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

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