### A periodic table of modes and MAP estimators

Ilja Klebanov and I have just uploaded a preprint of our paper “A ‘periodic table’ of modes and maximum a posteriori estimators” to the arXiv.

This paper forms part of the growing body of work on the ‘small balls’ theory of modes for probability measures on metric spaces, which is needed e.g. for the treatment of MAP estimation for Bayesian inverse problems with functional unknowns. There are already several versions in the literature: the strong mode, the weak mode, and the generalised strong mode. We take an axiomatic approach to the problem and identify a system of 17 essentially distinct notions of mode, proving implications between them and providing explicit counterexamples to distinguish them. From an axiomatic point of view, all these 17 seem to be ‘equally good’, suggesting that further research is needed in this area.

**Abstract.**
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 \( A, B \subseteq X\).

Published on Monday 17 July 2023 at 09:00 UTC #preprint #modes #map-estimators #klebanov