# Finding conditional distribution $P(X|Y)$ parameters from concatenated Gaussians $X$ and $Y$ joint distribution

I'm trying to find the parameters of a conditional Gaussian distribution given the joint distribution between two variables and use it in a MAP estimator.

Given two multivariate random variables $X$ and $Y$ defined on $IR^n$ normally distributed $\mathcal{N}(X;\mu_X,\Sigma_X)$ and $\mathcal{N}(Y;\mu_Y,\Sigma_Y)$ and a joint distribution built using their concatenation : $Z=\begin{bmatrix} X \\ Y \end{bmatrix}$, let's suppose for simplification's sake that the resulting distribution is also normally distributed $\mathcal{N}(Z;\mu_Z,\Sigma_Z)$ with :

$\mu_z=\begin{bmatrix} \mu_x \\ \mu_y \end{bmatrix}$ and $\Sigma_z=\begin{bmatrix} \Sigma_{xx} & \Sigma_{xy} \\ \Sigma_{yx} & \Sigma_{yy} \end{bmatrix}$

How do I find $P(X|Y)$ ? Is it $P(X|Y)=\frac{P(X,Y)}{P(Y)}=\frac{P(Z)}{P(Y)}$ ? The joint distribution $P(X,Y)$ is generally called $P(X$ and $Y)$, and I'm wondering if a concatenation count as an "and".

And is it right to use it in a MAP estimator as : $\hat{X}=argmax_X P(X|Y)=argmax_X P(Z) = argmax_X \mathcal{N}(Z;\mu_Z,\Sigma_Z)$ ?

Is it $P(X|Y)=\frac{P(X,Y)}{P(Y)}=\frac{P(Z)}{P(Y)}$?

Yes, of course, that's the definition of conditional probability.

What's not obvious is that that ratio gives -for any fixed $Y$- another Gaussian; the parameters of that Gaussian are given here or here.

The joint distribution $P(X,Y)$ is generally called $P(X \,{ \rm and}\, Y)$, and I'm wondering if a concatenation count as an "and".

Basically, yes. What you call "concatenation" is more usually called "joint probability". And be aware of some terminology that could lead to confusion. For example, to say "$X$ and $Y$ are Gaussian variables, would usually mean the same as "$X$ is Gaussian and $Y$ is Gaussian"... which is not the same as "$X,Y$ are jointly Gaussian" or "$(X,Y)$ is Gaussian".

let's suppose for simplification's sake that the resulting distribution is also normally distributed

Bear in mind that that assumption is not automatic, and it's essential.