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---
title: "Bayesian Unobserved Component Models"
author:
- name: Tomasz Woźniak
url: https://github.com/donotdespair
orcid: 0000-0003-2212-2378
- name: Stephan Berke
url: https://github.com/sb955/
orcid: 0009-0004-8443-1453
- name: Carl Buan
url: https://github.com/CarlB99
- name: Gezhi Chen
url: https://github.com/cora1ine
- name: Ben Gussen
url: https://github.com/bgussen
orcid: 0000-0003-4406-4076
- name: Inhye Kang
url: https://github.com/inhyekang-au
orcid: 0009-0007-8544-4952
- name: Zheyuan Li
url: https://github.com/lzyzero3
- name: Rui Liu
url: https://github.com/rruiliu1005
orcid: 0009-0008-9348-8581
- name: Qingqing Pang
url: https://github.com/comeseemeintheocean
- name: Nhu Pham
url: https://github.com/NhuPham254
- name: Pun Suparattanapinan
url: https://github.com/Punzbd
- name: Yansong Yang
url: https://github.com/russellyang728
- name: Yuqiao Yang
url: https://github.com/JimmyYang917
- name: Adam Wang
url: https://github.com/adamwang15
- name: Yufei Wu
url: https://github.com/yufeimichelle
orcid: 0009-0002-3521-2751
execute:
#echo: false
citation:
issued: 2024-05-01
url: https://donotdespair.github.io/Bayesian-Unobserved-Component-Models/
doi: 10.26188/25814617
bibliography: references.bib
---
> **Abstract.** We present the basics of Bayesian estimation and inference for unobserved component models on the example of a local-level model.
> The range of topics includes the conjugate prior analysis using normal-inverted-gamma 2 distribution and its extensions focusing on hierarchical modelling, conditional heteroskedasticity, and Student-t error terms.
> We scrutinise Bayesian forecasting and sampling from the predictive density.
>
> **Keywords.** Unobserved Component Models, Local-Level Model, State-Space Bayesian Inference, Forecasting, Heteroskedasticity, Hierarchical Modelling, Gibbs Sampler, Simulation Smoother, Precision Sampling
# Unobserved component models
Unobserved Component (UC) models are a popular class of models in macroeconometrics that use the state-space representation for unit-root nonstationary time series.
The simple formulation of the model equations decomposing the series into a non-stationary and stationary component facilitates economic interpretations and good forecasting performance.
# A simple local-level model
The model is set for a univariate time series whose observation at time
$t$ is denoted by $y_t$.
It decomposes the variable into a stochastic trend component, $\tau_t$, and a stationary error component, $\epsilon_t$.
The former follows a Gaussian random walk process with the conditional variance $\sigma_\eta^2$, and the latter is zero-mean normally distributed with the variance $\sigma^2$.
These are expressed as the model equations:
$$
\begin{align}
y_t &= \tau_t + \epsilon_t,\\
\tau_t &= \tau_{t-1} + \eta_t,\\
\epsilon_t &\sim\mathcal{N}\left(0, \sigma^2\right),\\
\eta_t &\sim\mathcal{N}\left(0, \sigma_\eta^2\right),
\end{align}
$$
where the initial condition $\tau_0$ is a parameter of the model.
## Matrix notation for the model
To simplify the notation and the derivations introduce matrix notation for the model. Let $T$ be the available sample size for the variable $y$.
Define a $T$-vector of zeros, $\mathbf{0}_T$, and of ones, $\boldsymbol\imath_T$, the identity matrix of order $T$, $\mathbf{I}_T$, as well as $T\times1$ vectors:
$$
\begin{align}
\mathbf{y} = \begin{bmatrix} y_1\\ \vdots\\ y_T \end{bmatrix},\quad
\boldsymbol\tau = \begin{bmatrix} \tau_1\\ \vdots\\ \tau_T \end{bmatrix},\quad
\boldsymbol\epsilon = \begin{bmatrix} \epsilon_1\\ \vdots\\ \epsilon_T \end{bmatrix},\quad
\boldsymbol\eta = \begin{bmatrix} \eta_1\\ \vdots\\ \eta_T \end{bmatrix},\qquad
\mathbf{i} = \begin{bmatrix} 1\\0\\ \vdots\\ 0 \end{bmatrix},
\end{align}
$$
and a $T\times T$ matrix $\mathbf{H}$ with the elements:
$$
\begin{align}
\mathbf{H} = \begin{bmatrix}
1 & 0 & \cdots & 0 & 0\\
-1 & 1 & \cdots & 0 & 0\\
0 & -1 & \cdots & 0 & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots\\
0 & 0 & \cdots & 1 & 0\\
0 & 0 & \cdots & -1 & 1
\end{bmatrix}.
\end{align}
$$
Then the model can be written in a concise notation as:
$$
\begin{align}
\mathbf{y} &= \mathbf{\tau} + \boldsymbol\epsilon,\\
\mathbf{H}\boldsymbol\tau &= \mathbf{i} \tau_0 + \boldsymbol\eta,\\
\boldsymbol\epsilon &\sim\mathcal{N}\left(\mathbf{0}_T, \sigma^2\mathbf{I}_T\right),\\
\boldsymbol\eta &\sim\mathcal{N}\left(\mathbf{0}_T, \sigma_\eta^2\mathbf{I}_T\right).
\end{align}
$$
## Likelihood function
The model equations imply the predictive density of the data vector
$\mathbf{y}$. To see this, consider the model equation as a linear
transformation of a normal vector $\boldsymbol\epsilon$. Therefore, the data vector follows a multivariate normal distribution given by:
$$
\begin{align}
\mathbf{y}\mid \boldsymbol\tau, \sigma^2 &\sim\mathcal{N}_T\left(\boldsymbol\tau, \sigma^2\mathbf{I}_T\right).
\end{align}
$$
This distribution determines the shape of the likelihood function that
is defined as the sampling data density:
$$
\begin{align}
L(\boldsymbol\tau,\sigma^2|\mathbf{y})\equiv p\left(\mathbf{y}\mid\boldsymbol\tau, \sigma^2 \right).
\end{align}
$$
The likelihood function that for the sake of the estimation of the parameters, and after plugging in data in place of $\mathbf{y}$, is considered a function of parameters $\boldsymbol\tau$ and $\sigma^2$ is given by:
$$
\begin{align}
L(\boldsymbol\tau,\sigma^2|\mathbf{y}) =
(2\pi)^{-\frac{T}{2}}\left(\sigma^2\right)^{-\frac{T}{2}}\exp\left\{-\frac{1}{2}\frac{1}{\sigma^2}(\mathbf{y} - \boldsymbol\tau)'(\mathbf{y} - \boldsymbol\tau)\right\}.
