# Show the posterior distribution is multivariate Gaussian

It is a Bayesian Regression, y and X are observed data, the response and the feature matrix respectively. $$\begin{eqnarray*} \mathbf y \mid \mathbf X,\mathbf w,\sigma^2 &\sim& N (\mathbf X \mathbf w, \sigma^2 \mathbf I ) \\ \mathbf w\mid \lambda &\sim &N(\mathbf 0, \lambda \mathbf I) \\ \sigma, \lambda >0 \end{eqnarray*}$$

How could I show that $$\mathbf w \mid \mathbf X , \mathbf y , \lambda , \sigma^2$$ is multivariate Gaussian? And what is the mean vector and variance-covariance matrix? I have been looking into this for many days and dont know how to solve it at all. Thanks!

• Please add the self-study tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. Please make these changes as just posting your homework & hoping someone will do it for you is grounds for closing. Nov 24, 2021 at 10:55

The posterior distribution for $$\textbf{w}$$ will come from the product

$$p(\textbf{w}|\textbf{y},\textbf{X},\lambda,\sigma^{2}) \propto N(\textbf{X} \textbf{w},\sigma^{2}I)N(0,\lambda I)$$

by discarding all the terms that do not include $$\textbf{w}$$.

If you do that you will end up with

$$e^{-\frac{1}{2\sigma^{2}}(-2\textbf{w}^{T}\textbf{X}\textbf{y}+\textbf{w}^{T}\textbf{X}^{T}\textbf{X}\textbf{w})-\frac{1}{2\lambda}\textbf{w}^{T}\textbf{w}} \ \ \ \ (*)$$

Now let's assume that we have a multivariate normal distribution $$\textbf{z}\sim N(\textbf{m},\Sigma)$$, if we expand each pdf and keep only terms that have $$\textbf{z}$$, we will end up with

$$e^{-\frac{1}{2}(\textbf{z}^{T}\Sigma^{-1}\textbf{z}-2\textbf{z}\Sigma^{-1}\textbf{m})} \ \ \ \ \ (**)$$

Hence, in order to identify the mean and covariance of $$\textbf{w}$$ try to bring $$(*)$$ into the form $$(**)$$ and identify which terms correspond to $$\Sigma^{-1}$$ and $$\textbf{m}$$