Posterior consistency in misspecified models for i.n.i.d response

TL;DR

This paper establishes conditions for Bayesian posterior consistency in misspecified models with independent but non-identically distributed responses, extending previous work to non-convex parametric families and demonstrating applicability to Bayesian quantile estimation.

Contribution

It provides a novel approach to posterior consistency in misspecified i.n.i.d models without requiring convexity of the parametric family, extending prior results to more general settings.

Findings

Derived conditions for posterior consistency in i.n.i.d models

Extended results from i.i.d to i.n.i.d responses

Applied findings to Bayesian quantile estimation

Abstract

We derive conditions for posterior consistency when the responses are independent but not identically distributed ($i.n.i.d$) and the model is "misspecified" to be a family of densities parametrized by a possibly infinite dimensional parameter. Our approach has connections to key ideas developed for $i.i.d$ models in Kleijn and van der Vaart(2006) and it's subsequent simplification in Ramamoorthi, et al.(2014) (unpublished manuscript). While key results in these two papers rely heavily on the convexity of the specified family of densities, parametric families are seldom convex. In this note, we take a direct approach to deriving posterior consistency with respect to natural topologies on the parameter space without having to impose conditions on the convex hull of the parametric family. We first derive our results for the case when the responses are $i.i.d$ and then extend it to the $i.n.i.d$ case. As an example, we demonstrate the applicability of the results to the Bayesian quantile estimation problem.