A 'periodic table' of modes and maximum a posteriori estimators

Abstract

The last decade has seen many attempts to generalise the definition of modes, or MAP estimators, of a probability distribution μ on a space X to the case that μ 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 μ|A, μ|B and μ|A∪B enjoy a straightforward relationship for well-separated positive-mass events A,B⊆X.