I have a question regarding the Bayes'theorem of Gaussians.
Assume we have the multivariate gaussian density function:
$$x=(X1, X2)$$
$X1$ and $X2$ are random variables
$$\Sigma = \begin{bmatrix}\sigma^2 & \alpha * \sigma^2 \\ \alpha * \sigma^2 & \sigma^2 \end{bmatrix}$$
$$ f(x|\mu,\sigma) =\frac{1}{\sqrt{(2\pi)^{2}|\Sigma|}}exp(-\frac{1}{2}(x)^t\Sigma^{-1}(x))$$
I want to calculate the joint probability p(X1, X2). I found out that this is pretty hard to integrate.
Now I have found the Bayes'theorem of Gaussians that could help me to achieve this result.
$$z = (X1, X2)$$ $$p(z) = N(z| m, R^{-1})$$
$$m = \begin{bmatrix}\mu \\ A*\mu + b \\\end{bmatrix}$$
$$ R^{-1}=\begin{bmatrix}\Lambda^{-1}& \Lambda^{-1} * A^{T} \\ A * \Lambda^{-1} & L^{-1}+A*\Lambda^{-1}*A^{T} \end{bmatrix}$$
This is probably the way I could calculate $p(X1, X2)$ but I still don't know how to calculate.
$\Lambda$ and $A$ and $b$.
I have $\mu_X1$ and $\mu_X2$ given as $0$. So I could probably calculate A and b from:
$$A*0+b=0$$ But then the A needs to be a scalar and not a matrix...
I am really confused by all the variables I don't know how to calculate.
Hopefully someone with more experience can help me :)