Index: The Book of Statistical ProofsStatistical Models ▷ Multivariate normal data ▷ Multivariate Bayesian linear regression ▷ Posterior distribution

Theorem: Let

$\label{eq:GLM} Y = X B + E, \; E \sim \mathcal{MN}(0, V, \Sigma)$

be a general linear model with measured $n \times v$ data matrix $Y$, known $n \times p$ design matrix $X$, known $n \times n$ covariance structure $V$ as well as unknown $p \times v$ regression coefficients $B$ and unknown $v \times v$ noise covariance $\Sigma$. Moreover, assume a normal-Wishart prior distribution over the model parameters $B$ and $T = \Sigma^{-1}$:

$\label{eq:GLM-NW-prior} p(B,T) = \mathcal{MN}(B; M_0, \Lambda_0^{-1}, T^{-1}) \cdot \mathcal{W}(T; \Omega_0^{-1}, \nu_0) \; .$

Then, the posterior distribution is also a normal-Wishart distribution

$\label{eq:GLM-NW-post} p(B,T|Y) = \mathcal{MN}(B; M_n, \Lambda_n^{-1}, T^{-1}) \cdot \mathcal{W}(T; \Omega_n^{-1}, \nu_n)$

and the posterior hyperparameters are given by

$\label{eq:GLM-NW-post-par} \begin{split} M_n &= \Lambda_n^{-1} (X^\mathrm{T} P Y + \Lambda_0 M_0) \\ \Lambda_n &= X^\mathrm{T} P X + \Lambda_0 \\ \Omega_n &= \Omega_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\ \nu_n &= \nu_0 + n \; . \end{split}$

Proof: According to Bayes’ theorem, the posterior distribution is given by

$\label{eq:GLM-NG-BT} p(B,T|Y) = \frac{p(Y|B,T) \, p(B,T)}{p(Y)} \; .$

Since $p(Y)$ is just a normalization factor, the posterior is proportional to the numerator:

$\label{eq:GLM-NG-post-JL} p(B,T|Y) \propto p(Y|B,T) \, p(B,T) = p(Y,B,T) \; .$

Equation \eqref{eq:GLM} implies the following likelihood function

$\label{eq:GLM-LF-Class} p(Y|B,\Sigma) = \mathcal{MN}(Y; X B, V, \Sigma) = \sqrt{\frac{1}{(2 \pi)^{nv} |\Sigma|^n |V|^v}} \, \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Sigma^{-1} (Y-XB)^\mathrm{T} V^{-1} (Y-XB) \right) \right]$

which, for mathematical convenience, can also be parametrized as

$\label{eq:GLM-LF-Bayes} p(Y|B,T) = \mathcal{MN}(Y; X B, P, T^{-1}) = \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \, \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T (Y-XB)^\mathrm{T} P (Y-XB) \right) \right]$

using the $v \times v$ precision matrix $T = \Sigma^{-1}$ and the $n \times n$ precision matrix $P = V^{-1}$.

Combining the likelihood function \eqref{eq:GLM-LF-Bayes} with the prior distribution \eqref{eq:GLM-NW-prior}, the joint likelihood of the model is given by

$\label{eq:GLM-NW-JL-s1} \begin{split} p(Y,B,T) = \; & p(Y|B,T) \, p(B,T) \\ = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \, \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T (Y-XB)^\mathrm{T} P (Y-XB) \right) \right] \cdot \\ & \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \, \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T (B-M_0)^\mathrm{T} \Lambda_0 (B-M_0) \right) \right] \cdot \\ & \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \; . \end{split}$

Collecting identical variables gives:

$\label{eq:GLM-NW-JL-s2} \begin{split} p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\ & \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (Y-XB)^\mathrm{T} P (Y-XB) + (B-M_0)^\mathrm{T} \Lambda_0 (B-M_0) \right] \right) \right] \; . \end{split}$

Expanding the products in the exponent gives:

$\label{eq:GLM-NW-JL-s3} \begin{split} p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\ & \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ Y^\mathrm{T} P Y - Y^\mathrm{T} P X B - B^\mathrm{T} X^\mathrm{T} P Y + B^\mathrm{T} X^\mathrm{T} P X B + \right. \right. \right. \\ & \hphantom{\exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ \right. \right. \right. \!\!\!} \; \left. \left. \left. B^\mathrm{T} \Lambda_0 B - B^\mathrm{T} \Lambda_0 M_0 - M_0^\mathrm{T} \Lambda_0 B + M_0^\mathrm{T} \Lambda_0 \mu_0 \right] \right) \right] \; . \end{split}$

Completing the square over $B$, we finally have

$\label{eq:GLM-NW-JL-s4} \begin{split} p(Y,B,T) = \; & \sqrt{\frac{|T|^n |P|^v}{(2 \pi)^{nv}}} \sqrt{\frac{|T|^p |\Lambda_0|^v}{(2 \pi)^{pv}}} \sqrt{\frac{|\Omega_0|^{\nu_0}}{2^{\nu_0 v}}} \frac{1}{\Gamma_v \left( \frac{\nu_0}{2} \right)} \cdot |T|^{(\nu_0-v-1)/2} \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_0 T \right) \right] \cdot \\ & \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (B-M_n)^\mathrm{T} \Lambda_n (B-M_n) + (Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n) \right] \right) \right] \; . \end{split}$

with the posterior hyperparameters

$\label{eq:GLM-NW-post-B-par} \begin{split} M_n &= \Lambda_n^{-1} (X^\mathrm{T} P Y + \Lambda_0 M_0) \\ \Lambda_n &= X^\mathrm{T} P X + \Lambda_0 \; . \end{split}$

Ergo, the joint likelihood is proportional to

$\label{eq:GLM-NW-JL-s5} p(Y,B,T) \propto |T|^{p/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( T \left[ (B-M_n)^\mathrm{T} \Lambda_n (B-M_n) \right] \right) \right] \cdot |T|^{(\nu_n-v-1)/2} \cdot \exp\left[ -\frac{1}{2} \mathrm{tr}\left( \Omega_n T \right) \right]$

with the posterior hyperparameters

$\label{eq:GLM-NW-post-T-par} \begin{split} \Omega_n &= \Omega_0 + Y^\mathrm{T} P Y + M_0^\mathrm{T} \Lambda_0 M_0 - M_n^\mathrm{T} \Lambda_n M_n \\ \nu_n &= \nu_0 + n \; . \end{split}$

From the term in \eqref{eq:GLM-NW-JL-s5}, we can isolate the posterior distribution over $B$ given $T$:

$\label{eq:GLM-NW-post-B} p(B|T,Y) = \mathcal{MN}(B; M_n, \Lambda_n^{-1}, T^{-1}) \; .$

From the remaining term, we can isolate the posterior distribution over $T$:

$\label{eq:GLM-NW-post-T} p(T|Y) = \mathcal{W}(T; \Omega_n^{-1}, \nu_n) \; .$

Together, \eqref{eq:GLM-NW-post-B} and \eqref{eq:GLM-NW-post-T} constitute the joint posterior distribution of $B$ and $T$.

Sources:

Metadata: ID: P160 | shortcut: mblr-post | author: JoramSoch | date: 2020-09-03, 08:37.