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.