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 modes property', under which the modes of $\mu|_{A}$, $\mu|_{B}$ and $\mu|_{A \cup B}$ enjoy a straightforward relationship for open, positive-mass $A,B \subseteq X$.

翻译：过去十年中，众多研究致力于将空间$X$上概率分布$\mu$的众数（即最大后验估计器）定义推广至$\mu$无连续勒贝格密度的情况，特别是推广至无穷维巴拿赫空间与希尔伯特空间$X$。本文系统考察了这些定义的性质及其内在关联。我们构建了众数的系统分类体系（即“元素周期表”），其中既包含现有概念，也涵盖此前未被探索的大类定义。我们建立了这些定义间的蕴含关系，并通过反例区分它们的差异性。同时，我们进一步区分了那些仅具有“语法正确性”的定义与那些满足众数“常识性”公理（如正确处理离散测度及具有连续勒贝格密度的测度）的“有意义”定义。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“众数合并性质”——即对于开集且具有正测度的$A,B \subseteq X$，$\mu|_{A}$、$\mu|_{B}$与$\mu|_{A \cup B}$的众数之间不存在简易关联关系。