Expert Information and Nonparametric Bayesian Inference of Rare Events

Abstract

Prior distributions are important in Bayesian inference of rare events because historical data information is scarce, and experts are an important source of information for elicitation of a prior distribution. I propose a method to incorporate expert information into nonparametric Bayesian inference on rare events when expert knowledge is elicited as moment conditions on a finite dimensional parameter $\theta$ only. I generalize the Dirichlet process mixture model to merge expert information into the Dirichlet process (DP) prior to satisfy expert’s moment conditions. Among all the priors that comply with expert knowledge, we use the one that is closest to the original DP prior in the Kullback–Leibler information criterion. The resulting prior distribution is given by exponentially tilting the DP prior along $\theta$. I provide a Metropolis–Hastings algorithm to implement this approach to sample from posterior distributions with exponentially tilted DP priors. The proposed method combines prior information from a statistician and an expert by finding the least-informative prior given expert information.