\end{align}
$$
## Prior distributions
The state equation for $\boldsymbol\tau$ can be rewritten as follows:
$$
\begin{gather}
\boldsymbol\tau = \mathbf{H}^{-1} \mathbf{i} \tau_0+\mathbf{H}^{-1}\boldsymbol\eta \\
\boldsymbol\eta \sim \mathcal{N}(\mathbf{0}_T, \sigma_\eta^2 \mathbf{I}_T) \\
\mathbf{H}^{-1} \boldsymbol\eta \sim \mathcal{N}(\mathbf{0}_T, \sigma_\eta^2\left(\mathbf{H}^{\prime} \mathbf{H}\right)^{-1})
\end{gather}
$$
Using the state equation for $\boldsymbol\tau$ above, we can derive the prior distribution of $\boldsymbol\tau$ as:
$$
\begin{align}
\boldsymbol\tau | \tau_0, \sigma_\eta^2 &\sim \mathcal{N}_T(\mathbf{H}^{-1} \mathbf{i} \tau_0, \sigma_\eta^2(\mathbf{H}^{\prime} \mathbf{H})^{-1})
\\ &\propto \exp \left\{-\frac{1}{2} \frac{1}{\sigma_\eta^2}\left(\boldsymbol\tau-\mathbf{H}^{-1} \mathbf{i} \tau_0\right)^{\prime}\left(\mathbf{H}^{\prime} \mathbf{H}\right)\left(\boldsymbol\tau-\mathbf{H}^{-1} \mathbf{i} \tau_0\right)\right\}
\end{align}
$$
Next, the prior distribution of $\tau_0$ is defined as uni-variate normal:
$$
\begin{align}
\tau_0 &\sim \mathcal{N}(\underline{\tau_0}, \underline{V_{\tau_{0}}})
\\ &\propto \exp \left\{-\frac{1}{2}(\tau_0-\underline{\tau_0})^{\prime} \underline{V_{\tau_{0}}}^{-1}(\tau_0-\underline{\tau_0})\right\}
\end{align}
$$
and the prior distribution of $\sigma_\eta^2$ is inverted gamma 2:
$$
\begin{align}
\sigma_\eta^2 &\sim \mathcal{IG}2(\underline{s_\eta}, \underline{v_\eta}) \\
&\propto (\sigma_\eta^2)^{-\frac{\underline{v}+2}{2}} \exp \left\{-\frac{1}{2} \frac{s}{\sigma_\eta^2}\right\}
\end{align}
$$
The prior distribution of $\sigma^2$ is the following:
$$
\begin{align}
\sigma^2 &\sim \mathcal{IG}2(\underline{s}, \underline{v}) \\
&\propto (\sigma^2)^{-\frac{\underline{v}+2}{2}} \exp \left\{-\frac{1}{2} \frac{s}{\sigma^2}\right\}
\end{align}
$$
The Joint prior distribution of the parameters of the model $\boldsymbol\tau$, $\tau_0$, $\sigma^2_{\eta}$, and $\sigma^2$ is then given by:
$$
\begin{aligned}
p(\boldsymbol\tau,\tau_{0},\sigma_{\eta}^{2},\sigma^{2}) = p(\boldsymbol\tau|\tau_{0},\sigma_{\eta}^{2}) \space p(\tau_{0}) \space p(\sigma_{\eta}^{2}) \space p(\sigma^{2})
\end{aligned}
$$
where the individual distributions on the RHS are as specified above.
Finally, given that $\boldsymbol\epsilon$ follows normal distribution,
$$
\begin{align}
\boldsymbol\epsilon\mid\sigma^2 &\sim \mathcal{N}(\mathbf{0}_T, \sigma^2 \mathbf{I}_T)
\end{align}
$$
the prior distribution of $\boldsymbol\epsilon$ given $\sigma^2$ is proportional to:
$$
\begin{align}
\boldsymbol\epsilon | \sigma^2 &\propto \exp \left\{-\frac{1}{2} \frac{1}{\sigma^2} \boldsymbol\epsilon^{\prime}\boldsymbol\epsilon\right\}
\end{align}
$$
# Bayesian estimation
## Derivation of full conditional posterior distributions
Full conditional posterior distribution could be estimated basing on Bayes' theorem and priors shown above. Therefore, the full conditional posterior distribution is proportional to the product of likelihood functions and priors.
$$
\begin{aligned}
p(\tau,\tau_{0},\sigma^{2},\sigma_{\eta}^{2}|\mathbf{y}) & \propto p(\mathbf{y}|\tau,\tau_{0},\sigma^{2},\sigma_{\eta}^{2}) \space p(\tau,\tau_{0},\sigma^{2},\sigma_{\eta}^{2})
\end{aligned}
$$
The joint prior is as specified in the section above.
Two Conditional likelihood functions to derive the **full Conditional posterior distribution** of $\tau,\tau_{0},\sigma^{2},\sigma_{\eta}^{2}$ is as follows:
$$
\begin{align}
L(\boldsymbol\tau,\sigma^{2}|\mathbf{y},\tau_{0},\sigma_{\eta}^{2}) \propto \left( \sigma^{2} \right)^{-\frac{T}{2}} \times\exp\left\{ -\frac{1}{2}\sigma^{-2}\left( \mathbf{y}-\boldsymbol\tau \right)'\left( \mathbf{y}-\boldsymbol\tau \right) \right\}
\end{align}
$$
and
$$
\begin{align}
L(\tau_{0},\sigma_{\eta}^{2}|\mathbf{y},\boldsymbol\tau,\sigma^{2}) \propto \left( \sigma_{\eta}^{2} \right)^{-\frac{T}{2}} \times\exp\left\{-\frac{1}{2}\sigma_{\eta}^{-2}\left(\mathbf{H}\boldsymbol\tau -\mathbf{i}\tau_{0}\right)'\left(\mathbf{H}\boldsymbol\tau-\mathbf{i}\tau_{0}\right)\right\}
\end{align}
$$
The full Conditional posterior distribution of $\boldsymbol\tau$ is derived as follows:
$$
\begin{align}
P(\boldsymbol\tau|\mathbf{y},\tau_{0},\sigma^{2},\sigma_{\eta}^{2})&\propto L(\boldsymbol\tau,\sigma^{2}|\mathbf{y},\tau_{0},\sigma_{\eta}^{2}) \space P(\boldsymbol\tau|\tau_{0},\sigma_{\eta}^{2})\\
\\&\propto \exp\left\{ -\frac{1}{2}\sigma^{-2}\left( \mathbf{y}-\boldsymbol\tau \right)'\left( \mathbf{y}-\boldsymbol\tau \right) \right\}\\
