# Mean and covariance of Multivariate Normal

I am trying to solve an equation from a journal. What will be the distribution of \begin{multline} p(\alpha|y,all) \propto \prod\limits_{i}N(\gamma_{i} \mid AX,\Sigma_{\gamma}) \times \prod\limits_{j}N(\alpha_{j}|0,\sigma^{2}_{\alpha}\mathbf{I}_{3}) \propto \Sigma_{\gamma}^{-1} exp[(\gamma_{i} - AX) ^{T} \Sigma_{\gamma}^{-1}(\gamma_{i} - AX)] \times exp[\alpha^{T}\alpha / \sigma^{2}]\\ here, \gamma_{i} = (\gamma_{1},\gamma_{2},\gamma_{3}), vector(A)= (\alpha_{1},\alpha_{2},\alpha_{3})^{T}, X= (x_{1},x_{2},x_{3}) \end{multline}

I know this will be Multivariate normal. But what will be the mean and covariance ?

• are you familiar with completing the square?
– jld
Jan 16 '18 at 3:41
• Note that the first part of the expression is the product of multivariate normals, not just one multivariate normal as you have written. This might involve the matrix variate normal (en.wikipedia.org/wiki/Matrix_normal_distribution), not really sure. Jan 16 '18 at 3:42
• @Chaconne : I am sorry, I am not familiar with that term.But I saw an article and trying to understand how it works. Jan 16 '18 at 3:44
• @marmle: yes. you are right. They said the $\alpha \sim N(\bar \mu_{\alpha},\bar\Sigma_{\alpha})$, where $\bar\mu_{\alpha} = \bar\Sigma_{\alpha}[(X^T \otimes \bar\Sigma_{\gamma}^{-1})\gamma]$ and $\bar\Sigma_{\alpha}=[X^TX \otimes \bar\Sigma_{\gamma}^{-1} + \sigma^{2}I_{n}] ^{-1}$, I am trying to understand how they got this answer. Jan 16 '18 at 3:46
• often problems like this come down to the same computation where you need to get something of the form $x^T A x + a^T x$ into a form like $(y - b)^T B(y - b) + c$. Here's one example from this site: stats.stackexchange.com/questions/33418/…
– jld
Jan 16 '18 at 3:51