Thomas Bayes' walk on manifolds

TL;DR

This paper investigates the convergence behavior of Bayes posterior measures when data lie on geometric structures like manifolds, proposing a geometric prior and deriving contraction rates.

Contribution

It introduces a geometric prior based on heat equation solutions and establishes bounds on posterior contraction rates in manifold settings.

Findings

Derived upper and lower bounds for posterior contraction rates.

Proposed a heat equation-based geometric prior.

Analyzed convergence of Bayes measures on manifolds.

Abstract

Convergence of the Bayes posterior measure is considered in canonical statistical settings where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions. A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived.