Bayesian analysis for proportions

In this section, we'll revisit inference for proportions, but from a Bayesian perspective. We will look at Bayesian methods for analyzing proportions of success in a group. This includes talking about computing credible intervals, and the Bayesian version of hypothesis testing for both one and two samples.

Conjugate priors are a class of prior probability distributions common in Bayesian statistics. A conjugate prior is a prior distribution such that the posterior distribution belongs to the same family of probability distributions as the prior. For binary data, the beta distribution is a conjugate prior. This is a distribution defined where only values in the (0, 1) interval have a chance of appearing. They are specified by two parameters. In a trial, if there are M successes out of N trials, then the posterior distribution is the prior distribution when we add M to the first parameter of the prior, and N - M to the second parameter of the prior. This concentrates the distribution to the observed population proportion.