mn>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>,</mo>\n",
      "<msub>\n",
      "<mi>y</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>1</mn>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mo>…<!-- … --></mo>\n",
      "<mo>,</mo>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<msub>\n",
      "<mi>x</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>n</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo>,</mo>\n",
      "<msub>\n",
      "<mi>y</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>n</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mo fence=\"false\" stretchy=\"false\">}</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle D=\\{(x_{1},y_{1})\\dots ,(x_{n},y_{n})\\}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle D=\\{(x_{1},y_{1})\\dots ,(x_{n},y_{n})\\}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/757190ec5d200f1ca40a0d23e32f090a12352b6d\" style=\"vertical-align: -0.838ex; width:27.049ex; height:2.843ex;\"/></span>, all sampled from the same joint distribution <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle P(x,y)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>P</mi>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>,</mo>\n",
      "<mi>y</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle P(x,y)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"P(x,y)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/d5b3d8f37f5458c22b61eaf26e5af0523acb63e2\" style=\"vertical-align: -0.838ex; width:7.074ex; height:2.843ex;\"/></span>. The three terms represent:\n",
      "</p>\n",
      "<ul><li>the square of the <i>bias</i> of the learning method, which can be thought of as the error caused by the simplifying assumptions built into the method. E.g., when approximating a non-linear function <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle f(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle f(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"f(x)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/202945cce41ecebb6f643f31d119c514bec7a074\" style=\"vertical-align: -0.838ex; width:4.418ex; height:2.843ex;\"/></span> using a learning method for <a href=\"/wiki/Linear_model\" title=\"Linear model\">linear models</a>, there will be error in the estimates <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\hat{f}(x)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/def60945c876ae17c5a91167971245424f2006f6\" style=\"vertical-align: -0.838ex; width:4.838ex; height:3.343ex;\"/></span> due to this assumption;</li>\n",
      "<li>the <i>variance</i> of the learning method, or, intuitively, how much the learning method <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\hat{f}(x)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/def60945c876ae17c5a91167971245424f2006f6\" style=\"vertical-align: -0.838ex; width:4.838ex; height:3.343ex;\"/></span> will move around its mean;</li>\n",
      "<li>the irreducible error <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\sigma ^{2}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msup>\n",
      "<mi>σ<!-- σ --></mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\sigma ^{2}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\sigma ^{2}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/53a5c55e536acf250c1d3e0f754be5692b843ef5\" style=\"vertical-align: -0.338ex; width:2.385ex; height:2.676ex;\"/></span>.</li></ul>\n",
      "<p>Since all three terms are non-negative, this forms a lower bound on the expected error on unseen samples.<sup class=\"reference\" id=\"cite_ref-islr_9-1\"><a href=\"#cite_note-islr-9\">[9]</a></sup><sup class=\"reference\" style=\"white-space:nowrap;\">:<span>34</span></sup>\n",
      "</p><p>The more complex the model <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\hat{f}(x)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/def60945c876ae17c5a91167971245424f2006f6\" style=\"vertical-align: -0.838ex; width:4.838ex; height:3.343ex;\"/></span> is, the more data points it will capture, and the lower the bias will be. However, complexity will make the model \"move\" more to capture the data points, and hence its variance will be larger.\n",
      "</p>\n",
      "<h3><span class=\"mw-headline\" id=\"Derivation\">Derivation</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=3\" title=\"Edit section: Derivation\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h3>\n",
      "<p>The derivation of the bias–variance decomposition for squared error proceeds as follows.<sup class=\"reference\" id=\"cite_ref-11\"><a href=\"#cite_note-11\">[11]</a></sup><sup class=\"reference\" id=\"cite_ref-12\"><a href=\"#cite_note-12\">[12]</a></sup> For notational convenience, we abbreviate <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle f=f(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>f</mi>\n",
      "<mo>=</mo>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle f=f(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"f = f(x)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/a8982ea0271b9a6b8036cd2a883000a59f5a1ab9\" style=\"vertical-align: -0.838ex; width:8.795ex; height:2.843ex;\"/></span>, <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}={\\hat {f}}(x;D)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo>=</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo>;</mo>\n",
      "<mi>D</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}={\\hat {f}}(x;D)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle {\\hat {f}}={\\hat {f}}(x;D)}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/5d13a9cdcbd8adcdb91352a02eb61129c669bf74\" style=\"vertical-align: -0.838ex; width:12.594ex; height:3.343ex;\"/></span> and we drop the <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle D}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>D</mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle D}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"D\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6\" style=\"vertical-align: -0.338ex; width:1.924ex; height:2.176ex;\"/></span> subscript on our expectation operators. First, recall that, by definition, for any random variable <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle X}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>X</mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle X}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"X\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab\" style=\"vertical-align: -0.338ex; width:1.98ex; height:2.176ex;\"/></span>, we have\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} [X^{2}]-{\\Big (}\\operatorname {E} [X]{\\Big )}^{2}.}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>Var</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>X</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<msup>\n",
      "<mi>X</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>X</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo>.</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} [X^{2}]-{\\Big (}\\operatorname {E} [X]{\\Big )}^{2}.}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {Var} [X]=\\operatorname {E} [X^{2}]-{\\Big (}\\operatorname {E} [X]{\\Big )}^{2}.}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/d297a102ff965a2b1f0c5cf7fd46d3fc2ae639d8\" style=\"vertical-align: -1.838ex; width:28.678ex; height:5.176ex;\"/></span></dd></dl>\n",
      "<p>Rearranging, we get:\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {E} [X^{2}]=\\operatorname {Var} [X]+{\\Big (}\\operatorname {E} [X]{\\Big )}^{2}.}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<msup>\n",
      "<mi>X</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi>Var</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>X</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>+</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>X</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo>.</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {E} [X^{2}]=\\operatorname {Var} [X]+{\\Big (}\\operatorname {E} [X]{\\Big )}^{2}.}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {E} [X^{2}]=\\operatorname {Var} [X]+{\\Big (}\\operatorname {E} [X]{\\Big )}^{2}.}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc35b8c4a5bfb7ac920b37b441b02732006cee0\" style=\"vertical-align: -1.838ex; width:28.678ex; height:5.176ex;\"/></span></dd></dl>\n",
      "<p>Since <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle f}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>f</mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle f}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"f\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61\" style=\"vertical-align: -0.671ex; width:1.279ex; height:2.509ex;\"/></span> is <a href=\"/wiki/Deterministic_algorithm\" title=\"Deterministic algorithm\">deterministic</a>, i.e. independent of <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle D}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>D</mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle D}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"D\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6\" style=\"vertical-align: -0.338ex; width:1.924ex; height:2.176ex;\"/></span>,\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {E} [f]=f.}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi>f</mi>\n",
      "<mo>.</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {E} [f]=f.}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {E} [f]=f.}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cc9a0668ccec054ec20f2d518495ea0aa46e61\" style=\"vertical-align: -0.838ex; width:9.179ex; height:2.843ex;\"/></span></dd></dl>\n",
      "<p>Thus, given <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle y=f+\\varepsilon }\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>y</mi>\n",
      "<mo>=</mo>\n",
      "<mi>f</mi>\n",
      "<mo>+</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle y=f+\\varepsilon }</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle y=f+\\varepsilon }\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e3b37184854b796cb4323f3f1c6285760c70d9\" style=\"vertical-align: -0.671ex; width:9.456ex; height:2.509ex;\"/></span> and <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {E} [\\varepsilon ]=0}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mn>0</mn>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {E} [\\varepsilon ]=0}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {E} [\\varepsilon ]=0}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/a9494698413723204253e82aaff8b02684a64ed6\" style=\"vertical-align: -0.838ex; width:8.221ex; height:2.843ex;\"/></span> (because <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\varepsilon }\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>ε<!-- ε --></mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\varepsilon }</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\varepsilon \" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173\" style=\"vertical-align: -0.338ex; width:1.083ex; height:1.676ex;\"/></span> is noise), implies <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {E} [y]=\\operatorname {E} [f+\\varepsilon ]=\\operatorname {E} [f]=f.}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>y</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>f</mi>\n",
      "<mo>+</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi>f</mi>\n",
      "<mo>.</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {E} [y]=\\operatorname {E} [f+\\varepsilon ]=\\operatorname {E} [f]=f.}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {E} [y]=\\operatorname {E} [f+\\varepsilon ]=\\operatorname {E} [f]=f.}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/fd5be0da3a9300d8822afa6080b3d404cf33bb18\" style=\"vertical-align: -0.838ex; width:27.487ex; height:2.843ex;\"/></span>\n",
      "</p><p>Also, since <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {Var} [\\varepsilon ]=\\sigma ^{2},}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>Var</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<msup>\n",
      "<mi>σ<!-- σ --></mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo>,</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {Var} [\\varepsilon ]=\\sigma ^{2},}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {Var} [\\varepsilon ]=\\sigma ^{2},}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/e1d4c278d7174310f125f7f1625b204d6c9ef6b7\" style=\"vertical-align: -0.838ex; width:12.324ex; height:3.176ex;\"/></span>\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\operatorname {Var} [y]=\\operatorname {E} [(y-\\operatorname {E} [y])^{2}]=\\operatorname {E} [(y-f)^{2}]=\\operatorname {E} [(f+\\varepsilon -f)^{2}]=\\operatorname {E} [\\varepsilon ^{2}]=\\operatorname {Var} [\\varepsilon ]+{\\Big (}\\operatorname {E} [\\varepsilon ]{\\Big )}^{2}=\\sigma ^{2}+0^{2}=\\sigma ^{2}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>Var</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>y</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>y</mi>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>y</mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<msup>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>y</mi>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mi>f</mi>\n",
      "<msup>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>f</mi>\n",
      "<mo>+</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mi>f</mi>\n",
      "<msup>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<msup>\n",
      "<mi>ε<!-- ε --></mi>\n",
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      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>=</mo>\n",
      "<mi>Var</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>+</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">(</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mi>ε<!-- ε --></mi>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.623em\" minsize=\"1.623em\">)</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo>=</mo>\n",
      "<msup>\n",
      "<mi>σ<!-- σ --></mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo>+</mo>\n",
      "<msup>\n",
      "<mn>0</mn>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "<mo>=</mo>\n",
      "<msup>\n",
      "<mi>σ<!-- σ --></mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>2</mn>\n",
