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Posterior distribution of the binomial parameter

In this example we consider the computation of the posterior distribution for the binomial paraméter.

This is the same problem considered by Bayes  in Proposition 9 of his essay.

We are given m observed successes and n observed failures in a binomial experiment.

The experiment may be tossing a coin, drawing a ball from an urn, or asking soméone their opinion, among many other possibilities.

What we know about the paraméter (let's call it a) is stated as the prior distribution, p(a).

For a given value of a,

the probability of m successes in m+n trials is

Since m and n are fixed, and a is unknown,

this is a likelihood function for a.

From the continuous form of the law of total probability we have

For some special choices of the prior distribution p(a),

the integral can be solved and the posterior takes a convenient form.