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

Abstract

Mixture models are well-known for their versatility, and the Bayesian paradigm is a suitable platform for mixture analysis, particularly when the number of components is unknown. Bhattacharya (2008) introduced a mixture model based on the Dirichlet process, where an upper bound on the unknown number of components is to be specified. Here we consider a Bayesian asymptotic framework for objectively specifying the upper bound, which we assume to depend on the sample size. In particular, we define a Bayesian analogue of the mean integrated squared error (Bayesian MISE), and select that form of the upper bound, and also that form of the precision parameter of the underlying Dirichlet process, for which Bayesian MISE of a specific density estimator, which is a suitable modification of the Polya-urn based prior predictive model, converges at a desired rate. As a byproduct of our approach, we investigate asymptotic choice of the precision parameter of the traditional Dirichlet process mixture model; the density estimator we consider here is a modification of the prior predictive distribution of Escobar & West (1995) associated with the Polya urn model. Various asymptotic issues related to the two aforementioned mixtures, including comparative performances, are also investigated.