Large Sample Asymptotic Analysis for Normalized Random Measures with Independent Increments

Abstract

Normalized random measures with independent increments (NRMIs) represent a large class of Bayesian nonparametric priors and are widely used in the Bayesian nonparametric framework. In this paper, we provide the posterior consistency analysis for these NRMIs through their characterizing Lévy intensities. Assumptions are introduced on the Lévy intensities to analyse the posterior consistency and are verified with multiple interesting examples. Another focus of the paper is the Bernstein-von Mises theorem for a particular subclass of NRMIs, namely the normalized generalized gamma processes (NGGP). When the Bernstein-von Mises theorem is applied to construct credible sets, in addition to the usual form, there will be an additional bias term on the left endpoint closely related to the number of atoms of the true distribution in the discrete case. We also discuss the effect of the estimators for the model parameters of the NGGP under the Bernstein-von Mises convergence. Finally, to further illustrate the impact of the bias correction term in the construction of credible sets, we present a numerical example to demonstrate numerically how the bias correction affects the coverage of the true value.