Bayesian Statistics: An Introduction, 4th Edition

3.1 The binomial distribution

3.1.1 Conjugate prior

In this section, the parameter of interest is the probability π of success in a number of trials which can result in success (S) or failure (F), the trials being independent of one another and having the same probability of success. Suppose that there is a fixed number of n trials, so that you have an observation x (the number of successes) such that

from a binomial distribution of index n and parameter π, and so

The binomial distribution was introduced in Section 1.3 on ‘Random Variables’ and its properties are of course summarized in Appendix A.

Figure 3.1 Examples of beta densities.

If your prior for π has the form

that is, if