Conjugate Prior for Multivariate Model
The intuitions for posterior parameters we studied from the univariate normal pretty much carry over to the multivariate case.
library(ggplot2) library(cowplot) library(reshape) Multivariate Normal Model Consider a bivariate normal random variable $[y_1, y_2]^T$. The density is written as ($p=2$)
$$ p(\mathbf{y}|\theta, \Sigma) = (\dfrac{1}{2\pi})^{-p/2}|\Sigma|^{-1/2} \exp{-\dfrac{1}{2}(\mathbf{y}-\theta)^T\Sigma^{-1}(\mathbf{y}-\theta)} $$
where the parameter is $\theta = \begin{pmatrix} E[y_1]\\\ E[y_2] \end{pmatrix}$ and $\Sigma = \begin{pmatrix} E[y_1^2]-E[y_1]^2 & E[y_1y_2]-E[y_1]E[y_2]\\\
E[y_2y_1]-E[y_2]E[y_1] & E[y_2^2]-E[y_2]^2 \end{pmatrix}$ $=\begin{pmatrix} \sigma_1^2 & \sigma_{12}\\\
\sigma_{21} & \sigma_2^1 \end{pmatrix}$.
Few things worth mentioning for multivariate normal model
the term in the exponent $(\mathbf{y}-\theta)^T\Sigma^{-1}(\mathbf{y}-\theta)$ is somewhat a measure of distance between mean and the data.
