I am trying to work with the Beta Binomial model for Machine Learning and Bayesian probability theory.
Here we have the Likelihood for the Beta-Binomial model:
Here we have the Prior:
And if we were to multiply them, we would get this:
My first question is this: The textbook says that the prior distribution is proportional to the likelihood times the prior itself:
But how is this possible? The $p(D | \theta)$ term is NOT CONSTANT - it varies as $\theta$ varies (and NOT necessarily proportionally with $p(\theta)$), correct?
And my second question is this: The textbook also says that the posterior is this:
I've already tried to derive this, but no luck. Could someone give me a couple steps in the right direction for this as well? Where do the 1 Bin and 2 Beta terms come from?
Thanks in advance