An order-theoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems

TL;DR

This paper introduces an order-theoretic approach to understanding modes and MAP estimators in Bayesian inverse problems, revealing new insights into their properties and pathologies.

Contribution

It applies order theory to analyze MAP estimation, providing new proof strategies and clarifying issues related to the existence and nature of modes.

Findings

Order-theoretic perspective clarifies mode definitions

Pathologies linked to greatest and maximal elements

Incomparable elements can be dense in the space

Abstract

It is often desirable to summarise a probability measure on a space $X$ in terms of a mode, or MAP estimator, i.e.\ a point of maximum probability. Such points can be rigorously defined using masses of metric balls in the small-radius limit. However, the theory is not entirely straightforward: the literature contains multiple notions of mode and various examples of pathological measures that have no mode in any sense. Since the masses of balls induce natural orderings on the points of $X$, this article aims to shed light on some of the problems in non-parametric MAP estimation by taking an order-theoretic perspective, which appears to be a new one in the inverse problems community. This point of view opens up attractive proof strategies based upon the Cantor and Kuratowski intersection theorems; it also reveals that many of the pathologies arise from the distinction between greatest and maximal elements of an order, and from the existence of incomparable elements of $X$, which we show can be dense in $X$, even for an absolutely continuous measure on $X = \mathbb{R}$.