      "</mrow>\n",
      "</msup>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {Var} [y]=\\operatorname {E} [(y-\\operatorname {E} [y])^{2}]=\\operatorname {E} [(y-f)^{2}]=\\operatorname {E} [(f+\\varepsilon -f)^{2}]=\\operatorname {E} [\\varepsilon ^{2}]=\\operatorname {Var} [\\varepsilon ]+{\\Big (}\\operatorname {E} [\\varepsilon ]{\\Big )}^{2}=\\sigma ^{2}+0^{2}=\\sigma ^{2}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {Var} [y]=\\operatorname {E} [(y-\\operatorname {E} [y])^{2}]=\\operatorname {E} [(y-f)^{2}]=\\operatorname {E} [(f+\\varepsilon -f)^{2}]=\\operatorname {E} [\\varepsilon ^{2}]=\\operatorname {Var} [\\varepsilon ]+{\\Big (}\\operatorname {E} [\\varepsilon ]{\\Big )}^{2}=\\sigma ^{2}+0^{2}=\\sigma ^{2}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/22a04d2cb3d3cd11aca31f8595439cb559d12975\" style=\"vertical-align: -1.838ex; width:99.852ex; height:5.176ex;\"/></span></dd></dl>\n",
      "<p>Thus, since <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle \\varepsilon }\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mi>ε<!-- ε --></mi>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\varepsilon }</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"\\varepsilon \" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173\" style=\"vertical-align: -0.338ex; width:1.083ex; height:1.676ex;\"/></span> and <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\hat {f}}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\hat {f}}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\hat {f}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/14ce989fd75da938ec6f95a0cdb71037b23a11cb\" style=\"vertical-align: -0.671ex; width:1.699ex; height:3.176ex;\"/></span> are independent, we can write\n",
      "</p>\n",
      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\begin{aligned}\\operatorname {E} {\\big [}(y-{\\hat {f}})^{2}{\\big ]}&amp;=\\operatorname {E} {\\big [}(f+\\varepsilon -{\\hat {f}})^{2}{\\big ]}\\\\[5pt]&amp;=\\operatorname {E} {\\big [}(f+\\varepsilon -{\\hat {f}}+\\operatorname {E} [{\\hat {f}}]-\\operatorname {E} [{\\hat {f}}])^{2}{\\big ]}\\\\[5pt]&amp;=\\operatorname {E} {\\big [}(f-\\operatorname {E} [{\\hat {f}}])^{2}{\\big ]}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}+2\\operatorname {E} {\\big [}(f-\\operatorname {E} [{\\hat {f}}])\\varepsilon {\\big ]}+2\\operatorname {E} {\\big [}\\varepsilon (\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}){\\big ]}+2\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})(f-\\operatorname {E} [{\\hat {f}}]){\\big ]}\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}+2(f-\\operatorname {E} [{\\hat {f}}])\\operatorname {E} [\\varepsilon ]+2\\operatorname {E} [\\varepsilon ]\\operatorname {E} {\\big [}\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}{\\big ]}+2\\operatorname {E} {\\big [}\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}{\\big ]}(f-\\operatorname {E} [{\\hat {f}}])\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {Var} [\\varepsilon ]+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\\\[5pt]&amp;=\\operatorname {Bias} [{\\hat {f}}]^{2}+\\operatorname {Var} [\\varepsilon ]+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\\\[5pt]&amp;=\\operatorname {Bias} [{\\hat {f}}]^{2}+\\sigma ^{2}+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\end{aligned}}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mtable columnalign=\"right left right left right left right left right left right left\" columnspacing=\"0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em\" displaystyle=\"true\" rowspacing=\"0.8em 0.8em 0.8em 0.8em 0.8em 0.8em 0.8em 0.3em\">\n",
      "<mtr>\n",
      "<mtd>\n",
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      "<mo>⁡<!-- ⁡ --></mo>\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>y</mi>\n",
      "<mo>−<!-- − --></mo>\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "</mrow>\n",
      "</mrow>\n",
      "<msup>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mtd>\n",
      "<mi></mi>\n",
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      "</mrow>\n",
      "<msup>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "</mtr>\n",
      "<mtr>\n",
      "<mtd></mtd>\n",
      "<mtd>\n",
      "<mi></mi>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mo stretchy=\"false\">(</mo>\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo>+</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">]</mo>\n",
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      "<mo stretchy=\"false\">)</mo>\n",
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      "</mtr>\n",
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      "<mtd></mtd>\n",
      "<mtd>\n",
      "<mi></mi>\n",
      "<mo>=</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>f</mi>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<msup>\n",
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      "</msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "</mrow>\n",
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      "<mo>+</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<msup>\n",
      "<mi>ε<!-- ε --></mi>\n",
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      "</msup>\n",
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      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">[</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mover>\n",
      "<mi>f</mi>\n",
      "<mo stretchy=\"false\">^<!-- ^ --></mo>\n",
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      "</mrow>\n",
      "</mrow>\n",
      "<msup>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "</mrow>\n",
      "</msup>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">]</mo>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo>+</mo>\n",
      "<mn>2</mn>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mo maxsize=\"1.2em\" minsize=\"1.2em\">[</mo>\n",
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      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>f</mi>\n",
      "<mo>−<!-- − --></mo>\n",
      "<mi mathvariant=\"normal\">E</mi>\n",
      "<mo>⁡<!-- ⁡ --></mo>\n",
      "<mo stretchy=\"false\">[</mo>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mover>\n",
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      "</mover>\n",
      "</mrow>\n",
      "</mrow>\n",
      "<mo stretchy=\"false\">]</mo>\n",
      "<mo stretchy=\"false\">)</mo>\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mi mathvariant=\"normal\">E</mi>\n",
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      "</mrow>\n",
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      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\begin{aligned}\\operatorname {E} {\\big [}(y-{\\hat {f}})^{2}{\\big ]}&amp;=\\operatorname {E} {\\big [}(f+\\varepsilon -{\\hat {f}})^{2}{\\big ]}\\\\[5pt]&amp;=\\operatorname {E} {\\big [}(f+\\varepsilon -{\\hat {f}}+\\operatorname {E} [{\\hat {f}}]-\\operatorname {E} [{\\hat {f}}])^{2}{\\big ]}\\\\[5pt]&amp;=\\operatorname {E} {\\big [}(f-\\operatorname {E} [{\\hat {f}}])^{2}{\\big ]}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}+2\\operatorname {E} {\\big [}(f-\\operatorname {E} [{\\hat {f}}])\\varepsilon {\\big ]}+2\\operatorname {E} {\\big [}\\varepsilon (\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}){\\big ]}+2\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})(f-\\operatorname {E} [{\\hat {f}}]){\\big ]}\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}+2(f-\\operatorname {E} [{\\hat {f}}])\\operatorname {E} [\\varepsilon ]+2\\operatorname {E} [\\varepsilon ]\\operatorname {E} {\\big [}\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}{\\big ]}+2\\operatorname {E} {\\big [}\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}{\\big ]}(f-\\operatorname {E} [{\\hat {f}}])\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {Var} [\\varepsilon ]+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\\\[5pt]&amp;=\\operatorname {Bias} [{\\hat {f}}]^{2}+\\operatorname {Var} [\\varepsilon ]+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\\\[5pt]&amp;=\\operatorname {Bias} [{\\hat {f}}]^{2}+\\sigma ^{2}+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\end{aligned}}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle {\\begin{aligned}\\operatorname {E} {\\big [}(y-{\\hat {f}})^{2}{\\big ]}&amp;=\\operatorname {E} {\\big [}(f+\\varepsilon -{\\hat {f}})^{2}{\\big ]}\\\\[5pt]&amp;=\\operatorname {E} {\\big [}(f+\\varepsilon -{\\hat {f}}+\\operatorname {E} [{\\hat {f}}]-\\operatorname {E} [{\\hat {f}}])^{2}{\\big ]}\\\\[5pt]&amp;=\\operatorname {E} {\\big [}(f-\\operatorname {E} [{\\hat {f}}])^{2}{\\big ]}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}+2\\operatorname {E} {\\big [}(f-\\operatorname {E} [{\\hat {f}}])\\varepsilon {\\big ]}+2\\operatorname {E} {\\big [}\\varepsilon (\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}){\\big ]}+2\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})(f-\\operatorname {E} [{\\hat {f}}]){\\big ]}\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}+2(f-\\operatorname {E} [{\\hat {f}}])\\operatorname {E} [\\varepsilon ]+2\\operatorname {E} [\\varepsilon ]\\operatorname {E} {\\big [}\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}{\\big ]}+2\\operatorname {E} {\\big [}\\operatorname {E} [{\\hat {f}}]-{\\hat {f}}{\\big ]}(f-\\operatorname {E} [{\\hat {f}}])\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {E} [\\varepsilon ^{2}]+\\operatorname {E} {\\big [}(\\operatorname {E} [{\\hat {f}}]-{\\hat {f}})^{2}{\\big ]}\\\\[5pt]&amp;=(f-\\operatorname {E} [{\\hat {f}}])^{2}+\\operatorname {Var} [\\varepsilon ]+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\\\[5pt]&amp;=\\operatorname {Bias} [{\\hat {f}}]^{2}+\\operatorname {Var} [\\varepsilon ]+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\\\[5pt]&amp;=\\operatorname {Bias} [{\\hat {f}}]^{2}+\\sigma ^{2}+\\operatorname {Var} {\\big [}{\\hat {f}}{\\big ]}\\end{aligned}}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/30dd0e093cc889a45834d04b04456688244b50fb\" style=\"vertical-align: -18.505ex; width:128.289ex; height:38.176ex;\"/></span></dd></dl>\n",
      "<p>Finally, MSE loss function (or negative log-likelihood) is obtained by taking the expectation value over <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle x\\sim P}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
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      "</mrow>\n",
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      "<dl><dd><span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle {\\text{MSE}}=\\operatorname {E} _{x}{\\bigg \\{}\\operatorname {Bias} _{D}[{\\hat {f}}(x;D)]^{2}+\\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}{\\bigg \\}}+\\sigma ^{2}}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
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      "<mtext>MSE</mtext>\n",
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      "<msub>\n",
      "<mi>Var</mi>\n",
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      "<mi>D</mi>\n",
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      "</mrow>\n",
      "<mo stretchy=\"false\">(</mo>\n",
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      "<mi>D</mi>\n",
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      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
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      "<mo maxsize=\"2.047em\" minsize=\"2.047em\">}</mo>\n",
      "</mrow>\n",
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      "<mi>σ<!-- σ --></mi>\n",
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      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle {\\text{MSE}}=\\operatorname {E} _{x}{\\bigg \\{}\\operatorname {Bias} _{D}[{\\hat {f}}(x;D)]^{2}+\\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}{\\bigg \\}}+\\sigma ^{2}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle {\\text{MSE}}=\\operatorname {E} _{x}{\\bigg \\{}\\operatorname {Bias} _{D}[{\\hat {f}}(x;D)]^{2}+\\operatorname {Var} _{D}{\\big [}{\\hat {f}}(x;D){\\big ]}{\\bigg \\}}+\\sigma ^{2}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/10b361231391b0297bcbaf5dc30688f349d64a0b\" style=\"vertical-align: -2.505ex; width:54.827ex; height:6.176ex;\"/></span></dd></dl>\n",
      "<p>*Note: An alternative proof of this Bias-Variance tradeoff can be found on the <a href=\"/wiki/Mean_squared_error\" title=\"Mean squared error\">Mean squared error</a> Wikipedia page, under the section \"Bias-Variance in Machine Learning\".\n",
      "</p>\n",
      "<h2><span class=\"mw-headline\" id=\"Application_to_regression\">Application to regression</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=4\" title=\"Edit section: Application to regression\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<p>The bias–variance decomposition forms the conceptual basis for regression <a href=\"/wiki/Regularization_(mathematics)\" title=\"Regularization (mathematics)\">regularization</a> methods such as <a href=\"/wiki/Lasso_(statistics)\" title=\"Lasso (statistics)\">Lasso</a> and <a class=\"mw-redirect\" href=\"/wiki/Ridge_regression\" title=\"Ridge regression\">ridge regression</a>. Regularization methods introduce bias into the regression solution that can reduce variance considerably relative to the <a href=\"/wiki/Ordinary_least_squares\" title=\"Ordinary least squares\">ordinary least squares (OLS)</a> solution.  Although the OLS solution provides non-biased regression estimates, the lower variance solutions produced by regularization techniques provide superior MSE performance.\n",
      "</p>\n",
      "<h2><span class=\"mw-headline\" id=\"Application_to_classification\">Application to classification</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=5\" title=\"Edit section: Application to classification\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<p>The bias–variance decomposition was originally formulated for least-squares regression. For the case of <a href=\"/wiki/Statistical_classification\" title=\"Statistical classification\">classification</a> under the <a class=\"mw-redirect\" href=\"/wiki/0-1_loss\" title=\"0-1 loss\">0-1 loss</a> (misclassification rate), it is possible to find a similar decomposition.<sup class=\"reference\" id=\"cite_ref-13\"><a href=\"#cite_note-13\">[13]</a></sup><sup class=\"reference\" id=\"cite_ref-14\"><a href=\"#cite_note-14\">[14]</a></sup> Alternatively, if the classification problem can be phrased as <a href=\"/wiki/Probabilistic_classification\" title=\"Probabilistic classification\">probabilistic classification</a>, then the expected squared error of the predicted probabilities with respect to the true probabilities can be decomposed as before.<sup class=\"reference\" id=\"cite_ref-15\"><a href=\"#cite_note-15\">[15]</a></sup>\n",
