Superiority of Bayes Estimators Over the MLE in High Dimensional Models on Compact Riemannian Manifolds and its Implication for Nonparametric Bayes Theory

Abstract

This research focuses on the performance of Bayes estimators, in comparison with the MLE, in multinomial models with a relatively large number of cells. The prior for the Bayes estimator is taken to be the conjugate Dirichlet, i.e., the multivariate Beta, with exchangeable distributions over the coordinates, including the non-informative uniform distribution. The choice of the multinomial is motivated by its many applications in business and industry, but also by its use in providing a simple nonparametric estimator of an unknown distribution on a compact Riemannian manifold. It is striking that the Bayes procedure outperforms the asymptotically efficient MLE over most of the parameter space for even moderately large dimensional parameter spaces and rather large sample sizes.