Student Probability Seminar
Contiguity, second moments, and overlap formulas
Speaker: Tim Kunisky, CIMS
Date: Wednesday, April 7, 2021, 9 a.m.
I will present a popular method for establishing that it is impossible to solve certain high-dimensional hypothesis testing problems, which reduces this to controlling the second moment of the associated likelihood ratio, and explain how this method relates to the notion of "contiguity" between sequences of probability measures. I will then show how, in problems where we have to distinguish between i.i.d. Gaussian observations and those shifted to have some random but structured mean, this furthermore reduces to controlling the random "overlap" or inner product between two independent draws of the mean, and give some applications of this observation to spiked models of random matrices. Finally, I will show how this whole story generalizes to a broader setting of observations in certain exponential families, which will involve seeing a beautiful characterization due to Morris of the "next-simplest" exponential families after the translates of a Gaussian, as well as a unified description of the orthogonal polynomials of those families, which enjoy variants of many of the convenient properties of Hermite polynomials.