A note on Bayesian robustness for count data

The usual Bayesian approach for count data is Gamma/Poisson conjugate analysis. However, in this conjugate analysis the influence of the prior distribution could be dominant even when prior and likelihood are in conflict. Our proposal is an analysis based on the Cauchy prior for natural parameter in exponential families. In this work, we show that the Cauchy/Poisson posterior model is a robust model for count data in contrast with the usual conjugate Bayesian approach Gamma/Poisson model. We use the polynomial tails comparison theorem given in (Bayesian Anal. 4 (2009) 817-843) that gives easy-to-check conditions to ensure prior robustness. In short, this means that when the location of the prior and the bulk of the mass of the likelihood get further apart (a situation of conflict between prior and likelihood information), Bayes theorem will cause the posterior distribution to discount the prior information. Finally, we analyze artificial data sets to investigate the robustness of the Cauchy/Poisson model.