SQUIDS Seminar - Classification of small-ball modes and maximum a posteriori estimators on metric spaces
| Dates: | 19 March 2025 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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| Speaker: | Tim J. Sullivan |
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A mode, or `most likely point', for a probability measure $\mu$ can be defined in various ways using the asymptotic behaviour of the $\mu$-mass of balls of radius $r \to 0$. Such points are of intrinsic interest in the local theory of measures on metric spaces and also arise naturally in the study of Bayesian inverse problems and diffusion processes. Building upon special cases already proposed in the literature, this paper undertakes a systematic study of possible definitions of modes using such small-ball probabilities. We propose `common-sense' axioms that such definitions should obey, e.g.\ correct handling of discrete and absolutely continuous $\mu$, as well as symmetry and invariance considerations. We show that there are exactly ten such definitions consistent with these axioms, and that they are partially but not totally ordered in strength, forming a complete, distributive lattice. We also show how this general system of ten mode types simplifies for well-behaved $\mu$, e.g.\ those dominated by a Gaussian measure, as is common in Bayesian inference and diffusion processes.
Joint work with Ilja Klebanov (FU Berlin) and Hefin Lambley (Warwick).
Speaker
Tim J. Sullivan
Organisation: University of Warwick
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G.113
Alan Turing Building
Manchester