&\qquad\times \exp\left\{ -\frac{1}{2}\sigma_{\eta}^{-2}\left(\boldsymbol\tau-\mathbf{H}^{-1} \mathbf{i}\tau_{0}\right)'\mathbf{H}'\mathbf{H}\left( \boldsymbol\tau-\mathbf{H}^{-1} \mathbf{i}\tau_{0} \right) \right\}
\end{align}
$$
that after completing th squares results in a multivariate normal full conditional posterior distribution:
$$
\begin{align}
P(\boldsymbol\tau &|\mathbf{y},\tau_{0},\sigma^{2},\sigma_{\eta}^{2}) \sim \mathcal{N}(\overline{\boldsymbol\tau},\overline{V_{\boldsymbol\tau}}) \\
\\ \overline{V_{\boldsymbol\tau}} &= \left[ \sigma^{-2}\mathbf{I}_T +\sigma_{\eta}^{-2}\mathbf{H}'\mathbf{H}\right]^{-1}\\
\overline{\boldsymbol\tau} &= \overline{V_{\tau}}\left[\sigma^{-2}\mathbf{y} + \sigma_{\eta}^{-2}\mathbf{H}'\mathbf{i}\tau_{0} \right]
\end{align}
$$
The full conditional posterior distribution of $\tau_{0}$ is the following:
$$
\begin{align}
P(\tau_{0}|\mathbf{y},\boldsymbol\tau,\sigma^{2},\sigma_{\eta}^{2}) &\propto L(\tau_{0}|\mathbf{y},\boldsymbol\tau,\sigma^{2},\sigma_{\eta}^{2}) \space P(\tau_{0}|\sigma_{\eta}^{2}) \\
\\ &\propto \exp\left\{ -\frac{1}{2}\sigma_{\eta}^{-2}\left(\mathbf{H}\boldsymbol\tau -\mathbf{i}\tau_{0}\right)'\left(\mathbf{H}\boldsymbol\tau-\mathbf{i}\tau_{0}\right)\right\}\\
&\times \exp\left\{ -\frac{1}{2}\left(\tau_{0}- \underline{\tau_0}\right)'\underline{V_{\tau_{0}}}^{-1}\left(\tau_{0}- \underline{\tau_0}\right) \right\}
\end{align}
$$
and is given by a uni-variate normal distribution:
$$
\begin{align}
P(\tau_{0} &|\mathbf{y},\boldsymbol\tau,\sigma^{2},\sigma_{\eta}^{2}) \sim N(\overline{\tau_{0}},\overline{V_{\tau_{0}}}) \\
\\ \overline{V_{\tau_{0}}}&=\left[ \sigma_{\eta}^{-2}\mathbf{i}'\mathbf{i} +\underline{V_{\tau_{0}}}^{-1}\right]^{-1} =\left[ \sigma_{\eta}^{-2} +\underline{V_{\tau_{0}}}^{-1}\right]^{-1} \\
\overline{\tau_{0}}&=\overline{V_{\tau_{0}}}\left[ \sigma_{\eta}^{-2} \mathbf{i}'\mathbf{H}\boldsymbol\tau +\underline{V_{\tau_{0}}}^{-1}\underline{\tau_{0}}\right]\\
&= \overline{V_{\tau_{0}}}\left[ \sigma_{\eta}^{-2} \tau_1 +\underline{V_{\tau_{0}}}^{-1}\underline{\tau_{0}}\right]
\end{align}
$$
The full conditional posterior distribution of $\sigma_{\eta}^{2}$ is derived as:
$$
\begin{align}
P(\sigma_{\eta}^{2}|\mathbf{y},\boldsymbol\tau,\tau_{0},\sigma^{2}) &\propto L(\sigma_{\eta}^{2}|\mathbf{y},\boldsymbol\tau,\tau_{0},\sigma^{2}) \space P(\sigma_{\eta}^{2}) \\
\\ &\propto \left( \sigma_{\eta}^{2} \right)^{-\frac{T}{2}} \exp\left\{ -\frac{1}{2}\sigma_{\eta}^{-2}\left(\mathbf{H}\boldsymbol\tau -\mathbf{i}\tau_{0}\right)'\left(\mathbf{H}\boldsymbol\tau -\mathbf{i}\tau_{0}\right)\right\}
\\ &\times \left( \sigma_{\eta}^{2} \right)^{-\frac{\underline{\upsilon}+2}{2}} \exp\left\{ -\frac{1}{2}\frac{\underline{s}}{\sigma_{\eta}^{2}}\right\}
\end{align}
$$
where
$$
\begin{align}
P(\sigma_{\eta}^{2}&|\mathbf{y},\boldsymbol\tau,\tau_{0},\sigma^{2}) \sim \mathcal{IG}2(\overline{s_{\eta}},\overline{\upsilon_{\eta}}) \\
\\ \overline{s_{\eta}}&=\underline{s} + \left(\mathbf{H}\boldsymbol\tau -\mathbf{i}\tau_{0}\right)'\left(\mathbf{H}\boldsymbol\tau -\mathbf{i}\tau_{0}\right) \\
\overline{\upsilon_{\eta}}&= \underline{\upsilon}+T
\end{align}
$$
Finally, the full conditional posterior distribution of $\sigma^{2}$ is the following:
$$
\begin{align}
P(\sigma^{2}|\mathbf{y},\boldsymbol\tau,\tau_{0},\sigma_{\eta}^{2})&=L(\sigma^{2}|\mathbf{y},\boldsymbol\tau,\tau_{0},\sigma_{\eta}^{2}) \space P(\sigma^{2}) \\
\\ &\propto \left( \sigma^{2} \right)^{-\frac{T}{2}} \exp\left\{ -\frac{1}{2}\sigma^{-2}\left(\mathbf{y}-\boldsymbol\tau\right)'\left(\mathbf{y}-\boldsymbol\tau\right)\right\}
\\ &\times \left( \sigma^{2} \right)^{-\frac{\underline{\upsilon}+2}{2}} \exp\left\{ -\frac{1}{2}\frac{\underline{s}}{\sigma^{2}}\right\}
\end{align}
$$
where
$$
\begin{align}
P(\sigma^{2}|&\mathbf{y},\boldsymbol\tau,\tau_{0},\sigma_{\eta}^{2}) \sim \mathcal{IG}2(\overline{s},\overline{\upsilon}) \\
\\ \overline{s}&=\underline{s} + \left(\mathbf{y}-\boldsymbol\tau\right)'\left(\mathbf{y}-\boldsymbol\tau\right) \\
\overline{\upsilon}&= \underline{\upsilon}+T
\end{align}
$$
## Gibbs sampler
A **Gibbs Sampler** can be applied using the following steps:
At each iteration $s$ where $s$ goes from 1 to $S$,
| Step 1. Draw $\tau_0^{(s)}$ from the $N({\tau_{0}}, {V_{\tau_{0}}})$ distribution and collect $\tau_0$.
| Step 2. Draw ${\sigma^2_{\eta}}^{(s)}$ from the $\mathcal{IG}2({S_\eta}, {v_\eta})$ distribution and collect $\sigma_\eta^2$.
| Step 3. Draw ${\sigma^2}^{(s)}$ from the $\mathcal{IG}2(S, v)$ distribution and collect $\sigma^2$.
| Step 4. Draw $\boldsymbol\tau^{(s)}$ from the $\mathcal{N}({\boldsymbol\tau},{V_{\tau}})$ distribution and collect $\boldsymbol\tau$.
