The Birthday Problem: Bayesian Inference with Multiple Discrete Hypotheses

Abstract

The “Birthday Problem” expands consideration from two hypotheses to multiple, discrete hypotheses. In this chapter, interest is in determining the posterior probability that a woman named Mary was born in a given month; there are twelve alternative hypotheses. Furthermore, consideration is given to assigning prior probabilities. The priors represent a priori probabilities that each alternative hypothesis is correct, where a priori means “prior to data collection,” and can be “informative” or “non-informative.” A Bayesian analysis cannot be conducted without using a prior distribution, whether that is an informative prior distribution or a non-informative prior distribution. The chapter discusses objective priors, subjective priors, and prior sensitivity analysis. In addition, the concept of likelihood is explored more deeply.