Both necessary and sufficient conditions for Bayesian exponential consistency

TL;DR

This paper establishes both necessary and sufficient conditions for Bayesian exponential consistency, improving understanding of when Bayesian procedures reliably converge quickly, and introduces a weaker, more practical sufficient condition.

Contribution

It provides a unified set of necessary and sufficient conditions for Bayesian exponential consistency, enhancing theoretical understanding and practical applicability.

Findings

Derived a simple, weaker sufficient condition for exponential consistency.

Identified the gap between KL support and exponential consistency.

Unified necessary and sufficient conditions for Bayesian exponential consistency.

Abstract

The last decade has seen a remarkable development in the theory of asymptotics of Bayesian nonparametric procedures. Exponential consistency has played an important role in this area. It is known that the condition of $f_0$ being in the Kullback-Leibler support of the prior cannot ensure exponential consistency of posteriors. Many authors have obtained additional sufficient conditions for exponential consistency of posteriors, see, for instance, Schwartz (1965), Barron, Schervish and Wasserman (1999), Ghosal, Ghosh and Ramamoorthi (1999), Walker (2004), Xing and Ranneby (2008). However, given the Kullback-Leibler support condition, less is known about both necessary and sufficient conditions. In this paper we give one type of both necessary and sufficient conditions. As a consequence we derive a simple sufficient condition on Bayesian exponential consistency, which is weaker than the previous sufficient conditions.