```{r Gibbs Sampler}
#| echo: true
# Gibbs sampler for a simple UC model using simulation smoother
############################################################
UC.Gibbs.sampler = function(S, starting.values, priors){
# Initialize the data
aux = starting.values
T = nrow(aux$Y)
i_matrix <- diag(T)
i <- matrix(0, T, 1)
i[1, 1] <- 1
# Posteriors list
posteriors = list(
tau = matrix(NA,T,S),
tau_0 = matrix(NA,1,S),
sigma = matrix(NA,2,S)
)
HH = crossprod(priors$H)
for (s in 1:S){
# Sampling tau_0
###########################
tau_0.v.inv = 1/priors$tau_0.v
V.tau_0.bar = 1/((1/aux$sigma[1]) + tau_0.v.inv )
tau_0.bar = V.tau_0.bar %*% ( (1/aux$sigma[1])*aux$tau[1] + tau_0.v.inv*priors$tau_0 )
tau_0.draw = rnorm(1,as.vector(tau_0.bar),sqrt(V.tau_0.bar))
aux$tau_0 = tau_0.draw
# Sampling sigma
###########################
# sigma of tau (sigma_eta)
sigma.eta.s = as.numeric(priors$sigma.s + crossprod(priors$H%*%aux$tau - i%*%priors$tau_0))
sigma.eta.nu = priors$sigma.nu + T
sigma.eta.draw = sigma.eta.s/rchisq(1,sigma.eta.nu)
# sigma of errors (sigma)
sigma.e.s = as.numeric(priors$sigma.s + crossprod(aux$Y - aux$tau))
sigma.e.nu = priors$sigma.nu + T
sigma.e.draw = sigma.e.s/rchisq(1,sigma.e.nu)
aux$sigma = c(sigma.eta.draw,sigma.e.draw)
# Sampling tau
###########################
V.tau.inv = (1/aux$sigma[2])*i_matrix + (1/aux$sigma[1])*HH
V.tau.inv = 0.5*(V.tau.inv + t(V.tau.inv))
b.tau = (1/aux$sigma[2])*aux$Y + (1/aux$sigma[1])*t(priors$H)%*%i*priors$tau_0
precision.L = t(bandchol(V.tau.inv))
epsilon = rnorm(T)
b.tau.tmp = forwardsolve(precision.L, b.tau)
tau.draw = backsolve(t(precision.L), b.tau.tmp + epsilon)
aux$tau = tau.draw
posteriors$tau[,s] = aux$tau
posteriors$tau_0[,s] = aux$tau_0
posteriors$sigma[,s] = aux$sigma
if (s%%1000==0){cat(" ",s)}
}
output = list(
posterior = posteriors,
last.draw = aux
)
return(output)
}
```
## Simulation smoother and precision sampler
```{r}
#| echo: false
#| message: false
#| warning: false
library(mgcv)
library(microbenchmark)
```
Simulation Smoother is used when drawing state variables in discrete time state-space models from their conditional distribution. This tool greatly improved Bayesian estimation process in conditional linear Gaussian state-space models. The benefit of using this method comes from the fact that one does not rely on an iterative procedure of forward filtering and backward smoothing. Instead, this step is performed all without a loop.
Consider sampling random draws from a multivariate normal distribution presented as:
$$
N_{T}(D^{-1}b,D^{-1})
$$
where: $D$ is a $T \times T$ precision matrix of covariance matrix $\Sigma$, such that $D = \Sigma^{-1}$, and $b$ is an $T\times1$ restriction vector to specify the **Mean** of multivariate normal distribution, where $M = \Sigma b$.
This parameterisation is particularly useful as matrices $D$ and $b$ can be computed easily. Additionally, matrix $D$ that in the simulation smoother for state-space models is a band matrix and often a tridiagonal matrix. The latter case arises for a one-lag autoregressive state-space process. Therefore, the computation time is much shorter if dedicated numerical algorithms are employed. For tridiagnal matrix $D$ in state space model, let $L$ be a lower-triangular matrix obtained by Cholesky decomposition, where $D = LL'$ (Assume that $L^{-1}$ can be compute efficiently).
Therefore, the mean of the normal distribution above can be presented as:
$$
M = D^{-1}b = (LL')^{-1}b = L'^{-1}L^{-1}b
$$
Let a $T \times 1$ vector x contain $T$ independent random draws from a standard normal distribution, $X\sim N_{T} ( 0_{T}, I_{T})$. Therefore, the target distribution, $Y\sim N_{T}(D^{-1}b,D^{-1})$, can be treated as:
$$
Y = D^{-1}b + \sqrt{D^{-1}} X = L'^{-1}L^{-1}b + L'^{-1} X
$$
Therefor a draw form the target distribution, $N_{T}(D^{-1}b,D^{-1})$, it's equivalent to:
$$
L'^{-1}L^{-1}b + L'^{-1}x = L'^{-1}(L^{-1}b + x)
$$
Following algorithm base on solving a linear equation by back-substitution rather than computation of the inverse matrix:
1. **Compute** $L = chol(D)$ such that $D = LL'$
2. **Sample** $x\sim N_{T} ( 0_{T}, I_{T})$
3. **Compute** a draw from the distribution via the affine transformation: $L '\smallsetminus (L \smallsetminus b + x)$
Note: Let $L \smallsetminus b$ denote the unique solution to the triangular system Lx = b obtained by forward (backward) substitution, that is, $L \smallsetminus b = L^{−1}b$.
### Simulation smoother algorithm in R
Compare the computational speed of generating random numbers from a multivariate normal distribution using dedicated numerical algorithms in function `rmvnorm.tridiag.precision` and function `solve` for the matrix inversion ignoring the tridiagonality of the precision matrix in function `rmvnorm.usual`:
```{R Simulation smoother in R comparison}
#| echo: true
#| cache: true
library(mgcv)
rmvnorm.tridiag.precision = function(n, D, b){
N = dim(D)[1]
lead.diag = diag(D)
sub.diag = sdiag(D, -1)
D.chol = trichol(ld = lead.diag, sd=sub.diag)
D.L = diag(D.chol$ld)
sdiag(D.L,-1) = D.chol$sd
x = matrix(rnorm(n*N), ncol=n)
a = forwardsolve(D.L, b)
draw = backsolve(t(D.L), matrix(rep(a,n), ncol=n) + x)
return(draw)
}
# Function of normal method
rmvnorm.usual = function(n, D, b){
N = dim(D)[1]
D.chol = t(chol(D))
variance.chol = solve(D.chol)
x = matrix(rnorm(n*N), ncol=n)
draw = t(variance.chol) %*%
(matrix(rep(variance.chol%*%b,n), ncol=n) + x)
return(draw)
}
# Comparison
set.seed(12345)
T = 300
md = rgamma(T, shape=10, scale=10)
od = rgamma(T-1, shape=10, scale=1)
D = 2*diag(md)
sdiag(D, 1) = -od
sdiag(D, -1) = -od
b = as.matrix(rnorm(T))
microbenchmark(
trid = rmvnorm.tridiag.precision(n=100, D=D, b=b),
usual = rmvnorm.usual(n=100, D=D, b=b),
check = "equal", setup=set.seed(123456)
)
```
Therefore, we could clearly see that the computational time drops substatially due to simulation smoothing method.
