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：通常希望用众数或最大后验估计（即概率最大的点）来概括空间 $X$ 上的概率测度。这些点可以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：文献中包含多种众数概念以及各种病态测度的例子，这些测度在任何意义上都不存在众数。由于球的质量导出了 $X$ 上点的自然序关系，本文旨在通过采用序理论视角（这在逆问题社区似乎是新颖的）来阐明非参数最大后验估计中的若干问题。该视角基于康托尔和库拉托夫斯基相交定理打开了有吸引力的证明策略；同时揭示了许多病态现象源于序中最大元与极大元之间的区别，以及 $X$ 中不可比元素的存在——我们证明，即使对于 $X = \mathbb{R}$ 上的绝对连续测度，这些元素也可能在 $X$ 中稠密。