      "</p>\n",
      "<h2><span class=\"mw-headline\" id=\"Application_to_reinforcement_learning\">Application to reinforcement learning</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=6\" title=\"Edit section: Application to reinforcement learning\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<p>Even though the bias–variance decomposition does not directly apply in reinforcement learning, a similar tradeoff can also characterize generalization. When an agent has limited information on its environment, the suboptimality of an RL algorithm can be decomposed into the sum of two terms: a term related to an asymptotic bias and a term due to overfitting. The asymptotic bias is directly related to the learning algorithm (independently of the quantity of data) while the overfitting term comes from the fact that the amount of data is limited.<sup class=\"reference\" id=\"cite_ref-16\"><a href=\"#cite_note-16\">[16]</a></sup>\n",
      "</p>\n",
      "<h2><span class=\"mw-headline\" id=\"Approaches\">Approaches</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=7\" title=\"Edit section: Approaches\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<p><a href=\"/wiki/Dimensionality_reduction\" title=\"Dimensionality reduction\">Dimensionality reduction</a> and <a href=\"/wiki/Feature_selection\" title=\"Feature selection\">feature selection</a> can decrease variance by simplifying models. Similarly, a larger training set tends to decrease variance. Adding features (predictors) tends to decrease bias, at the expense of introducing additional variance. Learning algorithms typically have some tunable parameters that control bias and variance; for example,\n",
      "</p>\n",
      "<ul><li><a href=\"/wiki/Linear_model\" title=\"Linear model\">linear</a>  and <a href=\"/wiki/Generalized_linear_model\" title=\"Generalized linear model\">Generalized linear</a> models can be <a href=\"/wiki/Regularization_(mathematics)\" title=\"Regularization (mathematics)\">regularized</a> to decrease their variance at the cost of increasing their bias.<sup class=\"reference\" id=\"cite_ref-17\"><a href=\"#cite_note-17\">[17]</a></sup></li>\n",
      "<li>In <a href=\"/wiki/Artificial_neural_network\" title=\"Artificial neural network\">artificial neural networks</a>, the variance increases and the bias decreases as the number of hidden units increase,<sup class=\"reference\" id=\"cite_ref-geman_3-1\"><a href=\"#cite_note-geman-3\">[3]</a></sup> although this classical assumption has been the subject of recent debate.<sup class=\"reference\" id=\"cite_ref-neal2018_8-2\"><a href=\"#cite_note-neal2018-8\">[8]</a></sup> Like in GLMs, regularization is typically applied.</li>\n",
      "<li>In <a class=\"mw-redirect\" href=\"/wiki/K-nearest_neighbor\" title=\"K-nearest neighbor\"><i>k</i>-nearest neighbor</a> models, a high value of <span class=\"texhtml mvar\" style=\"font-style:italic;\">k</span> leads to high bias and low variance (see below).</li>\n",
      "<li>In <a href=\"/wiki/Instance-based_learning\" title=\"Instance-based learning\">instance-based learning</a>, regularization can be achieved varying the mixture of <a href=\"/wiki/Prototype\" title=\"Prototype\">prototypes</a> and exemplars.<sup class=\"reference\" id=\"cite_ref-18\"><a href=\"#cite_note-18\">[18]</a></sup></li>\n",
      "<li>In <a href=\"/wiki/Decision_tree\" title=\"Decision tree\">decision trees</a>, the depth of the tree determines the variance. Decision trees are commonly pruned to control variance.<sup class=\"reference\" id=\"cite_ref-islr_9-2\"><a href=\"#cite_note-islr-9\">[9]</a></sup><sup class=\"reference\" style=\"white-space:nowrap;\">:<span>307</span></sup></li></ul>\n",
      "<p>One way of resolving the trade-off is to use <a class=\"mw-redirect\" href=\"/wiki/Mixture_models\" title=\"Mixture models\">mixture models</a> and <a href=\"/wiki/Ensemble_learning\" title=\"Ensemble learning\">ensemble learning</a>.<sup class=\"reference\" id=\"cite_ref-19\"><a href=\"#cite_note-19\">[19]</a></sup><sup class=\"reference\" id=\"cite_ref-20\"><a href=\"#cite_note-20\">[20]</a></sup> For example, <a href=\"/wiki/Boosting_(machine_learning)\" title=\"Boosting (machine learning)\">boosting</a> combines many \"weak\" (high bias) models in an ensemble that has lower bias than the individual models, while <a href=\"/wiki/Bootstrap_aggregating\" title=\"Bootstrap aggregating\">bagging</a> combines \"strong\" learners in a way that reduces their variance.  \n",
      "</p><p><a class=\"mw-redirect\" href=\"/wiki/Model_validation\" title=\"Model validation\">Model validation</a> methods such as <a href=\"/wiki/Cross-validation_(statistics)\" title=\"Cross-validation (statistics)\">cross-validation (statistics)</a> can be used to tune models so as to optimize the trade-off. \n",
      "</p>\n",
      "<h3><span class=\"mw-headline\" id=\"k-nearest_neighbors\"><i>k</i>-nearest neighbors</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=8\" title=\"Edit section: k-nearest neighbors\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h3>\n",
      "<p>In the case of <a href=\"/wiki/K-nearest_neighbors_algorithm\" title=\"K-nearest neighbors algorithm\"><span class=\"texhtml mvar\" style=\"font-style:italic;\">k</span>-nearest neighbors regression</a>, when the expectation is taken over the possible labeling of a fixed training set, a <a href=\"/wiki/Closed-form_expression\" title=\"Closed-form expression\">closed-form expression</a> exists that relates the bias–variance decomposition to the parameter <span class=\"texhtml mvar\" style=\"font-style:italic;\">k</span>:<sup class=\"reference\" id=\"cite_ref-ESL_10-1\"><a href=\"#cite_note-ESL-10\">[10]</a></sup><sup class=\"reference\" style=\"white-space:nowrap;\">:<span>37, 223</span></sup>\n",
      "</p>\n",
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      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle \\operatorname {E} [(y-{\\hat {f}}(x))^{2}\\mid X=x]=\\left(f(x)-{\\frac {1}{k}}\\sum _{i=1}^{k}f(N_{i}(x))\\right)^{2}+{\\frac {\\sigma ^{2}}{k}}+\\sigma ^{2}}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"{\\displaystyle \\operatorname {E} [(y-{\\hat {f}}(x))^{2}\\mid X=x]=\\left(f(x)-{\\frac {1}{k}}\\sum _{i=1}^{k}f(N_{i}(x))\\right)^{2}+{\\frac {\\sigma ^{2}}{k}}+\\sigma ^{2}}\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/46dd9ffaa7af7d8738d2799f9f91df7c00d2118a\" style=\"vertical-align: -3.171ex; width:64.367ex; height:8.009ex;\"/></span></dd></dl>\n",
      "<p>where <span class=\"mwe-math-element\"><span class=\"mwe-math-mathml-inline mwe-math-mathml-a11y\" style=\"display: none;\"><math alttext=\"{\\displaystyle N_{1}(x),\\dots ,N_{k}(x)}\" xmlns=\"http://www.w3.org/1998/Math/MathML\">\n",
      "<semantics>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mstyle displaystyle=\"true\" scriptlevel=\"0\">\n",
      "<msub>\n",
      "<mi>N</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mn>1</mn>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "<mo>,</mo>\n",
      "<mo>…<!-- … --></mo>\n",
      "<mo>,</mo>\n",
      "<msub>\n",
      "<mi>N</mi>\n",
      "<mrow class=\"MJX-TeXAtom-ORD\">\n",
      "<mi>k</mi>\n",
      "</mrow>\n",
      "</msub>\n",
      "<mo stretchy=\"false\">(</mo>\n",
      "<mi>x</mi>\n",
      "<mo stretchy=\"false\">)</mo>\n",
      "</mstyle>\n",
      "</mrow>\n",
      "<annotation encoding=\"application/x-tex\">{\\displaystyle N_{1}(x),\\dots ,N_{k}(x)}</annotation>\n",
      "</semantics>\n",
      "</math></span><img alt=\"N_1(x), \\dots, N_k(x)\" aria-hidden=\"true\" class=\"mwe-math-fallback-image-inline\" src=\"https://wikimedia.org/api/rest_v1/media/math/render/svg/5835207a9f81fe493e2b77a4a3ab0ca773e4fc50\" style=\"vertical-align: -0.838ex; width:17.332ex; height:2.843ex;\"/></span> are the <span class=\"texhtml mvar\" style=\"font-style:italic;\">k</span> nearest neighbors of <span class=\"texhtml mvar\" style=\"font-style:italic;\">x</span> in the training set. The bias (first term) is a monotone rising function of <span class=\"texhtml mvar\" style=\"font-style:italic;\">k</span>, while the variance (second term) drops off as <span class=\"texhtml mvar\" style=\"font-style:italic;\">k</span> is increased. In fact, under \"reasonable assumptions\" the bias of the first-nearest neighbor (1-NN) estimator vanishes entirely as the size of the training set approaches infinity.<sup class=\"reference\" id=\"cite_ref-geman_3-2\"><a href=\"#cite_note-geman-3\">[3]</a></sup>\n",
      "</p>\n",
      "<h2><span class=\"mw-headline\" id=\"Application_to_human_learning\">Application to human learning</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=9\" title=\"Edit section: Application to human learning\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<p>While widely discussed in the context of machine learning, the bias-variance dilemma has been examined in the context of <a href=\"/wiki/Cognitive_science\" title=\"Cognitive science\">human cognition</a>, most notably by <a href=\"/wiki/Gerd_Gigerenzer\" title=\"Gerd Gigerenzer\">Gerd Gigerenzer</a> and co-workers in the context of learned heuristics. They have argued (see references below) that the human brain resolves the dilemma in the case of the typically sparse, poorly-characterised training-sets provided by experience by adopting high-bias/low variance heuristics. This reflects the fact that a zero-bias approach has poor generalisability to new situations, and also unreasonably presumes precise knowledge of the true state of the world. The resulting heuristics are relatively simple, but produce better inferences in a wider variety of situations.<sup class=\"reference\" id=\"cite_ref-ReferenceA_7-1\"><a href=\"#cite_note-ReferenceA-7\">[7]</a></sup>\n",
      "</p><p><a href=\"/wiki/Stuart_Geman\" title=\"Stuart Geman\">Geman</a> et al.<sup class=\"reference\" id=\"cite_ref-geman_3-3\"><a href=\"#cite_note-geman-3\">[3]</a></sup> argue that the bias-variance dilemma implies that abilities such as generic <a class=\"mw-redirect\" href=\"/wiki/Object_recognition\" title=\"Object recognition\">object recognition</a> cannot be learned from scratch, but require a certain degree of “hard wiring”   that is later tuned by experience.  This is because model-free approaches to inference require impractically large training sets if they are to avoid high variance.\n",
      "</p>\n",
      "<h2><span class=\"mw-headline\" id=\"See_also\">See also</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=10\" title=\"Edit section: See also\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<div class=\"div-col columns column-width\" style=\"-moz-column-width: 25em; -webkit-column-width: 25em; column-width: 25em;\">\n",
      "<ul><li><a href=\"/wiki/Accuracy_and_precision\" title=\"Accuracy and precision\">Accuracy and precision</a></li>\n",
      "<li><a href=\"/wiki/Bias_of_an_estimator\" title=\"Bias of an estimator\">Bias of an estimator</a></li>\n",
      "<li><a href=\"/wiki/Gauss%E2%80%93Markov_theorem\" title=\"Gauss–Markov theorem\">Gauss–Markov theorem</a></li>\n",
      "<li><a href=\"/wiki/Hyperparameter_optimization\" title=\"Hyperparameter optimization\">Hyperparameter optimization</a></li>\n",
      "<li><a href=\"/wiki/Minimum-variance_unbiased_estimator\" title=\"Minimum-variance unbiased estimator\">Minimum-variance unbiased estimator</a></li>\n",
      "<li><a href=\"/wiki/Model_selection\" title=\"Model selection\">Model selection</a></li>\n",
      "<li><a class=\"mw-redirect\" href=\"/wiki/Regression_model_validation\" title=\"Regression model validation\">Regression model validation</a></li>\n",
      "<li><a href=\"/wiki/Supervised_learning\" title=\"Supervised learning\">Supervised learning</a></li></ul>\n",
      "</div>\n",
      "<h2><span class=\"mw-headline\" id=\"References\">References</span><span class=\"mw-editsection\"><span class=\"mw-editsection-bracket\">[</span><a href=\"/w/index.php?title=Bias%E2%80%93variance_tradeoff&amp;action=edit&amp;section=11\" title=\"Edit section: References\">edit</a><span class=\"mw-editsection-bracket\">]</span></span></h2>\n",
      "<div class=\"reflist\" style=\"list-style-type: decimal;\">\n",
      "<div class=\"mw-references-wrap mw-references-columns\"><ol class=\"references\">\n",
      "<li id=\"cite_note-1\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-1\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation journal cs1\" id=\"CITEREFKohaviWolpert1996\">Kohavi, Ron; Wolpert, David H. (1996). \"Bias Plus Variance Decomposition for Zero-One Loss Functions\". <i>ICML</i>. <b>96</b>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=ICML&amp;rft.atitle=Bias+Plus+Variance+Decomposition+for+Zero-One+Loss+Functions&amp;rft.volume=96&amp;rft.date=1996&amp;rft.aulast=Kohavi&amp;rft.aufirst=Ron&amp;rft.au=Wolpert%2C+David+H.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><style data-mw-deduplicate=\"TemplateStyles:r951705291\">.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:\"\\\"\"\"\\\"\"\"'\"\"'\"}.mw-parser-output .id-lock-free a,.mw-parser-output .citation .cs1-lock-free a{background-image:url(\"//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png\");background-image:linear-gradient(transparent,transparent),url(\"//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg\");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .id-lock-limited a,.mw-parser-output .id-lock-registration a,.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background-image:url(\"//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png\");background-image:linear-gradient(transparent,transparent),url(\"//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg\");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .id-lock-subscription a,.mw-parser-output .citation .cs1-lock-subscription a{background-image:url(\"//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png\");background-image:linear-gradient(transparent,transparent),url(\"//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg\");background-repeat:no-repeat;background-size:9px;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background-image:url(\"//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png\");background-image:linear-gradient(transparent,transparent),url(\"//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg\");background-repeat:no-repeat;background-size:12px;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}</style></span>\n",
