A new calibrated bayesian internal goodness-of-fit method: Sampled posterior p-values as simple and general p-values that allow double use of the data

Abstract

Background: Recent approaches mixing frequentist principles with Bayesian inference propose internal goodness-of-fit (GOF) p-values that might be valuable for critical analysis of Bayesian statistical models. However, GOF p-values developed to date only have known probability distributions under restrictive conditions. As a result, no known GOF p-value has a known probability distribution for any discrepancy function. Methodology/Principal Findings: We show mathematically that a new GOF p-value, called the sampled posterior p-value (SPP), asymptotically has a uniform probability distribution whatever the discrepancy function. In a moderate finite sample context, simulations also showed that the SPP appears stable to relatively uninformative misspecifications of the prior distribution. Conclusions/Significance: These reasons, together with its numerical simplicity, make the SPP a better canonical GOF p-value than existing GOF p-values. © 2011 Frédéric Gosselin.