## Analytical solution for a joint posterior distribution
Consider a simplified model with a fixed signal-to-noise ratio $c$ given by
\begin{align}
\text{measurement equation} && y &= \tau + \epsilon\\
\text{state equation} && \mathbf{H}\tau &= \eta\\
\text{error term} && \epsilon\mid\tau &\sim \mathcal{N}_T(\mathbf{0}_T, \sigma^2\mathbf{I}_T)\\
\text{innovation} && \eta &\sim \mathcal{N}_T(\mathbf{0}_T, c\sigma^2\mathbf{I}_T)
\end{align}
where $c$ is a constant, and $\tau_0 = 0$.
In this model the state equation for $\tau$ is given by a Gaussian random ralk presented as:
$$\begin{pmatrix}
1 & 0 & \cdots & 0 \\
-1 & 1 & \cdots & 0 \\
\vdots & \ddots & \ddots & \vdots \\
0 & 0 & -1 & 1
\end{pmatrix} \begin{pmatrix} \tau_1 \\ \tau_2 \\ \vdots \\ \tau_T \end{pmatrix} = \begin{pmatrix} \tau_1 \\ \tau_2-\tau_1 \\ \vdots \\ \tau_T - \tau_{T-1} \end{pmatrix} = \eta$$
The parameters to be estimated are: $\mathbf{\color{purple}{\tau, \sigma^2}}$
### Prior distributions
#### Prior distribution for $\tau$
Rewrite the state equation as
$$\tau = H^{-1}\eta$$
Then the prior distribution of $\tau$ is formed using
$$\mathbf{\color{purple}{\eta \sim \mathcal{N}(0_T, c \sigma^2 I_T) \implies H^{-1}\eta \sim \mathcal{N}(0_T, c\sigma^2(H^T H)^{-1})}}$$
since $Var(H^{-1}\eta) = H^{-1}Var(\eta)(H^{-1})^T =c\sigma^2(H^TH)^{-1}$
Then, the prior distribution of $\tau|\sigma^2$ is
$$\tau|\sigma^2 \sim \mathcal{N}(0_T, c\sigma^2(H^TH)^{-1}) \\ \propto (c\sigma^2)^{-\frac{T}{2}}exp\left(-\frac{\tau^TH^T H\tau}{2c\sigma^2}\right)
$$
#### Prior assumptions for $\sigma^2$
$$\sigma^2 \sim \mathcal{IG2}(s_{prior}, \nu_{prior}) \propto (\sigma^2)^\frac{-\nu_{prior}+2}{2}exp\left(-\frac{s_{prior}}{2\sigma^2}\right)$$
### The Joint Posterior Distribution
The likelihood function is constructed using the the measurement equation:
$$y = \tau + \epsilon$$
and the distributional assumption on its error term
$$\epsilon \sim \mathcal{N}(0_T, \sigma^2I_T)$$
which results in the likelihood function:
$$y|\tau, \sigma^2 \sim \mathcal{N}(\tau, \sigma^2I_T)\propto (\sigma^2)^{-\frac{T}{2}}exp\left(-\frac{1}{2\sigma^2}(y-\tau)^T(y-\tau)\right)$$
The likelihood is combined with the joint prior distribution of $\tau$ and $\sigma^2$ to derive the joint posterior distribution of $\tau$ and $\sigma^2$. It is derived as follows:
$$
p(\tau, \sigma^2 |y) = \frac{p(\tau, \sigma^2, y)}{p(y)} \propto p(\tau, \sigma^2, y) = p(y | \tau, \sigma^2)p(\tau, \sigma^2) =p(y | \tau, \sigma^2)p(\tau|\sigma^2)p(\sigma^2)
$$
This expression is proportional to
\begin{align}
&\propto (\sigma^2)^{-\frac{T}{2}}exp\left(-\frac{(y-\tau)^T(y-\tau)}{2\sigma^2}\right) \times
(\sigma^2)^{-\frac{T}{2}}\exp\left(-\frac{\tau^TH^T H\tau}{2c\sigma^2}\right)\\
&\qquad\times(\sigma^2)^{-\frac{\nu_{prior}+2}{2}} \exp(-\frac{s_{prior}}{2\sigma^2})\\
&\propto \exp(-\frac{y^Ty - 2\tau^T y + \tau^T\tau + c^{-1}\tau^TH^T H\tau}{2\sigma^2})\\ &\qquad\times \exp(-\frac{s_{prior}}{2\sigma^2})\times(\sigma^2)^{-\frac{2T+\nu_{prior}+2}{2}}\\
&= \exp(-\frac{\tau^T(c^{-1}H^T H + I_T)\tau - 2\tau^Ty}{2\sigma^2})\\
&\qquad\times\exp(-\frac{y^Ty + s_{prior}}{2\sigma^2})\times(\sigma^2)^{-\frac{2T+\nu_{prior}+2}{2}}
\end{align}
Let $\bar{\Sigma} = (c^{-1}H^T H + I_T)^{-1}$, then
\begin{align}
p(\tau, \sigma^2 |y) &\propto \exp(-\frac{\tau^T\bar{\Sigma}^{-1}\tau - 2\tau^Ty + y^T\bar{\Sigma}y}{2\sigma^2})\exp(-\frac{y^Ty + s_{prior}-y^T\bar{\Sigma}y}{2\sigma^2})\\
&\qquad\times(\sigma^2)^{-\frac{2T+\nu_{prior}+2}{2}}
\end{align}
In the expression above, we recognise the kernel of the $T$-variate normal inverted gamma 2 distribution. Therefore, the joint posterior of the parameters of the model is given by:
\begin{align}
\tau, \Sigma |y &\sim \mathcal{NIG2}(\bar{\tau}, \bar{\Sigma}, \bar{\nu}, \bar{s})\\[2ex]
\bar{\Sigma} &= (c^{-1}H^T H + I_T)^{-1}\\
\bar{\tau} &= \bar{\Sigma}y\\
\bar{\nu} &= 2T+\nu_{prior}\\
\bar{s} &= s_{prior}+y^Ty-y^T\bar{\Sigma}y
\end{align}
The function below implements the sampler from this joint distribution.
```{r joint_sampler}
UC.local.tau.sigma.Gibbs.sampler = function(S, starting.values, priors){
aux = starting.values
T = nrow(aux$Y)
posteriors = list(
tau = matrix(NA,T,S),
sigma = rep(NA,S)
)
for (s in 1:S){
V.tau.bar.inv = priors$c^(-1)*t(aux$H)%*%aux$H + diag(T)
V.tau.bar.inv = 0.5*(V.tau.bar.inv + t(V.tau.bar.inv))
# Sampling sigma
###########################
sigma.s = as.numeric(priors$sigma.s + t(aux$Y)%*%aux$Y - t(aux$tau)%*%V.tau.bar.inv%*%aux$tau)
sigma.nu = priors$sigma.nu + 2*T
sigma.draw= sigma.s/rchisq(1,sigma.nu)
aux$sigma = sigma.draw
# Sampling tau
###########################
b.tau = aux$Y
precision.L = t(bandchol(V.tau.bar.inv))
epsilon = rnorm(T)
b.tau.tmp = forwardsolve(precision.L, b.tau)
tau.draw = backsolve(t(precision.L), b.tau.tmp + epsilon)
aux$tau = tau.draw
posteriors$tau[,s] = aux$tau
posteriors$sigma[,s] = aux$sigma
}
output = list(
posterior = posteriors,
last.draw = aux
)
return(output)
}
```
# Hierarchical modeling
## Estimating gamma error term variance prior scale
To estimate the scale of inverted gamma 2 error term variance, we need to firstly put a prior on the prior scale of and present its full conditional posterior distribution.