      "</li>\n",
      "<li id=\"cite_note-2\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-2\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation journal cs1\" id=\"CITEREFLuxburgSchölkopf2011\">Luxburg, Ulrike V.; Schölkopf, B. (2011). \"Statistical learning theory: Models, concepts, and results\". <i>Handbook of the History of Logic</i>. <b>10</b>: Section 2.4.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Handbook+of+the+History+of+Logic&amp;rft.atitle=Statistical+learning+theory%3A+Models%2C+concepts%2C+and+results&amp;rft.volume=10&amp;rft.pages=Section+2.4&amp;rft.date=2011&amp;rft.aulast=Luxburg&amp;rft.aufirst=Ulrike+V.&amp;rft.au=Sch%C3%B6lkopf%2C+B.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-geman-3\"><span class=\"mw-cite-backlink\">^ <a href=\"#cite_ref-geman_3-0\"><sup><i><b>a</b></i></sup></a> <a href=\"#cite_ref-geman_3-1\"><sup><i><b>b</b></i></sup></a> <a href=\"#cite_ref-geman_3-2\"><sup><i><b>c</b></i></sup></a> <a href=\"#cite_ref-geman_3-3\"><sup><i><b>d</b></i></sup></a></span> <span class=\"reference-text\"><cite class=\"citation journal cs1\" id=\"CITEREFGemanBienenstockDoursat1992\"><a href=\"/wiki/Stuart_Geman\" title=\"Stuart Geman\">Geman, Stuart</a>; Bienenstock, Élie; Doursat, René (1992). <a class=\"external text\" href=\"http://web.mit.edu/6.435/www/Geman92.pdf\" rel=\"nofollow\">\"Neural networks and the bias/variance dilemma\"</a> <span class=\"cs1-format\">(PDF)</span>. <i>Neural Computation</i>. <b>4</b>: 1–58. <a class=\"mw-redirect\" href=\"/wiki/Doi_(identifier)\" title=\"Doi (identifier)\">doi</a>:<a class=\"external text\" href=\"https://doi.org/10.1162%2Fneco.1992.4.1.1\" rel=\"nofollow\">10.1162/neco.1992.4.1.1</a>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Neural+Computation&amp;rft.atitle=Neural+networks+and+the+bias%2Fvariance+dilemma&amp;rft.volume=4&amp;rft.pages=1-58&amp;rft.date=1992&amp;rft_id=info%3Adoi%2F10.1162%2Fneco.1992.4.1.1&amp;rft.aulast=Geman&amp;rft.aufirst=Stuart&amp;rft.au=Bienenstock%2C+%C3%89lie&amp;rft.au=Doursat%2C+Ren%C3%A9&amp;rft_id=http%3A%2F%2Fweb.mit.edu%2F6.435%2Fwww%2FGeman92.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-4\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-4\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation book cs1\" id=\"CITEREFSammutWebb2011\">Sammut, Claude; Webb, Geoffrey I., eds. (2011). \"Bias–Variance Decomposition\". <i>Encyclopedia of Machine Learning</i>. Springer. pp. 100–101. <a class=\"mw-redirect\" href=\"/wiki/Bibcode_(identifier)\" title=\"Bibcode (identifier)\">Bibcode</a>:<a class=\"external text\" href=\"https://ui.adsabs.harvard.edu/abs/2010eoml.book.....S\" rel=\"nofollow\">2010eoml.book.....S</a>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Bias%E2%80%93Variance+Decomposition&amp;rft.btitle=Encyclopedia+of+Machine+Learning&amp;rft.pages=100-101&amp;rft.pub=Springer&amp;rft.date=2011&amp;rft_id=info%3Abibcode%2F2010eoml.book.....S&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-nealThesis2019-5\"><span class=\"mw-cite-backlink\">^ <a href=\"#cite_ref-nealThesis2019_5-0\"><sup><i><b>a</b></i></sup></a> <a href=\"#cite_ref-nealThesis2019_5-1\"><sup><i><b>b</b></i></sup></a> <a href=\"#cite_ref-nealThesis2019_5-2\"><sup><i><b>c</b></i></sup></a></span> <span class=\"reference-text\"><cite class=\"citation arxiv cs1\" id=\"CITEREFNeal2019\">Neal, Brady (2019). \"On the Bias-Variance Tradeoff: Textbooks Need an Update\". <a class=\"mw-redirect\" href=\"/wiki/ArXiv_(identifier)\" title=\"ArXiv (identifier)\">arXiv</a>:<span class=\"cs1-lock-free\" title=\"Freely accessible\"><a class=\"external text\" href=\"//arxiv.org/abs/1912.08286\" rel=\"nofollow\">1912.08286</a></span> [<a class=\"external text\" href=\"//arxiv.org/archive/cs.LG\" rel=\"nofollow\">cs.LG</a>].</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=preprint&amp;rft.jtitle=arXiv&amp;rft.atitle=On+the+Bias-Variance+Tradeoff%3A+Textbooks+Need+an+Update&amp;rft.date=2019&amp;rft_id=info%3Aarxiv%2F1912.08286&amp;rft.aulast=Neal&amp;rft.aufirst=Brady&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-nealBlog2020-6\"><span class=\"mw-cite-backlink\">^ <a href=\"#cite_ref-nealBlog2020_6-0\"><sup><i><b>a</b></i></sup></a> <a href=\"#cite_ref-nealBlog2020_6-1\"><sup><i><b>b</b></i></sup></a> <a href=\"#cite_ref-nealBlog2020_6-2\"><sup><i><b>c</b></i></sup></a></span> <span class=\"reference-text\"><cite class=\"citation web cs1\" id=\"CITEREFNeal2020\">Neal, Brady (5 January 2020). <a class=\"external text\" href=\"https://www.bradyneal.com/bias-variance-tradeoff-textbooks-update\" rel=\"nofollow\">\"On the Bias-Variance Tradeoff: Textbooks Need an Update (Blog Post)\"</a><span class=\"reference-accessdate\">. Retrieved <span class=\"nowrap\">12 March</span> 2020</span>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=On+the+Bias-Variance+Tradeoff%3A+Textbooks+Need+an+Update+%28Blog+Post%29&amp;rft.date=2020-01-05&amp;rft.aulast=Neal&amp;rft.aufirst=Brady&amp;rft_id=https%3A%2F%2Fwww.bradyneal.com%2Fbias-variance-tradeoff-textbooks-update&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-ReferenceA-7\"><span class=\"mw-cite-backlink\">^ <a href=\"#cite_ref-ReferenceA_7-0\"><sup><i><b>a</b></i></sup></a> <a href=\"#cite_ref-ReferenceA_7-1\"><sup><i><b>b</b></i></sup></a></span> <span class=\"reference-text\"><cite class=\"citation journal cs1\" id=\"CITEREFGigerenzerBrighton2009\"><a href=\"/wiki/Gerd_Gigerenzer\" title=\"Gerd Gigerenzer\">Gigerenzer, Gerd</a>; Brighton, Henry (2009). \"Homo Heuristicus: Why Biased Minds Make Better Inferences\". <i>Topics in Cognitive Science</i>. <b>1</b> (1): 107–143. <a class=\"mw-redirect\" href=\"/wiki/Doi_(identifier)\" title=\"Doi (identifier)\">doi</a>:<a class=\"external text\" href=\"https://doi.org/10.1111%2Fj.1756-8765.2008.01006.x\" rel=\"nofollow\">10.1111/j.1756-8765.2008.01006.x</a>. <a class=\"mw-redirect\" href=\"/wiki/Hdl_(identifier)\" title=\"Hdl (identifier)\">hdl</a>:<span class=\"cs1-lock-free\" title=\"Freely accessible\"><a class=\"external text\" href=\"//hdl.handle.net/11858%2F00-001M-0000-0024-F678-0\" rel=\"nofollow\">11858/00-001M-0000-0024-F678-0</a></span>. <a class=\"mw-redirect\" href=\"/wiki/PMID_(identifier)\" title=\"PMID (identifier)\">PMID</a> <a class=\"external text\" href=\"//pubmed.ncbi.nlm.nih.gov/25164802\" rel=\"nofollow\">25164802</a>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Topics+in+Cognitive+Science&amp;rft.atitle=Homo+Heuristicus%3A+Why+Biased+Minds+Make+Better+Inferences&amp;rft.volume=1&amp;rft.issue=1&amp;rft.pages=107-143&amp;rft.date=2009&amp;rft_id=info%3Ahdl%2F11858%2F00-001M-0000-0024-F678-0&amp;rft_id=info%3Apmid%2F25164802&amp;rft_id=info%3Adoi%2F10.1111%2Fj.1756-8765.2008.01006.x&amp;rft.aulast=Gigerenzer&amp;rft.aufirst=Gerd&amp;rft.au=Brighton%2C+Henry&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-neal2018-8\"><span class=\"mw-cite-backlink\">^ <a href=\"#cite_ref-neal2018_8-0\"><sup><i><b>a</b></i></sup></a> <a href=\"#cite_ref-neal2018_8-1\"><sup><i><b>b</b></i></sup></a> <a href=\"#cite_ref-neal2018_8-2\"><sup><i><b>c</b></i></sup></a></span> <span class=\"reference-text\"><cite class=\"citation arxiv cs1\" id=\"CITEREFNealMittalBaratinTantia2018\">Neal, Brady; Mittal, Sarthak; Baratin, Aristide; Tantia, Vinayak; Scicluna, Matthew; Lacoste-Julien, Simon; Mitliagkas, Ioannis (2018). \"A Modern Take on the Bias-Variance Tradeoff in Neural Networks\". <a class=\"mw-redirect\" href=\"/wiki/ArXiv_(identifier)\" title=\"ArXiv (identifier)\">arXiv</a>:<span class=\"cs1-lock-free\" title=\"Freely accessible\"><a class=\"external text\" href=\"//arxiv.org/abs/1810.08591\" rel=\"nofollow\">1810.08591</a></span> [<a class=\"external text\" href=\"//arxiv.org/archive/cs.LG\" rel=\"nofollow\">cs.LG</a>].</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=preprint&amp;rft.jtitle=arXiv&amp;rft.atitle=A+Modern+Take+on+the+Bias-Variance+Tradeoff+in+Neural+Networks&amp;rft.date=2018&amp;rft_id=info%3Aarxiv%2F1810.08591&amp;rft.aulast=Neal&amp;rft.aufirst=Brady&amp;rft.au=Mittal%2C+Sarthak&amp;rft.au=Baratin%2C+Aristide&amp;rft.au=Tantia%2C+Vinayak&amp;rft.au=Scicluna%2C+Matthew&amp;rft.au=Lacoste-Julien%2C+Simon&amp;rft.au=Mitliagkas%2C+Ioannis&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-islr-9\"><span class=\"mw-cite-backlink\">^ <a href=\"#cite_ref-islr_9-0\"><sup><i><b>a</b></i></sup></a> <a href=\"#cite_ref-islr_9-1\"><sup><i><b>b</b></i></sup></a> <a href=\"#cite_ref-islr_9-2\"><sup><i><b>c</b></i></sup></a></span> <span class=\"reference-text\"><cite class=\"citation book cs1\" id=\"CITEREFJamesWittenHastieTibshirani2013\">James, Gareth; <a href=\"/wiki/Daniela_Witten\" title=\"Daniela Witten\">Witten, Daniela</a>; <a href=\"/wiki/Trevor_Hastie\" title=\"Trevor Hastie\">Hastie, Trevor</a>; <a href=\"/wiki/Robert_Tibshirani\" title=\"Robert Tibshirani\">Tibshirani, Robert</a> (2013). <a class=\"external text\" href=\"http://www-bcf.usc.edu/~gareth/ISL/\" rel=\"nofollow\"><i>An Introduction to Statistical Learning</i></a>. Springer.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=An+Introduction+to+Statistical+Learning&amp;rft.pub=Springer&amp;rft.date=2013&amp;rft.aulast=James&amp;rft.aufirst=Gareth&amp;rft.au=Witten%2C+Daniela&amp;rft.au=Hastie%2C+Trevor&amp;rft.au=Tibshirani%2C+Robert&amp;rft_id=http%3A%2F%2Fwww-bcf.usc.edu%2F~gareth%2FISL%2F&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-ESL-10\"><span class=\"mw-cite-backlink\">^ <a href=\"#cite_ref-ESL_10-0\"><sup><i><b>a</b></i></sup></a> <a href=\"#cite_ref-ESL_10-1\"><sup><i><b>b</b></i></sup></a></span> <span class=\"reference-text\"><cite class=\"citation book cs1\" id=\"CITEREFHastieTibshiraniFriedman2009\">Hastie, Trevor; Tibshirani, Robert; <a href=\"/wiki/Jerome_H._Friedman\" title=\"Jerome H. Friedman\">Friedman, Jerome H.</a> (2009). <a class=\"external text\" href=\"https://web.archive.org/web/20150126123924/http://statweb.stanford.edu/~tibs/ElemStatLearn/\" rel=\"nofollow\"><i>The Elements of Statistical Learning</i></a>. Archived from <a class=\"external text\" href=\"http://statweb.stanford.edu/~tibs/ElemStatLearn/\" rel=\"nofollow\">the original</a> on 2015-01-26<span class=\"reference-accessdate\">. Retrieved <span class=\"nowrap\">2014-08-20</span></span>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=The+Elements+of+Statistical+Learning&amp;rft.date=2009&amp;rft.aulast=Hastie&amp;rft.aufirst=Trevor&amp;rft.au=Tibshirani%2C+Robert&amp;rft.au=Friedman%2C+Jerome+H.&amp;rft_id=http%3A%2F%2Fstatweb.stanford.edu%2F~tibs%2FElemStatLearn%2F&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-11\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-11\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation web cs1\" id=\"CITEREFVijayakumar2007\"><a href=\"/wiki/Sethu_Vijayakumar\" title=\"Sethu Vijayakumar\">Vijayakumar, Sethu</a> (2007). <a class=\"external text\" href=\"http://www.inf.ed.ac.uk/teaching/courses/mlsc/Notes/Lecture4/BiasVariance.pdf\" rel=\"nofollow\">\"The Bias–Variance Tradeoff\"</a> <span class=\"cs1-format\">(PDF)</span>. <a href=\"/wiki/University_of_Edinburgh\" title=\"University of Edinburgh\">University of Edinburgh</a><span class=\"reference-accessdate\">. Retrieved <span class=\"nowrap\">19 August</span> 2014</span>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=The+Bias%E2%80%93Variance+Tradeoff&amp;rft.pub=University+of+Edinburgh&amp;rft.date=2007&amp;rft.aulast=Vijayakumar&amp;rft.aufirst=Sethu&amp;rft_id=http%3A%2F%2Fwww.inf.ed.ac.uk%2Fteaching%2Fcourses%2Fmlsc%2FNotes%2FLecture4%2FBiasVariance.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-12\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-12\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation web cs1\" id=\"CITEREFShakhnarovich2011\">Shakhnarovich, Greg (2011). <a class=\"external text\" href=\"https://web.archive.org/web/20140821063842/http://ttic.uchicago.edu/~gregory/courses/wis-ml2012/lectures/biasVarDecom.pdf\" rel=\"nofollow\">\"Notes on derivation of bias-variance decomposition in linear regression\"</a> <span class=\"cs1-format\">(PDF)</span>. Archived from <a class=\"external text\" href=\"http://ttic.uchicago.edu/~gregory/courses/wis-ml2012/lectures/biasVarDecom.pdf\" rel=\"nofollow\">the original</a> <span class=\"cs1-format\">(PDF)</span> on 21 August 2014<span class=\"reference-accessdate\">. Retrieved <span class=\"nowrap\">20 August</span> 2014</span>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Notes+on+derivation+of+bias-variance+decomposition+in+linear+regression&amp;rft.date=2011&amp;rft.aulast=Shakhnarovich&amp;rft.aufirst=Greg&amp;rft_id=http%3A%2F%2Fttic.uchicago.edu%2F~gregory%2Fcourses%2Fwis-ml2012%2Flectures%2FbiasVarDecom.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-13\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-13\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation conference cs1\" id=\"CITEREFDomingos2000\"><a href=\"/wiki/Pedro_Domingos\" title=\"Pedro Domingos\">Domingos, Pedro</a> (2000). <a class=\"external text\" href=\"http://homes.cs.washington.edu/~pedrod/bvd.pdf\" rel=\"nofollow\"><i>A unified bias-variance decomposition</i></a> <span class=\"cs1-format\">(PDF)</span>. ICML.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=conference&amp;rft.btitle=A+unified+bias-variance+decomposition&amp;rft.date=2000&amp;rft.aulast=Domingos&amp;rft.aufirst=Pedro&amp;rft_id=http%3A%2F%2Fhomes.cs.washington.edu%2F~pedrod%2Fbvd.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-14\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-14\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation journal cs1\" id=\"CITEREFValentiniDietterich2004\">Valentini, Giorgio; <a href=\"/wiki/Thomas_G._Dietterich\" title=\"Thomas G. Dietterich\">Dietterich, Thomas G.