The last decade has seen many attempts to generalise the definition of modes, or MAP estimators, of a probability distribution $\mu$ on a space $X$ to the case that $\mu$ has no continuous Lebesgue density, and in particular to infinite-dimensional Banach and Hilbert spaces $X$. This paper examines the properties of and connections among these definitions. We construct a systematic taxonomy -- or `periodic table' -- of modes that includes the established notions as well as large hitherto-unexplored classes. We establish implications between these definitions and provide counterexamples to distinguish them. We also distinguish those definitions that are merely `grammatically correct' from those that are `meaningful' in the sense of satisfying certain `common-sense' axioms for a mode, among them the correct handling of discrete measures and those with continuous Lebesgue densities. However, despite there being 17 such `meaningful' definitions of mode, we show that none of them satisfy the `merging property', under which the modes of $\mu|_{A}$, $\mu|_{B}$ and $\mu|_{A \cup B}$ enjoy a straightforward relationship for well-separated positive-mass events $A,B \subseteq X$.

翻译：过去十年间，人们多次尝试将空间 $X$ 上概率分布 $\mu$ 的众数（或最大后验估计量）的定义推广到 $\mu$ 不具有连续勒贝格密度的情况，特别是推广到无穷维巴拿赫空间与希尔伯特空间 $X$ 的情形。本文研究了这些定义的性质及其相互联系。我们构建了一套系统的分类学——或称“周期表”——其中既包含既有的众数概念，也涵盖大量此前未被探索的新类别。我们建立了这些定义之间的蕴含关系，并通过反例加以区分。同时，我们将那些仅“语法正确”的定义与那些在满足众数的若干“常识性”公理（例如正确处理离散测度及具有连续勒贝格密度的测度）意义上“有意义”的定义加以区分。然而，尽管存在17种此类“有意义的”众数定义，我们证明它们均不满足“合并性质”：对于充分分离的正质量事件 $A,B \subseteq X$，$\mu|_{A}$、$\mu|_{B}$ 与 $\mu|_{A \cup B}$ 的众数无法呈现直接的关系。