The hirercical prior structure is below:
```{=tex}
\begin{align*}
\sigma^2 \mid s &\sim \text{IG2}(s, \nu) \\
s &\sim \mathcal{G}(s_s, a)
\end{align*}
```
The full conditional posterior distribution of the prior scale $s$ is based on the prior $p(\sigma^2 \mid s)$:
\begin{align}
p(\sigma^2 \mid s) \propto s^{\frac{\nu}{2}}(\sigma^2)^{-\frac{\nu+2}{2}} \exp \left( -\frac{1}{2}\frac{s}{\sigma^2} \right)
\end{align}
and that for $s$:
\begin{align}
s \sim \mathcal{G}(s_s, a) \propto s^{a-1} \exp(-\frac{s}{s_s})
\end{align}
which results in the following full conditional posterior distribution:
\begin{align}
p(s \mid \mathbf{y}, \mathbf{\tau}, \sigma^2) &= \mathcal{G}(\bar{s}_s,\bar{a})\\
\bar{s}_s &= \left(s_s^{-1} + (2\sigma^2)^{-1}\right)^{-1}\\
\bar{a} &= a + \frac{\nu}{2}
\end{align}
The code below implements the sampler from the full conditional posterior for $s$
```{r}
sample_s = function(sigma2, s_s, a, nu) {
shape_s = a + nu/2
scale_s = 1/((1/s_s) + 1/(2*sigma2))
s = rgamma(1, shape = shape_s, scale = scale_s)
return(s)
}
```
## Estimating inverted-gamma 2 error term variance prior scale
In this section, we estimate the error term variance prior scale $s$ that follows a Inverted-gamma 2 distribution with scale $\underline{s}$ and shape $\underline{\mu}$: $s\sim\mathcal{IG2}(\underline{s},\underline{\mu})$. The probability density function equal to:
$$
p\left(s\right) = \Gamma(\frac{\underline{\mu}}{2})^{-1} (\frac{\underline{s}}{2})^{\frac{\underline{\mu}}{2}} s^{-\frac{\underline{\mu}+2}{2}}exp\{ -\frac{1}{2}\frac{\underline{s}}{s}\}
$$
As we assume $\sigma^2|s \sim \mathcal{IG2} (s,\underline{\nu})$
$$
p\left(\sigma^2|s\right) = \Gamma(\frac{\underline{\nu}}{2})^{-1} (\frac{s}{2})^{\frac{\underline{\nu}}{2}} \sigma^2{^{-\frac{\underline{\nu}+2}{2}}}exp\{-\frac{1}{2}\frac{s}{\sigma^2}\}
$$
In order to find full conditional posterior of $s$ write out its kernel as:
$$
\begin{aligned}
p(s|\tau, \sigma^2) &\propto L(\tau, \sigma^2|y) \cdot p(\epsilon|\sigma^2) \cdot p(\sigma^2|s) \cdot p(s) \\
&\propto p(\sigma^2|s) \cdot p(s) \\
&\propto (\frac{s}{2})^{\frac{\underline{\nu}}{2}} exp\{ -\frac{1}{2}\frac{s}{\sigma^2}\} \cdot s^{-\frac{\underline{\mu}+2}{2}}exp\{-\frac{1}{2}\frac{\underline{s}}{s}\} \\
&\propto (\frac{s}{2})^{\frac{\underline{\nu}}{2}} \cdot s^{-\frac{\underline{\mu}+2}{2}} \cdot exp \{-\frac{1}{2}(\frac{\underline{s}}{s} + \frac{s}{\sigma^2} )\} \\
&\propto (\frac{1}{2}^{\frac{\underline{\nu}}{2}}) \cdot s^{\frac{\underline{\nu}}{2}} \cdot s^{-\frac{\underline{\mu}+2}{2}} \cdot exp \{-\frac{1}{2}(\frac{\underline{s}}{s} + \frac{s}{\sigma^2} )\} \\
&\propto s^{\frac{\underline{\nu}-\underline{\mu}-2}{2}} exp\{ -\frac{\frac{1}{\sigma^2}s +\frac{s}{\underline{s}}}{2}\}
\end{aligned}
$$
from which we recognize a Generalized inverse Gaussian distribution: $\mathcal{GIG}(\overline{p}, \overline{a}, \overline{b})$ with parameters:
\begin{align}
\overline{p} &= \frac{\underline{\nu} - \underline{\mu}}{2}\\
\overline{a} &= \frac{1}{\sigma^2 }\\
\overline{b} &= \underline{s }
\end{align}
The following script illustrates sampling from the full conditional posterior distribution of $s$ and the function `s.sampler`:
```{r}
s.sampler = function(sigma2, prior){
# sigma2 - the current draw
# prior is a list containing:
# s.prior - a positive scalar
# mu.prior - a scalar
# nu.prior - a scalar
a.bar.s = 1/sigma2
b.bar.s = (prior$nu.prior - prior$mu.prior)/2
p.bar.s = prior$s.prior
s = GIGrvg::rgig(n = 1, lambda = p.bar.s, chi = b.bar.s, psi = a.bar.s)
return(s)
}
```
## Estimating the initial condition prior scale
In the univariate local-level model, the initial state $\tau_0$ and the
variance $s^2$ are key parameters determining the behavior of the
stochastic process. For the initial state $\tau_0$, a prior is specified
as a univariate normal distribution, denoted by
$$\tau_0\mid s^2 \sim \text{N}(\underline\tau_0, s^2)$$
reflecting our belief
about the distribution of $\tau_0$ before observing the data. While, for
the variance $s^2$, which controls the fluctuation of the process, we
use inverse gamma 2 distribution,
$$s^2 \sim \text{IG2}(\underline{s}, \underline{\nu})$$
The full posterior of $s^2$ is derived below:
\begin{align}
p(s^2|y,\sigma,\tau,\tau_0,\sigma_{\eta}, \underline{s}, \underline{\nu}) &\propto
