Stability, Fluctuations, and Non-Chaos in Gaussian Mixture NPMLEs

Abstract

This talk introduces the Gaussian mixture Non Parametric Likelihood Estimation (NPMLE) problem from first principles and explains why it is natural to view it as a random optimization problem. Recent results show that the NPMLE landscape is stable: approximate maximizers remain close to the truth in Hellinger and Kullback–Leibler loss, with guarantees that tolerate finite-time optimization error. The talk also discusses fluctuation results for the optimized likelihood, their consequences for differential entropy estimation, and a non-chaos result showing that small perturbations of the data do not produce macroscopically different fitted densities. The final part interprets these results through a statistical-mechanics lens, connecting likelihood landscapes to ideas such as ground states, free energy, asymptotic essential uniqueness, superconcentration, and chaos. This perspective suggests a useful common language for robustness, fluctuations, and stability in non-parametric likelihood problems.

Biography

Satyaki Mukherjee is a postdoctoral researcher at National University of Singapore, working with Subhroshekhar Ghosh. He earned his PhD in Mathematics in 2021 from University of California at Berkeley focusing on Spectral problems in Linear Algebra. Following his doctoral studies, he was a postdoctoral fellow at TUM in the Theoretical Foundations of Machine Learning group. His current research interests lie in the intersection of probability theory, theoretical statistics and machine learning.