</a> (2004). <a class=\"external text\" href=\"http://www.jmlr.org/papers/volume5/valentini04a/valentini04a.pdf\" rel=\"nofollow\">\"Bias–variance analysis of support vector machines for the development of SVM-based ensemble methods\"</a> <span class=\"cs1-format\">(PDF)</span>. <i><a href=\"/wiki/Journal_of_Machine_Learning_Research\" title=\"Journal of Machine Learning Research\">Journal of Machine Learning Research</a></i>. <b>5</b>: 725–775.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+Machine+Learning+Research&amp;rft.atitle=Bias%E2%80%93variance+analysis+of+support+vector+machines+for+the+development+of+SVM-based+ensemble+methods&amp;rft.volume=5&amp;rft.pages=725-775&amp;rft.date=2004&amp;rft.aulast=Valentini&amp;rft.aufirst=Giorgio&amp;rft.au=Dietterich%2C+Thomas+G.&amp;rft_id=http%3A%2F%2Fwww.jmlr.org%2Fpapers%2Fvolume5%2Fvalentini04a%2Fvalentini04a.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-15\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-15\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation book cs1\" id=\"CITEREFManningRaghavanSchütze2008\">Manning, Christopher D.; Raghavan, Prabhakar; Schütze, Hinrich (2008). <a class=\"external text\" href=\"http://nlp.stanford.edu/IR-book/\" rel=\"nofollow\"><i>Introduction to Information Retrieval</i></a>. Cambridge University Press. pp. 308–314.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Introduction+to+Information+Retrieval&amp;rft.pages=308-314&amp;rft.pub=Cambridge+University+Press&amp;rft.date=2008&amp;rft.aulast=Manning&amp;rft.aufirst=Christopher+D.&amp;rft.au=Raghavan%2C+Prabhakar&amp;rft.au=Sch%C3%BCtze%2C+Hinrich&amp;rft_id=http%3A%2F%2Fnlp.stanford.edu%2FIR-book%2F&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-16\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-16\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation journal cs1\" id=\"CITEREFFrancois-LavetRabusseauPineauErnst2019\">Francois-Lavet, Vincent; Rabusseau, Guillaume; Pineau, Joelle; Ernst, Damien; Fonteneau, Raphael (2019). <a class=\"external text\" href=\"https://jair.org/index.php/jair/article/view/11478\" rel=\"nofollow\">\"On Overﬁtting and Asymptotic Bias in Batch Reinforcement Learning with Partial Observability\"</a>. <i>Journal of AI Research</i>. <b>65</b>: 1–30. <a class=\"mw-redirect\" href=\"/wiki/Doi_(identifier)\" title=\"Doi (identifier)\">doi</a>:<span class=\"cs1-lock-free\" title=\"Freely accessible\"><a class=\"external text\" href=\"https://doi.org/10.1613%2Fjair.1.11478\" rel=\"nofollow\">10.1613/jair.1.11478</a></span>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Journal+of+AI+Research&amp;rft.atitle=On+Over%EF%AC%81tting+and+Asymptotic+Bias+in+Batch+Reinforcement+Learning+with+Partial+Observability&amp;rft.volume=65&amp;rft.pages=1-30&amp;rft.date=2019&amp;rft_id=info%3Adoi%2F10.1613%2Fjair.1.11478&amp;rft.aulast=Francois-Lavet&amp;rft.aufirst=Vincent&amp;rft.au=Rabusseau%2C+Guillaume&amp;rft.au=Pineau%2C+Joelle&amp;rft.au=Ernst%2C+Damien&amp;rft.au=Fonteneau%2C+Raphael&amp;rft_id=https%3A%2F%2Fjair.org%2Findex.php%2Fjair%2Farticle%2Fview%2F11478&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-17\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-17\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation book cs1\" id=\"CITEREFBelsley1991\">Belsley, David (1991). <i>Conditioning diagnostics : collinearity and weak data in regression</i>. New York (NY): Wiley. <a class=\"mw-redirect\" href=\"/wiki/ISBN_(identifier)\" title=\"ISBN (identifier)\">ISBN</a> <a href=\"/wiki/Special:BookSources/978-0471528890\" title=\"Special:BookSources/978-0471528890\"><bdi>978-0471528890</bdi></a>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=book&amp;rft.btitle=Conditioning+diagnostics+%3A+collinearity+and+weak+data+in+regression&amp;rft.place=New+York+%28NY%29&amp;rft.pub=Wiley&amp;rft.date=1991&amp;rft.isbn=978-0471528890&amp;rft.aulast=Belsley&amp;rft.aufirst=David&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-18\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-18\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation journal cs1\" id=\"CITEREFGagliardi2011\">Gagliardi, Francesco (May 2011). <a class=\"external text\" href=\"https://www.researchgate.net/publication/51173579\" rel=\"nofollow\">\"Instance-based classifiers applied to medical databases: diagnosis and knowledge extraction\"</a>. <i>Artificial Intelligence in Medicine</i>. <b>52</b> (3): 123–139. <a class=\"mw-redirect\" href=\"/wiki/Doi_(identifier)\" title=\"Doi (identifier)\">doi</a>:<a class=\"external text\" href=\"https://doi.org/10.1016%2Fj.artmed.2011.04.002\" rel=\"nofollow\">10.1016/j.artmed.2011.04.002</a>. <a class=\"mw-redirect\" href=\"/wiki/PMID_(identifier)\" title=\"PMID (identifier)\">PMID</a> <a class=\"external text\" href=\"//pubmed.ncbi.nlm.nih.gov/21621400\" rel=\"nofollow\">21621400</a>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=Artificial+Intelligence+in+Medicine&amp;rft.atitle=Instance-based+classifiers+applied+to+medical+databases%3A+diagnosis+and+knowledge+extraction&amp;rft.volume=52&amp;rft.issue=3&amp;rft.pages=123-139&amp;rft.date=2011-05&amp;rft_id=info%3Adoi%2F10.1016%2Fj.artmed.2011.04.002&amp;rft_id=info%3Apmid%2F21621400&amp;rft.aulast=Gagliardi&amp;rft.aufirst=Francesco&amp;rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpublication%2F51173579&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-19\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-19\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation book cs1\" id=\"CITEREFTingVijaykumarSchaal2011\">Ting, Jo-Anne; Vijaykumar, Sethu; Schaal, Stefan (2011). \"Locally Weighted Regression for Control\".  In Sammut, Claude; Webb, Geoffrey I. (eds.). <a class=\"external text\" href=\"http://homepages.inf.ed.ac.uk/svijayak/publications/ting-EMLDM2016.pdf\" rel=\"nofollow\"><i>Encyclopedia of Machine Learning</i></a> <span class=\"cs1-format\">(PDF)</span>. Springer. p. 615. <a class=\"mw-redirect\" href=\"/wiki/Bibcode_(identifier)\" title=\"Bibcode (identifier)\">Bibcode</a>:<a class=\"external text\" href=\"https://ui.adsabs.harvard.edu/abs/2010eoml.book.....S\" rel=\"nofollow\">2010eoml.book.....S</a>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=bookitem&amp;rft.atitle=Locally+Weighted+Regression+for+Control&amp;rft.btitle=Encyclopedia+of+Machine+Learning&amp;rft.pages=615&amp;rft.pub=Springer&amp;rft.date=2011&amp;rft_id=info%3Abibcode%2F2010eoml.book.....S&amp;rft.aulast=Ting&amp;rft.aufirst=Jo-Anne&amp;rft.au=Vijaykumar%2C+Sethu&amp;rft.au=Schaal%2C+Stefan&amp;rft_id=http%3A%2F%2Fhomepages.inf.ed.ac.uk%2Fsvijayak%2Fpublications%2Fting-EMLDM2016.pdf&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "<li id=\"cite_note-20\"><span class=\"mw-cite-backlink\"><b><a href=\"#cite_ref-20\">^</a></b></span> <span class=\"reference-text\"><cite class=\"citation web cs1\" id=\"CITEREFFortmann-Roe2012\">Fortmann-Roe, Scott (2012). <a class=\"external text\" href=\"http://scott.fortmann-roe.com/docs/BiasVariance.html\" rel=\"nofollow\">\"Understanding the Bias–Variance Tradeoff\"</a>.</cite><span class=\"Z3988\" title=\"ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rft.genre=unknown&amp;rft.btitle=Understanding+the+Bias%E2%80%93Variance+Tradeoff&amp;rft.date=2012&amp;rft.aulast=Fortmann-Roe&amp;rft.aufirst=Scott&amp;rft_id=http%3A%2F%2Fscott.fortmann-roe.com%2Fdocs%2FBiasVariance.html&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3ABias%E2%80%93variance+tradeoff\"></span><link href=\"mw-data:TemplateStyles:r951705291\" rel=\"mw-deduplicated-inline-style\"/></span>\n",
      "</li>\n",
      "</ol></div></div>\n",
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     "text": [
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      "<h1 class=\"firstHeading\" id=\"firstHeading\" lang=\"en\">Category:Machine learning algorithms</h1>\n",
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      "<div class=\"noprint\" id=\"siteSub\">From Wikipedia, the free encyclopedia</div>\n",
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      "<td class=\"mbox-text plainlist\">Wikimedia Commons has media related to <i><b><a class=\"extiw\" href=\"https://commons.wikimedia.org/wiki/Category:Machine_learning_algorithms\" title=\"commons:Category:Machine learning algorithms\"><span style=\"\">Machine learning algorithms</span></a></b></i>.</td></tr>\n",
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      "<div class=\"hatnote navigation-not-searchable selfref\" role=\"note\">See also: the categories <a href=\"/wiki/Category:Data_mining\" title=\"Category:Data mining\">Data mining</a>, <a href=\"/wiki/Category:Classification_algorithms\" title=\"Category:Classification algorithms\">Classification algorithms</a>, and <a href=\"/wiki/Category:Decision_trees\" title=\"Category:Decision trees\">Decision trees</a>.</div>\n",
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      "</div><div class=\"mw-category-generated\" dir=\"ltr\" lang=\"en\"><div id=\"mw-subcategories\">\n",
      "<h2>Subcategories</h2>\n",
      "<p>This category has only the following subcategory.\n",
      "</p><div class=\"mw-content-ltr\" dir=\"ltr\" lang=\"en\"><h3>G</h3>\n",
      "<ul><li><div class=\"CategoryTreeSection\"><div class=\"CategoryTreeItem\"><span class=\"CategoryTreeBullet\"><span class=\"CategoryTreeToggle\" data-ct-state=\"collapsed\" data-ct-title=\"Genetic_algorithms\">►</span> </span> <a href=\"/wiki/Category:Genetic_algorithms\" title=\"Category:Genetic algorithms\">Genetic algorithms</a>‎ <span dir=\"ltr\" title=\"Contains 2 subcategories, 43 pages, and 0 files\">(2 C, 43 P)</span></div><div class=\"CategoryTreeChildren\" style=\"display:none\"></div></div></li></ul></div>\n",
      "</div><div id=\"mw-pages\">\n",
      "<h2><span id=\"Pages_in_category\"></span>Pages in category \"Machine learning algorithms\"</h2>\n",
      "<p>The following 68 pages are in this category, out of  68 total. This list may not reflect recent changes (<a href=\"/wiki/Wikipedia:FAQ/Categorization#Why_might_a_category_list_not_be_up_to_date?\" title=\"Wikipedia:FAQ/Categorization\">learn more</a>).\n",
      "</p><div class=\"mw-content-ltr\" dir=\"ltr\" lang=\"en\"><div class=\"mw-category\"><div class=\"mw-category-group\"><h3>A</h3>\n",
      "<ul><li><span class=\"redirect-in-category\"><a class=\"mw-redirect\" href=\"/wiki/Adagrad\" title=\"Adagrad\">Adagrad</a></span></li>\n",
      "<li><a href=\"/wiki/Algorithms_of_Oppression\" title=\"Algorithms of Oppression\">Algorithms of Oppression</a></li>\n",
      "<li><a href=\"/wiki/Almeida%E2%80%93Pineda_recurrent_backpropagation\" title=\"Almeida–Pineda recurrent backpropagation\">Almeida–Pineda recurrent backpropagation</a></li>\n",
      "<li><a href=\"/wiki/Augmented_Analytics\" title=\"Augmented Analytics\">Augmented Analytics</a></li></ul></div><div class=\"mw-category-group\"><h3>B</h3>\n",
      "<ul><li><a href=\"/wiki/Backpropagation\" title=\"Backpropagation\">Backpropagation</a></li>\n",
      "<li><a href=\"/wiki/Bioz\" title=\"Bioz\">Bioz</a></li>\n",
      "<li><a href=\"/wiki/Bootstrap_aggregating\" title=\"Bootstrap aggregating\">Bootstrap aggregating</a></li></ul></div><div class=\"mw-category-group\"><h3>C</h3>\n",
      "<ul><li><a href=\"/wiki/Cancer_Likelihood_in_Plasma\" title=\"Cancer Likelihood in Plasma\">Cancer Likelihood in Plasma</a></li>\n",
      "<li><a href=\"/wiki/CN2_algorithm\" title=\"CN2 algorithm\">CN2 algorithm</a></li>\n",
      "<li><a href=\"/wiki/Constructing_skill_trees\" title=\"Constructing skill trees\">Constructing skill trees</a></li></ul></div><div class=\"mw-category-group\"><h3>D</h3>\n",
      "<ul><li><a href=\"/wiki/Deep_reinforcement_learning\" title=\"Deep reinforcement learning\">Deep reinforcement learning</a></li>\n",
      "<li><a href=\"/wiki/Dehaene%E2%80%93Changeux_model\" title=\"Dehaene–Changeux model\">Dehaene–Changeux model</a></li>\n",
      "<li><a href=\"/wiki/Diffusion_map\" title=\"Diffusion map\">Diffusion map</a></li>\n",
      "<li><a href=\"/wiki/Dominance-based_rough_set_approach\" title=\"Dominance-based rough set approach\">Dominance-based rough set approach</a></li>\n",
      "<li><a href=\"/wiki/Dynamic_time_warping\" title=\"Dynamic time warping\">Dynamic time warping</a></li></ul></div><div class=\"mw-category-group\"><h3>E</h3>\n",
      "<ul><li><a href=\"/wiki/Elastic_net_regularization\" title=\"Elastic net regularization\">Elastic net regularization</a></li>\n",
      "<li><a href=\"/wiki/Error-driven_learning\" title=\"Error-driven learning\">Error-driven learning</a></li>\n",
      "<li><a href=\"/wiki/Evolutionary_multimodal_optimization\" title=\"Evolutionary multimodal optimization\">Evolutionary multimodal optimization</a></li>\n",
      "<li><a href=\"/wiki/Expectation%E2%80%93maximization_algorithm\" title=\"Expectation–maximization algorithm\">Expectation–maximization algorithm</a></li>\n",
      "<li><a href=\"/wiki/Extremal_Ensemble_Learning\" title=\"Extremal Ensemble Learning\">Extremal Ensemble Learning</a></li></ul></div><div class=\"mw-category-group\"><h3>F</h3>\n",
      "<ul><li><a href=\"/wiki/FastICA\" title=\"FastICA\">FastICA</a></li>\n",
      "<li><a href=\"/wiki/Forward%E2%80%93backward_algorithm\" title=\"Forward–backward algorithm\">Forward–backward algorithm</a></li></ul></div><div class=\"mw-category-group\"><h3>G</h3>\n",
      "<ul><li><a href=\"/wiki/GeneRec\" title=\"GeneRec\">GeneRec</a></li>\n",
      "<li><a href=\"/wiki/Genetic_Algorithm_for_Rule_Set_Production\" title=\"Genetic Algorithm for Rule Set Production\">Genetic Algorithm for Rule Set Production</a></li>\n",
      "<li><a href=\"/wiki/Growing_self-organizing_map\" title=\"Growing self-organizing map\">Growing self-organizing map</a></li></ul></div><div class=\"mw-category-group\"><h3>H</h3>\n",
      "<ul><li><a href=\"/wiki/Hyper_basis_function_network\" title=\"Hyper basis function network\">Hyper basis function network</a></li></ul></div><div class=\"mw-category-group\"><h3>I</h3>\n",
      "<ul><li><a href=\"/wiki/IDistance\" title=\"IDistance\">IDistance</a></li>\n",
      "<li><a href=\"/wiki/Incremental_learning\" title=\"Incremental learning\">Incremental learning</a></li></ul></div><div class=\"mw-category-group\"><h3>K</h3>\n",
      "<ul><li><a href=\"/wiki/K-nearest_neighbors_algorithm\" title=\"K-nearest neighbors algorithm\">K-nearest neighbors algorithm</a></li>\n",
