Ok, let me rephrase the question.
if ##p(\theta|y)## is the distribution of interest, then what good is ##p(y|\theta)p(\theta)## if the mean and variance aren't the same?
Help me understand something.
I get that the posterior ##p(\theta|y) \propto p(y|\theta)p(\theta)## should be normalized by ##\frac{1}{p(y)}## for the probability to sum to 1, but what about the mean and variance?
Am I not right understanding that sampling from the un-normalized posterior...