Exact Bayesian Analysis of a 2 Times 2 Contingency Table, and Fisher's “Exact” Significance Test

Abstract

A relationship is derived between the posterior probability of negative association of rows and columns of a 2 times 2 contingency table and Fisher's “exact” probability, as given in existing tables for testing the hypothesis of no association of rows and columns. The result for the 2 times 2 table is generalized to provide the posterior probability that one discrete-valued random variable is stochastically larger than another.