      "<li><a href=\"/wiki/Kernel_methods_for_vector_output\" title=\"Kernel methods for vector output\">Kernel methods for vector output</a></li>\n",
      "<li><a href=\"/wiki/Kernel_principal_component_analysis\" title=\"Kernel principal component analysis\">Kernel principal component analysis</a></li></ul></div><div class=\"mw-category-group\"><h3>L</h3>\n",
      "<ul><li><a href=\"/wiki/Label_propagation_algorithm\" title=\"Label propagation algorithm\">Label propagation algorithm</a></li>\n",
      "<li><a href=\"/wiki/Lasso_(statistics)\" title=\"Lasso (statistics)\">Lasso (statistics)</a></li>\n",
      "<li><a href=\"/wiki/Leabra\" title=\"Leabra\">Leabra</a></li>\n",
      "<li><a href=\"/wiki/Linde%E2%80%93Buzo%E2%80%93Gray_algorithm\" title=\"Linde–Buzo–Gray algorithm\">Linde–Buzo–Gray algorithm</a></li>\n",
      "<li><a href=\"/wiki/Local_outlier_factor\" title=\"Local outlier factor\">Local outlier factor</a></li>\n",
      "<li><a href=\"/wiki/Logic_learning_machine\" title=\"Logic learning machine\">Logic learning machine</a></li>\n",
      "<li><a href=\"/wiki/LogitBoost\" title=\"LogitBoost\">LogitBoost</a></li>\n",
      "<li><a href=\"/wiki/Loss_functions_for_classification\" title=\"Loss functions for classification\">Loss functions for classification</a></li></ul></div><div class=\"mw-category-group\"><h3>M</h3>\n",
      "<ul><li><a href=\"/wiki/Manifold_alignment\" title=\"Manifold alignment\">Manifold alignment</a></li>\n",
      "<li><a href=\"/wiki/Minimum_redundancy_feature_selection\" title=\"Minimum redundancy feature selection\">Minimum redundancy feature selection</a></li>\n",
      "<li><a href=\"/wiki/Mixture_of_experts\" title=\"Mixture of experts\">Mixture of experts</a></li>\n",
      "<li><a href=\"/wiki/Multiple_kernel_learning\" title=\"Multiple kernel learning\">Multiple kernel learning</a></li></ul></div><div class=\"mw-category-group\"><h3>N</h3>\n",
      "<ul><li><a href=\"/wiki/Non-negative_matrix_factorization\" title=\"Non-negative matrix factorization\">Non-negative matrix factorization</a></li></ul></div><div class=\"mw-category-group\"><h3>O</h3>\n",
      "<ul><li><a href=\"/wiki/Online_machine_learning\" title=\"Online machine learning\">Online machine learning</a></li>\n",
      "<li><a href=\"/wiki/Out-of-bag_error\" title=\"Out-of-bag error\">Out-of-bag error</a></li></ul></div><div class=\"mw-category-group\"><h3>P</h3>\n",
      "<ul><li><a href=\"/wiki/Prefrontal_cortex_basal_ganglia_working_memory\" title=\"Prefrontal cortex basal ganglia working memory\">Prefrontal cortex basal ganglia working memory</a></li>\n",
      "<li><a href=\"/wiki/Prototype_methods\" title=\"Prototype methods\">Prototype methods</a></li>\n",
      "<li><a href=\"/wiki/PVLV\" title=\"PVLV\">PVLV</a></li></ul></div><div class=\"mw-category-group\"><h3>Q</h3>\n",
      "<ul><li><a href=\"/wiki/Q-learning\" title=\"Q-learning\">Q-learning</a></li>\n",
      "<li><a href=\"/wiki/Quadratic_unconstrained_binary_optimization\" title=\"Quadratic unconstrained binary optimization\">Quadratic unconstrained binary optimization</a></li>\n",
      "<li><a href=\"/wiki/Query-level_feature\" title=\"Query-level feature\">Query-level feature</a></li>\n",
      "<li><a href=\"/wiki/Quickprop\" title=\"Quickprop\">Quickprop</a></li></ul></div><div class=\"mw-category-group\"><h3>R</h3>\n",
      "<ul><li><a href=\"/wiki/Radial_basis_function_network\" title=\"Radial basis function network\">Radial basis function network</a></li>\n",
      "<li><a href=\"/wiki/Randomized_weighted_majority_algorithm\" title=\"Randomized weighted majority algorithm\">Randomized weighted majority algorithm</a></li>\n",
      "<li><a href=\"/wiki/Repeated_incremental_pruning_to_produce_error_reduction_(RIPPER)\" title=\"Repeated incremental pruning to produce error reduction (RIPPER)\">Repeated incremental pruning to produce error reduction (RIPPER)</a></li>\n",
      "<li><a href=\"/wiki/Rprop\" title=\"Rprop\">Rprop</a></li>\n",
      "<li><a href=\"/wiki/Rule-based_machine_learning\" title=\"Rule-based machine learning\">Rule-based machine learning</a></li></ul></div><div class=\"mw-category-group\"><h3>S</h3>\n",
      "<ul><li><a href=\"/wiki/Skill_chaining\" title=\"Skill chaining\">Skill chaining</a></li>\n",
      "<li><a href=\"/wiki/Sparse_PCA\" title=\"Sparse PCA\">Sparse PCA</a></li>\n",
      "<li><a href=\"/wiki/State%E2%80%93action%E2%80%93reward%E2%80%93state%E2%80%93action\" title=\"State–action–reward–state–action\">State–action–reward–state–action</a></li>\n",
      "<li><a href=\"/wiki/Stochastic_gradient_descent\" title=\"Stochastic gradient descent\">Stochastic gradient descent</a></li>\n",
      "<li><a href=\"/wiki/Structured_kNN\" title=\"Structured kNN\">Structured kNN</a></li></ul></div><div class=\"mw-category-group\"><h3>T</h3>\n",
      "<ul><li><a href=\"/wiki/T-distributed_stochastic_neighbor_embedding\" title=\"T-distributed stochastic neighbor embedding\">T-distributed stochastic neighbor embedding</a></li>\n",
      "<li><a href=\"/wiki/Triplet_loss\" title=\"Triplet loss\">Triplet loss</a></li></ul></div><div class=\"mw-category-group\"><h3>W</h3>\n",
      "<ul><li><a href=\"/wiki/Wake-sleep_algorithm\" title=\"Wake-sleep algorithm\">Wake-sleep algorithm</a></li>\n",
      "<li><a href=\"/wiki/Weighted_majority_algorithm_(machine_learning)\" title=\"Weighted majority algorithm (machine learning)\">Weighted majority algorithm (machine learning)</a></li></ul></div><div class=\"mw-category-group\"><h3>Z</h3>\n",
      "<ul><li><a href=\"/wiki/Zero-shot_learning\" title=\"Zero-shot learning\">Zero-shot learning</a></li></ul></div></div></div>\n",
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      "<div class=\"catlinks\" data-mw=\"interface\" id=\"catlinks\"><div class=\"mw-normal-catlinks\" id=\"mw-normal-catlinks\"><a href=\"/wiki/Help:Category\" title=\"Help:Category\">Categories</a>: <ul><li><a href=\"/wiki/Category:Machine_learning\" title=\"Category:Machine learning\">Machine learning</a></li><li><a href=\"/wiki/Category:Algorithms\" title=\"Category:Algorithms\">Algorithms</a></li></ul></div><div class=\"mw-hidden-catlinks mw-hidden-cats-ns-shown\" id=\"mw-hidden-catlinks\">Hidden categories: <ul><li><a href=\"/wiki/Category:Commons_category_link_is_on_Wikidata\" title=\"Category:Commons category link is on Wikidata\">Commons category link is on Wikidata</a></li></ul></div></div>\n",
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     ]
    }
   ],
   "source": [
    "print(soup_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
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      "<span class=\"player-bar-source player-bar-source--spotify\">Playing via Spotify</span>\n",
      "<span class=\"player-bar-source player-bar-source--youtube\">Playing via YouTube</span>\n",
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      "<h3>Scrobble from Spotify?</h3>\n",
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      "<p>\n",
      "<a class=\"btn-primary js-setup-spotify-scrobbling\" href=\"/settings/applications\">\n",
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      "<h1 class=\"header-title\"><a href=\"/tag/black+metal\">Black metal music</a></h1>\n",
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      "<li class=\"navlist-item secondary-nav-item secondary-nav-item--overview\"><a class=\"secondary-nav-item-link\" href=\"/tag/black+metal\">\n",
      "                            Overview\n",
      "                            \n",
      "                        </a></li><li class=\"navlist-item secondary-nav-item secondary-nav-item--artists\"><a class=\"secondary-nav-item-link secondary-nav-item-link--active\" href=\"/tag/black+metal/artists\">\n",
      "                            Artists\n",
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      "                                <span class=\"sr-only\">(current section)</span>\n",
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      "                            Shouts\n",
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      "<div class=\"container content-top-lower\">\n",
      "<h1 class=\"content-top-header\">\n",
      "                            \n",
      "    Black Metal Artists\n",
      "\n",
      "                        </h1>\n",
      "</div>\n",
      "</div>\n",
      "</div>\n",
      "<div class=\"container page-content\">\n",
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      "<section class=\"grid-items-section\">\n",
      "<ol class=\"big-artist-list\">\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Burzum\" itemprop=\"url\">Burzum</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Burzum\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/37aa1ab6ce5f448881bcf54dfaef0dc3.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Burzum\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/37aa1ab6ce5f448881bcf54dfaef0dc3.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                385,325 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Burzum is a Norwegian music project founded by Varg Vikernes in 1991. Although Burzum never played live performances, it became a part of the…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Burzum\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" data-ad-container=\"\">\n",
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      "<p class=\"mpu-subscription-upsell mpu-subscription-upsell--mobile_pos_1\">\n",
      "                Don't want to see ads? <a data-analytics-action=\"SubscribeClick\" data-analytics-label=\"AdUpsellCTA\" href=\"/pro\">Upgrade Now</a>\n",
      "</p>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Immortal\" itemprop=\"url\">Immortal</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Immortal\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/a607729aa86d4e0e81e0a53213e2564f.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Immortal\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/a607729aa86d4e0e81e0a53213e2564f.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                275,170 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Immortal is a black metal band from Bergen, Norway, founded in 1991 by frontman and guitarist Abbath Doom Occulta (Olve Eikemo) and guitarist…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Immortal\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Dimmu+Borgir\" itemprop=\"url\">Dimmu Borgir</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Dimmu Borgir\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/09595e2955f0d06370fa3f237f102fa2.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Dimmu Borgir\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/09595e2955f0d06370fa3f237f102fa2.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                514,766 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Dimmu Borgir is a symphonic black metal band from Oslo, Norway, formed in 1993. The name is derived from Dimmuborgir, a volcanic formation in Iceland,…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Dimmu+Borgir\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Darkthrone\" itemprop=\"url\">Darkthrone</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Darkthrone\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/7c9cea72770146ada56fe7633d3bd744.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Darkthrone\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/7c9cea72770146ada56fe7633d3bd744.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                262,447 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Darkthrone is an influential Norwegian black metal band. They formed in 1986 as a death metal group in Kolbotn, Norway. After embracing the black…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Darkthrone\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Satyricon\" itemprop=\"url\">Satyricon</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Satyricon\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/e2effd7535ea7c5f9375b589700803bc.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Satyricon\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/e2effd7535ea7c5f9375b589700803bc.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                293,928 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Satyricon is an influential Norwegian black metal band, whose line-up over the years has constantly revolved around a stable core of…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Satyricon\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" data-ad-container=\"\">\n",
      "<div class=\"mobile_pos_2 lastfm-ad-mobile\" data-ad=\"mobile_pos_2\" data-ads-placement='{\"id\": \"mobile_pos_2\", \"sizes\": [[300, 250]], \"is_mobile\": true, \"targeting\": {\"pos\": \"2\"}, \"adblock_message\": true}' id=\"mobile_pos_2\"></div>\n",
      "<p class=\"mpu-subscription-upsell mpu-subscription-upsell--mobile_pos_2\">\n",
      "                Don't want to see ads? <a data-analytics-action=\"SubscribeClick\" data-analytics-label=\"AdUpsellCTA\" href=\"/pro\">Upgrade Now</a>\n",
