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

摘要

Mixture models are well-known for their versatility, and the Bayesianparadigm is a suitable platform for mixture analysis, particularly when thenumber of components is unknown. Bhattacharya (2008) introduced a mixture modelbased on the Dirichlet process, where an upper bound on the unknown number ofcomponents is to be specified. Here we consider a Bayesian asymptotic frameworkfor objectively specifying the upper bound, which we assume to depend on thesample size. In particular, we define a Bayesian analogue of the meanintegrated squared error (Bayesian MISE), and select that form of the upperbound, and also that form of the precision parameter of the underlyingDirichlet 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 investigateasymptotic choice of the precision parameter of the traditional Dirichletprocess mixture model; the density estimator we consider here is a modificationof the prior predictive distribution of Escobar & West (1995) associated withthe Polya urn model. Various asymptotic issues related to the twoaforementioned mixtures, including comparative performances, are alsoinvestigated.