The Robustness and Sensitivity of the Mixed-Dirichlet Bayesian Test for "Independence" in Contingency Tables

Abstract

A mixed-Dirichlet prior was previously used to model the hypotheses of "independence" and "dependence" in contingency tables, thus leading to a Bayesian test for independence. Each Dirichlet has a main hyperparameter $\kappa$ and the mixing is attained by assuming a hyperprior for $\kappa$. This hyperparameter can be regarded as a flattening or shrinking constant. We here review the method, generalize it and check the robustness and sensitivity with respect to variations in the hyperpriors and in their hyperhyperparameters. The hyperpriors examined included generalized log-Students with various numbers of degrees of freedom $\nu$. When $\nu$ is as large as 15 this hyperprior approximates a log-normal distribution and when $\nu = 1$ it is a log-Cauchy. Our experiments caused us to recommend the log-Cauchy hyperprior (or of course any distribution that closely approximates it). The user needs to judge values for the upper and lower quartiles, or any two quantiles, of $\kappa$, but we find that the outcome is robust with respect to fairly wide variations in these judgments.