Bayesian MISE convergence rates of Polya urn based density estimators: asymptotic comparisons and choice of prior parameters

Abstract

ABSTRACT Bhattacharya [Gibbs sampling based Bayesian analysis of mixtures with unknown number of components. Sankhya B. 2008;70:133–155] introduced a mixture model based on the Dirichlet process, where an upper bound on the unknown number of components is to be specified. Defining a Bayesian analogue of the mean integrated squared error (Bayesian MISE), here we consider a Bayesian asymptotic density estimation framework forobjectively specifying the upper bound, as well as the precision parameter of the Dirichlet process, such that the Bayesian MISE converges at a desired rate. As a byproduct of our approach, we also investigate Bayesian MISE convergence rate of the traditional Dirichlet process mixture model, which leads to asymptotic specification of the precision parameter. Various asymptotic issues related to the two aforementioned mixtures, including comparative performances, are also investigated. The theoretical studies, supplemented with simulation experiments, bring out the superiority of the approach of Bhattacharya (2008).