Large sample asymptotic analysis for normalized random measures with independent increments

TL;DR

This paper analyzes the asymptotic behavior and posterior consistency of normalized random measures with independent increments (NRMIs), focusing on the Bernstein-von Mises theorem for the normalized generalized gamma process and the impact of bias correction in credible sets.

Contribution

It provides new theoretical results on posterior consistency and the Bernstein-von Mises theorem for NRMIs, including bias correction effects in credible set construction.

Findings

Posterior consistency of NRMIs established under specific Levy intensity conditions.

Bernstein-von Mises theorem derived for the normalized generalized gamma process.

Bias correction significantly improves credible set coverage when the true distribution is discrete.

Abstract

Normalized random measures with independent increments represent a large class of Bayesian nonaprametric priors and are widely used in the Bayesian nonparametric framework. In this paper, we provide the posterior consistency analysis for normalized random measures with independent increments (NRMIs) through the corresponding Levy intensities used to characterize the completely random measures in the construction of NRMIs. Assumptions are introduced on the Levy intensities to analyze the posterior consistency of NRMIs and are verified with multiple interesting examples. A focus of the paper is the Bernstein-von Mises theorem for the normalized generalized gamma process (NGGP) when the true distribution of the sample is discrete or continuous. 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 when it is discrete. We also discuss the affect of the estimators for the model parameters of the NGGP under the Bernstein-von Mises convergences. Finally, to further explain the necessity of adding the bias correction in constructing credible sets, we illustrate numerically how the bias correction affects the coverage of the true value by the credible sets when the true distribution is discrete.