The Shark Attack Problem: The Gamma-Poisson Conjugate

Abstract

This chapter introduces the gamma-Poisson conjugate. Many Bayesian analyses consider alternative parameter values as hypotheses. The prior distribution for an unknown parameter can be represented by a continuous probability density function when the number of hypotheses is infinite. There are special cases where a Bayesian prior probability distribution for an unknown parameter of interest can be quickly updated to a posterior distribution of the same form as the prior. In the “Shark Attack Problem,” a gamma distribution is used as the prior distribution of λ‎, the mean number of shark attacks in a given year. Poisson data are then collected to determine the number of attacks in a given year. The prior distribution is updated to the posterior distribution in light of this new information. In short, a gamma prior distribution + Poisson data → gamma posterior distribution. The gamma distribution is said to be “conjugate to” the Poisson distribution.