Statistics and Computational Methods Seminar Series - Fall 2026

Speaker: Francesca Crucinio

Title: Gradient flows for statisticians

Abstract

Many statistical methods can be formulated as optimization problems, but the underlying optimization takes place on spaces with very different structures. We develop a unified perspective based on gradient flows, showing how the geometry of the underlying space determines the resulting dynamics and algorithms. We formulate maximum likelihood, marginal and nonparametric maximum likelihood, Bayesian and post-Bayesian inference as optimization problems, and derive the corresponding gradient flows on finite- and infinite-dimensional spaces. We focus on the Wasserstein and Fisher–Rao geometries, which induce fundamentally different mechanisms through transport and reweighting of probability mass. Discretizing these flows recovers or motivates familiar statistical algorithms, including gradient and mirror descent, the unadjusted Langevin algorithm, and methods for nonparametric maximum likelihood. This perspective provides a unified framework for analysing and developing statistical algorithms.

Link Teams: https://teams.microsoft.com/meet/3544053560130854?p=t2cqYX8frW0UOALXTK