conditional distribution question

Question

I have a joint distribution which factorizes as follows:

$$ P(y, w, \beta) = P(y|\beta, w) P(w) P(\beta) $$

Now, I want to write the conditional distribution for $P(w|y, \beta)$, so this should be written as:

$$ P(w|y, \beta) = \frac{P(y, w, \beta)}{P(y, \beta)} = \frac{P(y|\beta, w) P(w) P(\beta)}{P(y|\beta) P(\beta)} \propto P(y|\beta, w) P(w) $$

I am not sure of the last step but I am guessing the denominator does not depend on $w$, we can treat it like a constant. My question is whether I can find the maxima for $P(w|y, \beta)$ by finding the maxima of $P(y|\beta, w) P(w)$ i.e. treat the $P(y|\beta)$ denominator as a constant. I am trying to do this in the context of figuring out some variational bayes updates where I need to compute some expectations and need the conditional distributions to do that.

Answer 1

I was wondering if I could find the maxima of $P(w|y, \beta)$ by optimizing just $P(y|\beta, w) P(w)$.

You can because you are doing a discriminate task and you don't need to care about the denominator which is a constant for every element you are going to compare. If you are doing a generative task you need to calculate the exact result of the $P(w|y, \beta)$ and hence to work out the $p(y, \beta)$.

So in your case $$ P(w|y, \beta) = \frac{P(y, w, \beta)}{P(y, \beta)} = P(y|\beta, w) P(w) * \frac{P(\beta)}{P(y, \beta)} \propto P(y|\beta, w) P(w) $$