Getting the posterior

Let's remember that Bayes' theorem (equation 1.4) says the posterior is proportional to the likelihood times the prior. So, for our problem, we have to multiply the binomial and the beta distributions:

We can simplify this expression. For our practical concerns, we can drop all the terms that do not depend on  and our results will still be valid. Accordingly, we can write:

Reordering it, we get:

If we pay attention, we will see that this expression has the same functional form of a beta distribution (except for the normalization term) with  and . In fact, the posterior distribution for our problem is the beta distribution: