Superconcentration, stability and chaos in NPMLEs

Bangalore Probability Seminar

Talk 1

Title: Superconcentration, stability and chaos in NPMLEs

Abstract: NPMLE or non-parametric models are widely studied in the literature as models which estimate the underlying densities from independent samples of the same.We study log-likelihoods under the NPMLE model from a statistical mechanical perspective. In particular we draw parallels between the log likelihood of NPMLEs and  the energy function in Gaussian polymers. We begin by proving a fluctuation result for the maximum empirical log likelihood. Using this we prove that a phenomenon known as superconcentration does not hold. In principle what it really says is that the Poincare inequality for the log likelihood is tight. Chatterjee in his monograph on Superconcentration, Chaos and Multiple Valleys argues that this should in principle always be interlinked. We show that indeed in the the NPMLE case the inverse of all three of this is true, thus leading to the observation that NPMLe problems are remarkably stable under perturbations.