Seminar (Talk by Geoffrey Wolfer, AIP Approximate Bayesian Inference Team)

This is an online seminar. Registration is required.
【Date】2023/September/1 (Fri) 13:00-14:00(JST)
【Speaker】Geoffrey Wolfer, AIP Approximate Bayesian Inference Team

Title: From Distributions to Markov Chains: Recent Advances in Inference and Geometry

Abstract:
In this talk, we will explore recent progress in addressing fundamental questions in finite-sample statistical inference, spanning both iid and Markovian settings.
1/ We will begin by designing fully empirical confidence balls for distribution learning on countable spaces in total variation, along with an application to entropy estimation. We will subsequently delve into more general spaces, by considering kernel mean embeddings, and quantify learning via maximum mean discrepancy. Throughout, our emphasis will be on bypassing known minimax lower bounds by harnessing more information from the data.
2/ Continuing, we will discuss extensions of the well-established inference frameworks of estimation and identity testing of probability distributions. These extensions take us to single-trajectory models of Markov chains, for which I will provide an overview of the current state-of-the-art. In this segment of the talk, it will become apparent how the dependence in our observations and our ability to navigate the state space can lead to significant bottlenecks.
3/ In a seamless continuation, we will examine the central problem of estimating the mixing time of an ergodic Markov chain, which quantifies convergence towards its stationary distribution. A concise survey of applications in statistical learning theory, reinforcement learning, and Markov Chain Monte Carlo diagnostics will be provided. We will analyze the problem from a minimax perspective, with a primary focus on the challenges that arise within the non-reversible setting, and further construct fully data-dependent confidence intervals.
4/ In the final part, I will give a succinct overview of key structural findings within the realm of information geometry of stochastic matrices and infinitesimal generator matrices. If time permits, I will establish connections between this framework and practical applications, such as principles to guide the design of Markov Chain Monte Carlo proposal kernels and reduction techniques for some of our aforementioned inference problems.