Bayesian Consistency with the Supremum Metric

TL;DR

This paper establishes simple, weaker conditions for Bayesian consistency in the supremum metric, leveraging weak convergence and density smoothing to improve upon existing $ ext{L}_1$ consistency criteria.

Contribution

It introduces a novel approach using a triangle inequality and weak convergence to achieve supremum consistency under less restrictive conditions.

Findings

Supremum consistency achieved with weaker conditions than $ ext{L}_1$ consistency.

Utilizes a triangle inequality to connect weak convergence with supremum metric.

Provides explicit conditions for Bayesian consistency in the supremum metric.

Abstract

We present simple conditions for Bayesian consistency in the supremum metric. The key to the technique is a triangle inequality which allows us to explicitly use weak convergence, a consequence of the standard Kullback--Leibler support condition for the prior. A further condition is to ensure that smoothed versions of densities are not too far from the original density, thus dealing with densities which could track the data too closely. A key result of the paper is that we demonstrate supremum consistency using weaker conditions compared to those currently used to secure $\mathbb{L}_1$ consistency.