p(y|\tau,\sigma)p(\tau|\tau_0,\sigma_{\eta}^2)p(\tau_0|s^2)p(s^2) \\
&\propto p(\tau_0|s^2)p(s^2|\underline{s}, \underline{\nu}) \\
&= (s^2)^{-\frac{1}{2}}\exp\left\{-\frac{1}{2}(\tau_0-\underline{\tau_0})'(s^2)^{-1}(\tau_0-\underline{\tau_0})\right\}\\
&\qquad\times (s^2)^{-\frac{\underline{\nu}+2}{2}}\exp\left\{-\frac{1}{2}\frac{\underline{s}}{s^2}\right\} \\
&= (s^2)^{-\frac{1+\underline{\nu}+2}{2}}
\exp \left\{-\frac{1}{2}\frac{(\tau_0-\underline{\tau_0})^2+\underline{s}}{s^2}\right\}
\end{align}
Hence, full conditional posterior of $s^2$ in a form of inverse
gamma 2 distribution, where
\begin{align}
s^2\mid \mathbf{y}, \mathbf{\tau}, \tau_0 &\sim\mathcal{IG}2(\overline{s}, \overline{\nu})\\
\overline{\nu} &= 1+\underline{\nu} \\
\overline{s} &= (\tau_0-\underline{\tau_0})^2+\underline{s}
\end{align}
This R code below implements a sampler from the full-conditional posterior distribution of $s^2$:
```{r}
s2_sampler <- function(tau_0, tau_0_prior, s_prior, nu_prior) {
nu_posterior <- nu_prior + 1
s_posterior <- (tau_0 - tau_0_prior)^2 + s_prior
s2 <- s_posterior / rchisq(1, nu_posterior)
return(s2)
}
```
## Student-t prior for the trend component
## Estimating Student-t degrees of freedom parameter
The Student-t distribution is commonly used in statistical modeling to handle data with heavier tails than the normal distribution. An essential parameter of the Student-t distribution is the degrees of freedom $\nu$, which controls the tail heaviness. In this note, we present the Bayesian estimation of the degrees of freedom parameter for a $T$-variate Student-t distribution represented as an inverted gamma 2 scale mixture of normals:
\begin{align}
\mathbf{y} \mid \boldsymbol\tau, \boldsymbol\lambda, \sigma^2 &\sim \mathcal{N}_T(\boldsymbol\tau, \sigma^2diag(\boldsymbol\lambda))\\
\boldsymbol\lambda &= (\lambda_1, \dots, \lambda_T)'\\
\lambda_t \mid \nu &\sim \mathcal{IG2}(\nu - 2, \nu)
\end{align}
Therefore the joint prior for vector $\boldsymbol\lambda$ is proportional to:
$$
\Gamma\left(\frac{\nu}{2}\right)^{-T}\left(\frac{\nu-2}{2}\right)^{\frac{T\nu}{2}}\left(\prod_{t=1}^T\lambda_t\right)^{-\frac{\nu+2}{2}}\exp\left\{-\frac{\nu-2}{2}\sum_{t=1^T}\lambda_t^{-1}\right\}
$$
Assume the following Gamma prior for $\nu$:
$$
p(\nu) \sim \text{Gamma}(a, b) \propto \nu^{a-1}\exp\{-b\nu\}
$$
Therefore the kernel of the full conditional posterior of $\nu$ is defined by:
$$
p(\nu\mid\mathbf{y},\boldsymbol\tau,\boldsymbol\lambda,\sigma^2, \nu) \propto p(,\boldsymbol\lambda\mid\nu)p(\nu)
$$
This expression does not lead to a full conditional posterior distribution of a known typ, so we use the Metropolis-Hastings algorithm to sample from this posterior. We employ a random walk proposal distribution with a truncated normal distribution centred at the current value of $\nu$ with a fixed standard deviation.
```{r}
log_kernel_nu <- function(nu, lambda, a, b) {
T = length(lambda)
log_c = -T * lgamma(nu / 2) + (T * nu / 2) * log((nu - 2) / 2)
log_k = -((nu + 2) / 2) * sum(log(lambda)) - ((nu - 2) / 2) * sum(1 / lambda)
log_nu = (a - 1) * log(nu) - b * nu
return(log_c + log_k + log_nu)
}
sample_nu <- function(nu, lambda, a, b, proposal_sd) {
T = length(lambda)
nu_proposal = RcppTN::rtn(nu, proposal_sd, 0, Inf)
kernel_ratio = exp(log_kernel_nu(nu_proposal, lambda, a, b) - log_kernel_nu(nu, lambda, a, b))
candidate_ratio = RcppTN::dtn(nu_proposal, nu, proposal_sd, 0, Inf) / RcppTN::dtn(nu, nu_proposal, proposal_sd, 0, Inf)
accept_ratio = kernel_ratio * candidate_ratio
if (runif(1) < accept_ratio) {
nu <- nu_proposal
}
return(nu)
}
```
## Laplace prior for the trend component
To apply the Laplace prior for our trend component, we have $\lambda_t$ following an exponential distribution with mean alpha, where $\lambda_t \sim exp({\frac{1}{\alpha}})$, with the prior of the trend component is $\tau|\eta,\lambda \sim \mathcal{N}(H^{-1}i\tau_0, \sigma_\eta^2 H^{-1}\Omega(H^{-1})')$, then we have the marginal distribution of $\tau$ is Laplace distribution.
The model is where $\Omega=diag(\lambda_1,\lambda_2,...,\lambda_t)$, each lambda independently drawn from an exponential distribution.
Due to the change prior distribution of the trend component, now we have different full posterior distributions for parameters and in this note we only focus on the derivation for $\tau$ and $\lambda_t$.
