Computing unconditional probabilities at a children node

[bernardokp]

Hello,

I need to compute probabilities in a BN and for a bn = BayesNet() with 4 nodes (nodes 1 2 and 3 are parents of node 4, all Bernoulli r.v.) I was able to

1. Compute the joint distribution:
   pdf(bn, :node1=>true, :node2=>true, :node3=>true, :node4=>true)

2. Compute the conditional distribution at node 4:
   cpd4(:node1=>false, :node2=>true, :node3=>true)

What I could not do is to compute

Prob(:node4 = false)

I know that mathematically the answer is

\sum_{s\in S_4} Prob(s) Prob (node 4 = false | s) (1)

where S_4 is the set of states associated with node 4 (8 of them, TTT, TTF, TFF, etc).

Is there an automatic way of computing that without actually constructing equation (1)?

Thanks in advance!

Bernardo

[tawheeler]

Hello Bernardo.

I believe that what you are looking for is the inference section of our documentation.

For example:

ϕ = infer(bn, :c, evidence=Assignment(:b=>1))

will give you the posterior distribution over c given that b is 1. Note that this means we marginalize over any variables that are not b or c.

As such, for your example, just run ϕ = infer(bn, :node4, evidence=Assignment()) and see what the resulting factor has for :node4 = False.

[bernardokp]

Hello Tim,

thanks for the reply. I believe you are right, this is the function I need. However, I tried the command you suggested but it did not work. I went back to the tutorial and I understood the inference example.

Maybe it is because I am using StaticCPD? I don't know.
Here is my code (sorry to bother, and thanks in advance):

using DataFrames
using Random
Random.seed!(0) 
using BayesNets
using TikzGraphs 
using TikzPictures

bn = BayesNet()
d2 = StaticCPD(:node2, Bernoulli(0.8))
push!(bn, d2)
d1 = StaticCPD(:node1, Bernoulli(0.8))
push!(bn, d1)
d3 = StaticCPD(:node3, Bernoulli(0.8))
push!(bn, d3)

cpd4 = FunctionalCPD{Bernoulli}(:node4, [:node1, :node2, :node3], a->Bernoulli(
    if (a[:node1] == true  &&  a[:node2] == true && a[:node3] == true)
        1.0 
    elseif  (a[:node1] == true   &&  a[:node2] == true  && a[:node3] == false)
        2/3
    elseif  (a[:node1] == true   &&  a[:node2] == false && a[:node3] == true)
        2/3
    elseif  (a[:node1] == true   &&  a[:node2] == false && a[:node3] == false)
        1/3
    elseif  (a[:node1] == false  &&  a[:node2] == true  && a[:node3] == true)
        2/3
    elseif  (a[:node1] == false  &&  a[:node2] == true  && a[:node3] == false)
        1/3
    elseif  (a[:node1] == false  &&  a[:node2] == false && a[:node3] == true)
        1/3
    else 
        0.0
    end
    )
)
push!(bn, cpd4)
ϕ = infer(bn, :node4, evidence=Assignment())

I got this error:
MethodError: no method matching infer(::BayesNet{CPD}, ::Symbol; evidence=Dict{Symbol, Any}())

Best,
Bernardo

[tawheeler]

Ah, yes, I think the inference methods require using a DiscreteBayesNet - otherwise users could have continuous variables, non-integer variables, etc.

Here is the equivalent approach:

bn = DiscreteBayesNet()
push!(bn, DiscreteCPD(:node2, [0.2,0.8])) # 1 -> false, 2 -> true
push!(bn, DiscreteCPD(:node1, [0.2,0.8])) # 1 -> false, 2 -> true
push!(bn, DiscreteCPD(:node3, [0.2,0.8])) # 1 -> false, 2 -> true
push!(bn, DiscreteCPD(:node4, [:node1, :node2, :node3], [2,2,2],
        [Categorical([1.0,0.0]), # node1, node2, node3 = FFF
         Categorical([2/3, 1/3]), # FFT
         Categorical([2/3,1/3]), # FTF
         Categorical([1/3,2/3]), # FTT
         Categorical([2/3,1/3]), # TFF
         Categorical([1/3,2/3]), # TFT
         Categorical([1/3,2/3]), # TTF
         Categorical([0.0,1.0]), # TTT
        ]))

julia> convert(DataFrame, infer(bn, :node4, evidence=Assignment()))
2×2 DataFrame
 Row  │ node4  potential 
          │ Int64  Float64   
────┼──────────────────
   1     │     1        0.2
   2     │     2        0.8

which tells us that P(:node4 = true) is 0.8

[bernardokp]

Awesome Tim, thanks for saving me again.