Conditional probability distribution and prior In a linear Gaussian model, when I multiply a prior distribution $p(x)$ with the conditional $p(y|x)$ (here x and y are vectors), which one do I get:


*

*The joint distribution p(z) where z is the concatenation of x and y 

*Or p(y)?


In Bayes scenario, we have posterior p(x|y) is proportional to p(x)*p(y|x). Does this mean that p(x|y) is proportional to p(z)?

We see that
$$P(Y|X) \cdot P(X) = \frac{P(Y,X)}{P(X)}\cdot P(X) = P(Y,X) = P(Z),$$
Where $Z$ is defined as the variable over the cartesian product of the supports of $Y$ and $X$ (as in, a realization $z$ of $Z$ is in $S_Y \times S_X$). Hence, 1. is the more appropriate answer.
