Bayesian MISE Convergence Rates of Mixture Models Based on the Polya Urn Model: Asymptotic Comparisons and Choice of Prior Parameters

We study the asymptotic properties of Bayesian density estimators constructed using normal mixtures induced by the Polya urn scheme (Blackwell and McQueen (1973)). Within our marginalized mixture framework, we consider two separate density estimators; that of Escobar and West (1995) and that intro- duced by Bhattacharya (2008). The latter mixture model specifies a bound to the number of mixture components, preventing it from growing arbitrarily large with the sample size. We study the prior and the posterior rates of convergence of the mean integrated squared error (M ISE) for both kinds of mixtures and show that the M ISE corresponding to Bhattacharya (2008) converges to zero at a much faster rate compared to that of Escobar and West (1995) with respect to the posterior. We also show that with proper, but plausible, choices of the free parameters of our M ISE bounds the rate of convergence can be made smaller than the best rate of Ghosal and van der Vaart (2007) given by n-2/5 (log n)4/5 and in fact, can be made smaller that the optimal frequentist rate n-2/5 . Apart from these we study and compare the M ISE convergence rates of the two models in the case of the "large p small n" problem. Furthermore, we show that while the model of Escobar and West (1995) can converge to a wrong model under certain conditions, much stonger conditions are necessary for Bhattacharya (2008) to converge to a wrong model. We also consider a modified version of Bhattacharya (2008) but demonstrate that all the results remain same under the modified version.