Remarks on consistency of posterior distributions

TL;DR

This paper discusses the concept of posterior consistency in Bayesian analysis, comparing various approaches and presenting new results on non-exponential consistency, improper priors, and non-i.i.d. data with illustrative examples.

Contribution

It provides a comprehensive overview of posterior consistency issues and introduces new findings on challenging scenarios like improper priors and non-i.i.d. observations.

Findings

Analysis of non-exponential posterior consistency

Results on improper priors and their impact

Insights into non-i.i.d. data scenarios

Abstract

In recent years, the literature in the area of Bayesian asymptotics has been rapidly growing. It is increasingly important to understand the concept of posterior consistency and validate specific Bayesian methods, in terms of consistency of posterior distributions. In this paper, we build up some conceptual issues in consistency of posterior distributions, and discuss panoramic views of them by comparing various approaches to posterior consistency that have been investigated in the literature. In addition, we provide interesting results on posterior consistency that deal with non-exponential consistency, improper priors and non i.i.d. (independent but not identically distributed) observations. We describe a few examples for illustrative purposes.