An ordertheoretic perspective on modes and maximum a posteriori estimation in Bayesian inverse problems
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 smallradius 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 nonparametric MAP estimation by taking an ordertheoretic 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 $X = \mathbb{R}$.
 Publication:

arXiv eprints
 Pub Date:
 September 2022
 DOI:
 10.48550/arXiv.2209.11517
 arXiv:
 arXiv:2209.11517
 Bibcode:
 2022arXiv220911517L
 Keywords:

 Mathematics  Statistics Theory;
 Mathematics  Probability;
 Statistics  Methodology;
 06F99;
 28A75;
 28C15;
 60B05;
 62F10;
 62R20
 EPrint:
 38 pages