A note on nonexistence of posterior moments

Abstract

Bayesian analyses often take for granted the assumption that the posterior distribution has at least a first moment. They often include computed or estimated posterior means. In this note, the authors show an example of a Weibull distribution parameter where the theoretical posterior mean fails to exist for commonly used proper semi–conjugate priors. They also show that posterior moments can fail to exist with commonly used noninformative priors including Jeffreys, reference and matching priors, despite the fact that the posteriors are proper. Moreover, within a broad class of priors, the predictive distribution also has no mean. The authors illustrate the problem with a simulated example. Their results demonstrate that the unwitting use of estimated posterior means may yield unjustified conclusions.