      "</p>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Mayhem\" itemprop=\"url\">Mayhem</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Mayhem\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/aff9721ad61941d4b788e85c13d0d035.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Mayhem\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/aff9721ad61941d4b788e85c13d0d035.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                313,149 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>There are multiple artists named Mayhem;</p><p>1) Mayhem is a Norwegian black metal band formed in 1984 in Oslo, Norway. They were one of the founders…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Mayhem\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Emperor\" itemprop=\"url\">Emperor</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Emperor\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/5a872361d06cbaa9f7603dbebc5a077a.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Emperor\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/5a872361d06cbaa9f7603dbebc5a077a.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                281,081 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>There are multiple artists with this name:</p><p>1) Emperor is a Norwegian black metal band formed in 1991, regarded as highly influential by critics and…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Emperor\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Bathory\" itemprop=\"url\">Bathory</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Bathory\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/830af597977b42cca97bd86698be188d.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Bathory\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/830af597977b42cca97bd86698be188d.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                248,454 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Bathory was a Swedish black metal/viking metal/thrash metal band formed in Vällingby in 1983 and named after the infamous Hungarian countess…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Bathory\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Behemoth\" itemprop=\"url\">Behemoth</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Behemoth\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/ae854eaee8eb270a867b89cb5ea791ab.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Behemoth\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/ae854eaee8eb270a867b89cb5ea791ab.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                426,744 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>There are multiple artists with this name:</p><p>1) Behemoth is a Polish blackened death metal band from Gdańsk. They are considered to have played an…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Behemoth\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Cradle+of+Filth\" itemprop=\"url\">Cradle of Filth</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Cradle of Filth\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/249a0caa769183a45deca5dbe86b9425.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Cradle of Filth\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/249a0caa769183a45deca5dbe86b9425.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                613,848 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Cradle of Filth are an English extreme metal band, formed in Suffolk in 1991. The band's musical style evolved originally from black metal to a…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Cradle+of+Filth\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Marduk\" itemprop=\"url\">Marduk</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Marduk\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/8fabdf873a9047139c9d7ebec110f08f.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Marduk\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/8fabdf873a9047139c9d7ebec110f08f.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                196,945 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Marduk is a Swedish black metal band. Marduk's predominant lyrical topics are Satanism, anti-Christianity, death, war and the Third Reich. The latter…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Marduk\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Gorgoroth\" itemprop=\"url\">Gorgoroth</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Gorgoroth\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/cec3bd611efa7bc99f2f4c9da24c1541.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Gorgoroth\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/cec3bd611efa7bc99f2f4c9da24c1541.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                194,243 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Gorgoroth, meaning “land of great despair”, is a black metal band from Sunnfjord, meaning \"healthy fjord\", named after the dark plains of Mordor…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Gorgoroth\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Ulver\" itemprop=\"url\">Ulver</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Ulver\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/7304f66fd7711ed250068732226af580.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Ulver\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/7304f66fd7711ed250068732226af580.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                395,794 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Ulver (Norwegian for “wolves”) is a multi-disciplinary musical group founded in Oslo, Norway and currently based in London, England. The band was…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Ulver\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/1349\" itemprop=\"url\">1349</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for 1349\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/f78de8975668409a94c00893da6e86f6.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for 1349\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/f78de8975668409a94c00893da6e86f6.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                137,692 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>1349 is a Norwegian black metal band from Oslo, Norway, formed in 1997. The band's name comes from the year the Black Death reached Norway. </p><p>In…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/1349\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Enslaved\" itemprop=\"url\">Enslaved</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Enslaved\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/fdf4b47ce86095dec711419203b82267.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Enslaved\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/fdf4b47ce86095dec711419203b82267.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                207,477 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Enslaved is a metal band from Bergen, Norway. Their style has changed over the years. Originally being an old-school black metal/viking metal act,…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Enslaved\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Dark+Funeral\" itemprop=\"url\">Dark Funeral</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Dark Funeral\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/54ddab74d1a34e86aed95a1fdea89b50.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Dark Funeral\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/54ddab74d1a34e86aed95a1fdea89b50.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                170,600 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Dark Funeral is a Swedish black metal band from Stockholm, founded by guitarists Blackmoon and Lord Ahriman in 1993. They emerged during the…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Dark+Funeral\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Carpathian+Forest\" itemprop=\"url\">Carpathian Forest</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Carpathian Forest\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/a62777c2118142d0853e599358927842.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Carpathian Forest\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/a62777c2118142d0853e599358927842.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                150,096 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Carpathian Forest (previously known as \"The Childmolesters\", then \"Enthrone\") emerged with the demo tape \"Bloodlust and Perversion\" in 1992. In…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Carpathian+Forest\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Venom\" itemprop=\"url\">Venom</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Venom\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/29ef10728f37438cc310d7ca2cc29e01.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Venom\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/29ef10728f37438cc310d7ca2cc29e01.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                329,137 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Venom are an English metal band formed in 1978 in Newcastle upon Tyne. Coming to prominence towards the end of the new wave of British heavy…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Venom\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Anaal+Nathrakh\" itemprop=\"url\">Anaal Nathrakh</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Anaal Nathrakh\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/7bfdfc820f6d317501a2cd6b15f3f6a9.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Anaal Nathrakh\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/7bfdfc820f6d317501a2cd6b15f3f6a9.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                143,345 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Anaal Nathrakh was created for one purpose - to be the soundtrack for Armageddon, the audial essence of evil, hatred and violence, the true…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Anaal+Nathrakh\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Summoning\" itemprop=\"url\">Summoning</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Summoning\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/860c7ca9a9fd49ed1b82d4c5f60c2a70.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Summoning\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/860c7ca9a9fd49ed1b82d4c5f60c2a70.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                143,814 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>Summoning is an Austrian black metal band based in Vienna. Since their formation in 1993 they have released eight albums and two EPs on Napalm…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Summoning\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "<li class=\"big-artist-list-wrap\" itemscope=\"\" itemtype=\"http://schema.org/MusicGroup\">\n",
      "<div class=\"big-artist-list-item js-link-block link-block\">\n",
      "<h3 class=\"big-artist-list-title\" itemprop=\"name\"><a class=\"link-block-target\" href=\"/music/Dissection\" itemprop=\"url\">Dissection</a></h3>\n",
      "<span class=\"avatar big-artist-list-avatar-mobile\">\n",
      "<img alt=\"Avatar for Dissection\" src=\"https://lastfm.freetls.fastly.net/i/u/avatar70s/624f53502a4247229e66fc39aea4c2be.jpg\"/>\n",
      "</span>\n",
      "<span class=\"avatar big-artist-list-avatar-desktop\">\n",
      "<img alt=\"Avatar for Dissection\" src=\"https://lastfm.freetls.fastly.net/i/u/270x205/624f53502a4247229e66fc39aea4c2be.jpg\"/>\n",
      "</span>\n",
      "<p class=\"big-artist-list-listeners\">\n",
      "        \n",
      "            \n",
      "                190,223 <span class=\"stat-name\">listeners</span>\n",
      "</p>\n",
      "<div class=\"big-artist-list-bio\" itemprop=\"description\">\n",
      "<p>There are multiple bands called Dissection:</p><p>1) Dissection was Melodic Death Metal/Black Metal later (2006-Reinkaos) Conceptual Melodic Death/Thrash…</p>\n",
      "</div>\n",
      "<a aria-hidden=\"true\" class=\"js-link-block-cover-link link-block-cover-link\" href=\"/music/Dissection\" tabindex=\"-1\"></a>\n",
      "</div>\n",
      "</li>\n",
      "</ol>\n",
      "</section>\n",
      "<nav class=\"pagination\">\n",
      "<h2 class=\"sr-only\">Pagination</h2>\n",
      "<ul class=\"pagination-list\">\n",
      "<li aria-current=\"page\" class=\"pagination-page\">\n",
      "<span>1</span>\n",
      "</li>\n",
      "<li class=\"pagination-page\">\n",
      "<a href=\"?page=2\">2</a>\n",
      "</li>\n",
      "<li class=\"pagination-page\">\n",
      "<a href=\"?page=3\">3</a>\n",
      "</li>\n",
      "<li class=\"pagination-page\">\n",
      "<a href=\"?page=4\">4</a>\n",
      "</li>\n",
      "<li class=\"pagination-page pagination-page--hide-mobile\">\n",
      "<a href=\"?page=5\">5</a>\n",
      "</li>\n",
      "<li class=\"pagination-page pagination-page--hide-mobile\">\n",
      "<a href=\"?page=6\">6</a>\n",
      "</li>\n",
      "<li class=\"pagination-page pagination-page--ellipsis\">\n",
      "<span>…</span>\n",
      "</li>\n",
      "<li class=\"pagination-page\">\n",
      "<a href=\"?page=48\">48</a>\n",
      "</li>\n",
      "<li class=\"pagination-next\" data-pagination-next-link=\"\">\n",
      "<a href=\"?page=2\">Next</a>\n",
      "</li>\n",
      "</ul>\n",
      "</nav>\n",
      "</div>\n",
      "<div class=\"col-sidebar\">\n",
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      "<p class=\"mpu-subscription-upsell mpu-subscription-upsell--mpu\">\n",
      "                Don't want to see ads? <a data-analytics-action=\"SubscribeClick\" data-analytics-label=\"AdUpsellCTA\" href=\"/pro\">Upgrade Now</a>\n",
      "</p>\n",
      "<div class=\"tonefuze tonefuze--desktop hidden-xs\" data-require=\"ads/tonefuze\" data-tonefuze-artist=\"Frank Ocean\" data-tonefuze-id=\"100000054\" id=\"tonefuze-desktop-3db06544-ef80-48bc-93db-1736f5bad113\"></div>\n",
      "<div class=\"tonefuze tonefuze--mobile visible-xs\" data-require=\"ads/tonefuze\" data-tonefuze-artist=\"Radiohead\" data-tonefuze-id=\"100000055\" id=\"tonefuze-mobile-1edd47c3-7f79-4553-a4c7-de4fe82641a4\"></div>\n",
      "<section class=\"share-desktop\">\n",
      "<h2 class=\"share-heading\">\n",
      "        \n",
      "            Share this tag:\n",
      "        \n",
      "    </h2>\n",
      "<ul class=\"share-options\">\n",
      "<li class=\"share-option\">\n",
      "<a class=\"share-facebook\" data-analytics-social=\"2\" href=\"https://www.facebook.com/sharer/sharer.php?ref=facebooklink&amp;u=https://www.last.fm/tag/black%2Bmetal/artists?page=1\" target=\"_blank\">\n",
      "                Facebook\n",
      "            </a>\n",
      "</li>\n",
      "<li class=\"share-option\">\n",
      "<a class=\"share-twitter\" data-analytics-social=\"8\" href=\"https://twitter.com/intent/tweet?url=https://www.last.fm/tag/black%2Bmetal/artists?page=1&amp;via=lastfm&amp;lang=en&amp;text=Check+out+black+metal\" target=\"_blank\">\n",
      "                Twitter\n",
      "            </a>\n",
      "</li>\n",
      "</ul>\n",
      "</section>\n",
      "<div class=\"mobile_pos_3 lastfm-ad-mobile\" data-ad=\"mobile_pos_3\" data-ads-placement='{\"id\": \"mobile_pos_3\", \"sizes\": [[300, 250], [320, 50]], \"is_mobile\": true, \"targeting\": {\"pos\": \"3\"}}' id=\"mobile_pos_3\"></div>\n",