The change of the prior distribution for $\tau$ gives the followings:
$$\tau =y - \epsilon$$
$$\tau = H^{-1}i\tau_0 + H^{-1} \eta $$
$$\tau|\eta,\lambda \sim \mathcal{N}(H^{-1}i\tau_0, \sigma_\eta^2 H^{-1}\Omega(H^{-1})') $$
$$\lambda_t \sim exp({\frac{1}{\alpha}})$$
### Full-conditional posterior distribution for $\tau|\tau_0,\sigma^2,\Omega$
**Conditional Likelihood:**
$$\tau =y - \epsilon$$
$$\epsilon \sim \mathcal{N}(0_T, \sigma^2I_T)$$
$$L(\tau|y,\sigma^2) = exp({-\frac{1}{2}\frac{1}{\sigma^2}(\mathbf{y} - \boldsymbol\tau)'(\mathbf{y} - \boldsymbol\tau)})$$
The **prior distribution** of $\tau$is:
$$p(\tau|\tau_0,\sigma_\eta^2,\Omega) \propto exp(-\frac{1}{2}\frac{1}{\sigma_\eta^2}(\tau - H^{-1}i\tau_0)'H'\Omega^{-1}H(\tau - H^{-1}i\tau_0))$$
**Full-conditional posterior distribution** of ${\tau}|y,\sigma_\eta^2,\Omega$:
\begin{align}
P(\tau|y,\tau_0,\sigma^2,\Omega) &\propto L(\tau|y,\sigma^2)\times p(\tau|\tau_0,\sigma_\eta^2,\Omega)\\
&\propto exp({-\frac{1}{2}}\sigma^{-2}\ \times (\tau'\tau -2y'\tau+y'y))\\
&\times exp({-\frac{1}{2}\sigma_\eta^{-2}\ (\tau'H'\Omega^{-1}H\tau - 2\tau'H'\Omega^{-1}H(H^{-1}i\tau_0) +(H^{-1}i\tau_0)'H\Omega^{-1}H(H^{-1}i\tau_0)}\\
&\propto exp({-\frac{1}{2}}\sigma^{-2}\sigma_\eta^{-2}[\tau'(\sigma_\eta^2+\sigma^2H'\Omega^{-1}H)\tau-2\tau'(\sigma_\eta^2y+\sigma^2H'\Omega^{-1}H(H^{-1}i\tau_0))])\\
&= exp({-\frac{1}{2}}(\tau'(\sigma^{-2}+\sigma_\eta^{-2}H'\Omega^{-1}H\tau-2\tau'(\sigma^{-2}y+\sigma_\eta^{-2}H'\Omega^{-1}H(H^{-1}i\tau_0))
\end{align}
which results in
\begin{align}
\tau|y,\tau_0,\sigma^2,\Omega&\sim N(\bar{V_\tau},\bar{\tau})\\[1ex]
\bar{V_\tau} &= (\sigma^{-2}+\sigma_\eta^{-2}H'\Omega^{-1}H)^{-1}\\
\bar{\tau} &=\bar{V_\tau} (\sigma^{-2}y+\sigma_\eta^{-2}H'\Omega^{-1}i\tau_0)
\end{align}
### Full-conditional posterior distribution for $\lambda_t|y,\tau,\sigma_\eta^2$
**Conditional Likelihood:**
Define: $H\tau - i\tau_o=\eta$
\begin{align}
L(\lambda_t|y,\tau_0,\tau,\sigma_\eta^2)&\propto (\prod^{T}_{i = 1} \lambda_t^{-\frac{1}{2}}) exp({-\frac{1}{2}\frac{1}{\sigma^2_\eta}(i\tau_0-H\tau)'\Omega^{-1}(i\tau_0-H\tau)})\\
&= (\prod^{T}_{i = 1} \lambda_t^{-\frac{1}{2}}exp({-\frac{1}{2}\frac{1}{\sigma^2_\eta}\frac{1}{\lambda_t}\eta_t'\eta_t}))
\end{align}
The **prior distribution** of $\lambda_t$ is:
$$
P(\lambda_t) ={\frac{1}{\alpha}}exp({{-\frac{1}{\alpha}}\lambda_t)})
$$
**Full-conditional posterior distribution** of $\lambda_t$is:
\begin{align}
P(\lambda_t|y,\tau,\sigma_\eta^2) &\propto \lambda_t^{-\frac{1}{2}+1-1}exp(-\frac{1}{2}({\frac{\eta_t'\sigma_\eta^{-2}\eta_t}{\lambda_t}+{\frac{2}{\alpha}}\lambda_t)}) \\
\end{align}
The above expression can be rearranged in the form of a Generalized inverse Gaussian distribution kernel as follows:
\begin{align}
\lambda_t|Y,A,\Sigma &\sim GIG(a,b,p) \\
\\
a &=\frac{2}{\alpha} \\
b &= \eta_t'\sigma_\eta^{-2}\eta_t \\
p &= -\frac{1}{2}+1
\end{align}
### R Function for Gibbs Sampler
```{r}
UC.AR.Gibbs.sampler = function(S, starting.values, priors){
aux = starting.values
p = length(aux$alpha)
T = nrow(aux$Y)
i <- matrix(0, T, 1)
i[1, 1] <- 1
posteriors = list(
tau = matrix(NA,T,S),
epsilon = matrix(NA,T,S),
tau_0 = matrix(NA,1,S),
sigma = matrix(NA,2,S)
)
alpha <- 2
lambda.0 <- rexp(T, rate = 1/alpha)
lambda.priors = list(alpha = 2)
lambda.posterior.draws = array(NA,c(T,S+1))
for (s in 1:S){
if (s == 1) {
lambda.s = lambda.0
} else {
lambda.s = lambda.posterior.draws[,s]
}
Omega = (diag(lambda.s))
Omega.inv = diag(1/lambda.s)
# Sampling tau
###########################
V.tau.inv = (1/aux$sigma[2]) + (1/aux$sigma[1])*t(H) %*% Omega.inv %*% H
V.tau.inv = 0.5*(V.tau.inv + t(V.tau.inv))
b.tau = (1/aux$sigma[2])%*%aux$Y + (1/aux$sigma[1])*crossprod(H, Omega.inv%*%i%*%priors$tau_0)
precision.L = t(bandchol(V.tau.inv))
epsilon = rnorm(T)
b.tau.tmp = forwardsolve(precision.L, b.tau)
tau.draw = backsolve(t(precision.L), b.tau.tmp + epsilon)
aux$tau = tau.draw
# Sampling tau_0
###########################
tau_0.v.inv = diag(1/priors$tau_0.v)
V.tau_0.bar = 1/((1/aux$sigma[1])*crossprod(i,Omega.inv%*%i) + tau_0.v.inv)
tau_0.bar = V.tau_0.bar %*% ( (1/aux$sigma[1])%*%t(i)%*%Omega.inv%*%H%*%aux$tau + tau_0.v.inv * priors$tau_0.m )
tau_0.draw = rnorm(1,as.numeric(tau_0.bar), sqrt(V.tau_0.bar))
aux$tau_0 = as.vector(tau_0.draw)
# Sampling sigma_epsilon
###########################
sigma.eta.s = as.numeric(priors$sigma.s + crossprod((c(aux$tau_0[1],diff(aux$tau_0)) - i%*%aux$tau_0),(priors$H%*%aux$tau - i%*%aux$tau_0)))
sigma.eta.nu = priors$sigma.nu + T
sigma.eta.draw= sigma.eta.s/rchisq(1,sigma.eta.nu)
sigma.eta.draw.inv=1/(sigma.eta.draw)
sigma.e.s = as.numeric(priors$sigma.s + crossprod(aux$tau-aux$Y))
sigma.e.nu = priors$sigma.nu + T
sigma.e.draw = sigma.e.s/rchisq(1,sigma.e.nu)
aux$sigma = c(sigma.eta.draw,sigma.e.draw)
# Sampling lambda
###########################
u.t = H%*%aux$tau[,,s]-i%*%aux$tau_0[,,s]
# ---- loop lambda posterior ---- #
c = -1/2 + 1
a = 2 / lambda.priors$alpha
for (x in 1:T){
b = t((u.t)[x,])%*%(1/aux$sigma[1][,,s])%*%(u.t)[x,]
lambda.posterior.draws[x,s+1] = GIGrvg::rgig(1, lambda = c, chi = b, psi = a)
}
aux$lambda = lambda.posterior.draws
posteriors$tau[,s] = aux$tau
posteriors$tau_0[,s] = aux$tau_0
posteriors$lambda[,s] = aux$lambda
posteriors$sigma[,s] = aux$sigma
}
output = list(
posterior = posteriors,
last.draw = aux
)