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 $X = \mathbb{R}$.

翻译：通常需要以众数或最大后验（MAP）估计（即概率最大点）的形式来概括空间$X$上的概率测度。此类点可通过小半径极限下度量球的质量严格定义。然而，该理论并非完全直接：文献中存在多种众数概念，以及各种在任何意义上均无众数的病态测度实例。由于度量球的质量在$X$的点上诱导出自然序关系，本文旨在通过采取序理论视角（这在反问题领域似属新颖）阐明非参数MAP估计中的若干问题。该视角基于康托尔和库拉托夫斯基交定理开辟了引人入胜的证明策略；同时揭示出许多病态源于序中的最大元与极大元之区别，以及$X$中不可比元素的存在性——我们证明即使对于$X = \mathbb{R}$上的绝对连续测度，此类元素也可能在$X$中稠密。