      "<div class=\"related-content\" data-require=\"ads/taboola\" data-taboola-layout=\"None\" id=\"taboola-right-rail-thumbnails\"></div>\n",
      "<div class=\"mobile_pos_4 lastfm-ad-mobile\" data-ad=\"mobile_pos_4\" data-ads-placement='{\"id\": \"mobile_pos_4\", \"sizes\": [[300, 250], [320, 50]], \"is_mobile\": true, \"targeting\": {\"pos\": \"4\"}}' id=\"mobile_pos_4\"></div>\n",
      "<div class=\"related-ads\">\n",
      "<div data-searchlinks-desktop=\"\"></div>\n",
      "</div>\n",
      "<div class=\"mpu_bottom lastfm-ad\" data-ad=\"mpu_bottom\" data-ads-placement='{\"id\": \"mpu_bottom\", \"sizes\": [[300, 250]], \"targeting\": {\"pos\": \"bottom\"}}' id=\"mpu_bottom\"></div>\n",
      "</div>\n",
      "</div>\n",
      "</div>\n",
      "<div class=\"container\">\n",
      "<div class=\"leader_bottom lastfm-ad\" data-ad=\"leader_bottom\" data-ads-placement='{\"id\": \"leader_bottom\", \"sizes\": [[728, 90], [970, 66], [728, 92]], \"targeting\": {\"pos\": \"bottom\"}}' id=\"leader_bottom\"></div>\n",
      "</div>\n",
      "<div class=\"container page-content visible-xs\">\n",
      "<div class=\"related-ads\">\n",
      "<div data-searchlinks-mobile=\"\"></div>\n",
      "</div>\n",
      "</div>\n",
      "</div>\n",
      "<div id=\"qa-page-load\"></div>\n",
      "<div class=\"lazy-features-footer\" data-lazy-load-content=\"/features/partial/footer\" data-lazy-load-inverted-bg=\"\" data-lazy-load-when-on-screen=\"\" data-require=\"components/lazy-load-v3\"></div>\n",
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      "<link href=\"https://www.last.fm\" itemprop=\"url\"/>\n",
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      "</ul>\n",
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      "<div class=\"footer-top-col\">\n",
      "<h2 class=\"footer-heading\">Goodies</h2>\n",
      "<ul class=\"footer-links\">\n",
      "<li><a href=\"/about/trackmymusic\">Download Scrobbler</a></li>\n",
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      "<h2 class=\"footer-heading\">Account</h2>\n",
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      "</ul>\n",
      "</div>\n",
      "<div class=\"footer-top-col\">\n",
      "<h2 class=\"footer-heading\">Follow Us</h2>\n",
      "<ul class=\"footer-links\">\n",
      "<li><a href=\"https://www.facebook.com/lastfm\" itemprop=\"sameAs\" target=\"_blank\">Facebook</a></li>\n",
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      "<li><a href=\"https://www.youtube.com/user/lastfm\" itemprop=\"sameAs\" target=\"_blank\">YouTube</a></li>\n",
      "</ul>\n",
      "</div>\n",
      "</div>\n",
      "</div>\n",
      "</div>\n",
      "<div class=\"footer-bottom\">\n",
      "<div class=\"container\">\n",
      "<div class=\"row\">\n",
      "<div class=\"col-xs-12 col-sm-9\">\n",
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      "<li class=\"footer-language footer-language--active\">\n",
      "<strong>English</strong><span class=\"sr-only\">current language</span>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"de\"/>\n",
      "<button class=\"mimic-link\" name=\"de\" type=\"submit\">Deutsch</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"es\"/>\n",
      "<button class=\"mimic-link\" name=\"es\" type=\"submit\">Español</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"fr\"/>\n",
      "<button class=\"mimic-link\" name=\"fr\" type=\"submit\">Français</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"it\"/>\n",
      "<button class=\"mimic-link\" name=\"it\" type=\"submit\">Italiano</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"ja\"/>\n",
      "<button class=\"mimic-link\" name=\"ja\" type=\"submit\">日本語</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"pl\"/>\n",
      "<button class=\"mimic-link\" name=\"pl\" type=\"submit\">Polski</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"pt\"/>\n",
      "<button class=\"mimic-link\" name=\"pt\" type=\"submit\">Português</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"ru\"/>\n",
      "<button class=\"mimic-link\" name=\"ru\" type=\"submit\">Русский</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"sv\"/>\n",
      "<button class=\"mimic-link\" name=\"sv\" type=\"submit\">Svenska</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"tr\"/>\n",
      "<button class=\"mimic-link\" name=\"tr\" type=\"submit\">Türkçe</button>\n",
      "</form>\n",
      "</li>\n",
      "<li class=\"footer-language\">\n",
      "<form action=\"/i18n/setlang/\" class=\"footer-language-form\" method=\"post\">\n",
      "<input name=\"csrfmiddlewaretoken\" type=\"hidden\" value=\"PC4C2kTYTmLTLMfXLxKADt5U3vqJPKoG\"/>\n",
      "<input name=\"language\" type=\"hidden\" value=\"zh\"/>\n",
      "<button class=\"mimic-link\" name=\"zh\" type=\"submit\">简体中文</button>\n",
      "</form>\n",
      "</li>\n",
      "</ul>\n",
      "<p class=\"footer-timezone\">\n",
      "                            \n",
      "                                Time zone: <strong>Asia/Novosibirsk</strong>\n",
      "</p>\n",
      "<div class=\"footer-legal\">\n",
      "<ul>\n",
      "<li class=\"footer-legal-item\"><a href=\"http://www.cbsinteractive.com/\" rel=\"nofollow\">CBS Interactive</a> © 2020 <span itemprop=\"name\">Last.fm</span> Ltd. All rights reserved</li>\n",
      "<li class=\"footer-legal-item\"><a href=\"/legal/terms\">Terms of Use</a></li>\n",
      "<li class=\"footer-legal-item\"><a href=\"/legal/privacy\">Privacy Policy</a></li>\n",
      "<li class=\"footer-legal-item\"><a href=\"/legal\">Legal Policies</a></li>\n",
      "<li class=\"footer-legal-item\"><a href=\"https://ca.privacy.cbs\">California Privacy/Info We Collect</a></li>\n",
      "<li class=\"footer-legal-item\"><a href=\"https://ca.privacy.cbs/donotsell\">California Do Not Sell My Info</a></li>\n",
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      "<li class=\"footer-legal-item\"><a href=\"http://www.last.fm\" rel=\"nofollow\">Last.fm Music</a></li>\n",
      "</ul>\n",
      "</div>\n",
      "</div>\n",
      "<div class=\"logo col-xs-12 col-sm-3\">\n",
      "<p>Audioscrobbler</p>\n",
      "<img alt=\"Last.fm 'as' logo\" height=\"20\" src=\"/static/images/footer_logo@2x.49ca51948b0a.png\" width=\"37\"/>\n",
      "</div>\n",
      "</div>\n",
      "</div>\n",
      "</div>\n",
      "<div class=\"feedback-message\">\n",
      "<div class=\"feedback-message-inner\">\n",
      "<p>\n",
      "<span class=\"hidden-xs\">\n",
      "                Let us know what you think of the Last.fm website.\n",
      "            </span>\n",
      "<a href=\"http://cbsi.force.com/CBSi/template_lfmfbk?p_userid=\" target=\"_blank\">Leave feedback</a>\n",
      "</p>\n",
      "</div>\n",
      "</div>\n",
      "</footer>\n",
      "<section class=\"debug-log\">\n",
      "<h2 class=\"debug-log-heading\">API Calls</h2>\n",
      "</section>\n",
      "</div>\n",
      "<div data-ads-beacon=\"\" data-ads-targeting='{\"geo\": \"RU\", \"lang\": \"en\", \"beta\": \"yes\", \"ptype\": \"tag_artists\", \"tag\": \"blackmetal,metal,atmosphericblackmetal,rawblackmetal,deathmetal,depressiveblackmetal,melodicblackmetal\"}' data-ads-unitname=\"/8264/row-lastfm/tag/artists\" data-ads-unitname-mobile=\"/8264/maw-lastfm/tag/artists\"></div>\n",
      "</div>\n",
      "<div data-require=\"player/providers/spotify/spotify-provider\" data-spotify-auth-modal=\"/player/modal/spotify\" data-spotify-inactive-modal=\"/player/modal/spotify-inactive\" data-spotify-initial-device-modal=\"/player/modal/spotify-initial-device-detection\" data-spotify-premium-modal=\"/player/modal/spotify-premium\"></div>\n",
      "<div aria-hidden=\"true\" class=\"video-player\" data-require=\"player/providers/youtube/youtube-provider\" id=\"youtube-video\">\n",
      "<div class=\"video-player-inner-wrap\">\n",
      "<div class=\"video-player-embed js-player\"></div>\n",
      "<button class=\"video-player-close js-close-player\">Close</button>\n",
      "</div>\n",
      "</div>\n",
      "<div class=\"intromercial lastfm-ad\" data-ad=\"intromercial\" data-ads-placement='{\"id\": \"intromercial\", \"out_of_page\": true}' id=\"intromercial\"></div>\n",
      "<div class=\"mobile_intromercial lastfm-ad-mobile\" data-ad=\"mobile_intromercial\" data-ads-placement='{\"id\": \"mobile_intromercial\", \"is_mobile\": true, \"out_of_page\": true}' id=\"mobile_intromercial\"></div>\n",
      "<script class=\"js-modal-template\" type=\"text/template\">\n",
      "            <div class=\"modal-dialog\" role=\"dialog\" aria-labelledby=\"modal-label\">\n",
      "                <div class=\"modal-content\">\n",
      "                    <button class=\"modal-dismiss js-close\">Close</button>\n",
      "                    <h2\n",
      "                        id=\"modal-label\"\n",
      "                        class=\"modal-title js-modal-title\"\n",
      "                    ></h2>\n",
      "                    <div class=\"modal-body js-modal-body\"></div>\n",
      "                </div>\n",
      "            </div>\n",
      "        </script>\n",
      "<script class=\"js-adblock-message-template\" type=\"text/template\">\n",
      "                \n",
      "    \n",
      "    <div><div>\n",
      "        <section class=\"\n",
      "            adblock-message\n",
      "            adblock-message--adfree\n",
      "            link-block\n",
      "            js-link-block\n",
      "        \">\n",
      "            <p class=\"adblock-message-body\">\n",
      "                \n",
      "                    <strong>Browse ad-free</strong> with a premium Last.fm account\n",
      "                \n",
      "            </p>\n",
      "            \n",
      "            <a\n",
      "                class=\"adblock-message-button link-block-target\"\n",
      "                href=\"/pro\"\n",
      "                data-analytics-action=\"SubscribeClick\"\n",
      "                data-analytics-label=\"AdblockMessageAdfree\"\n",
      "            >Find out more</a>\n",
      "            <a\n",
      "                href=\"/pro\"\n",
      "                class=\"\n",
      "                    js-link-block-cover-link\n",
      "                    link-block-cover-link\n",
      "                \"\n",
      "                tabindex=\"-1\"\n",
      "                aria-hidden=\"true\"\n",
      "                data-analytics-action=\"SubscribeClick\"\n",
      "                data-analytics-label=\"AdblockMessageAdfree\"\n",
      "            ></a>\n",
      "        </section>\n",
      "    </div></div>\n",
      "\n",
      "            </script>\n",
      "<script class=\"js-adblock-message-template\" type=\"text/template\">\n",
      "                \n",
      "    \n",
      "    <div><div>\n",
      "        <section class=\"\n",
      "            adblock-message\n",
      "            adblock-message--support\n",
      "            link-block\n",
      "            js-link-block\n",
      "        \">\n",
      "            <p class=\"adblock-message-body\">\n",
      "                \n",
      "                    <strong>Show your support</strong> with a premium Last.fm account\n",
      "                \n",
      "            </p>\n",
      "            \n",
      "            <a\n",
      "                class=\"adblock-message-button link-block-target\"\n",
      "                href=\"/pro\"\n",
      "                data-analytics-action=\"SubscribeClick\"\n",
      "                data-analytics-label=\"AdblockMessageSupport\"\n",
      "            >Find out more</a>\n",
      "            <a\n",
      "                href=\"/pro\"\n",
      "                class=\"\n",
      "                    js-link-block-cover-link\n",
      "                    link-block-cover-link\n",
      "                \"\n",
      "                tabindex=\"-1\"\n",
      "                aria-hidden=\"true\"\n",
      "                data-analytics-action=\"SubscribeClick\"\n",
      "                data-analytics-label=\"AdblockMessageSupport\"\n",
      "            ></a>\n",
      "        </section>\n",
      "    </div></div>\n",
      "\n",
      "            </script>\n",
      "<script class=\"js-adblock-message-template\" type=\"text/template\">\n",
      "                \n",
      "    \n",
      "    <div><div>\n",
      "        <section class=\"\n",
      "            adblock-message\n",
      "            adblock-message--benefits\n",
      "            link-block\n",
      "            js-link-block\n",
      "        \">\n",
      "            <p class=\"adblock-message-body\">\n",
      "                \n",
      "                    <strong>Get exclusive benefits</strong> with a premium Last.fm account\n",
      "                \n",
      "            </p>\n",
      "            \n",
      "            <a\n",
      "                class=\"adblock-message-button link-block-target\"\n",
      "                href=\"/pro\"\n",
      "                data-analytics-action=\"SubscribeClick\"\n",
      "                data-analytics-label=\"AdblockMessageBenefits\"\n",
      "            >Find out more</a>\n",
      "            <a\n",
      "                href=\"/pro\"\n",
      "                class=\"\n",
      "                    js-link-block-cover-link\n",
      "                    link-block-cover-link\n",
      "                \"\n",
      "                tabindex=\"-1\"\n",
      "                aria-hidden=\"true\"\n",
      "                data-analytics-action=\"SubscribeClick\"\n",
      "                data-analytics-label=\"AdblockMessageBenefits\"\n",
      "            ></a>\n",
      "        </section>\n",
      "    </div></div>\n",
      "\n",
      "            </script>\n",
      "<script>\n",
      "            window.lfmDomReadyCallback && window.lfmDomReadyCallback();\n",
      "            window.lfmDomReady = true;\n",
      "        </script>\n",
      "<link as=\"style\" charset=\"utf-8\" data-require=\"shim/rel-preload\" href=\"/static/styles/build/e4d726bdb6.e4d726bdb6f8.css\" media=\"(min-width: 768px)\" rel=\"preload\"/>\n",
      "</body>\n",
      "</html>\n"
     ]
    }
   ],
   "source": [
    "print(soup_3[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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