Mengxin Xi

Title: Extrapolation of Tempered Posteriors
Abstract: Tempering is a popular tool in Bayesian computation, being used to transform a complicated posterior distribution into one that is more easily sampled. The idea is to construct a sequence (pt)0≤t≤1, with p0 typically representing the prior and p1 representing the posterior, and then to numerically approximate terms in this sequence, starting with p0 and proceeding through intermediate distributions until an approximation to p1 is obtained. Our contribution reveals that high-quality approximation of terms up to p1 is not essential, as knowledge of the intermediate distributions enables posterior quantities of interest to be extrapolated. Specifically, we establish weak sufficient conditions under which tempered expectations are not merely smooth as a function of t, but analytic, implying that knowledge of the tempered expectation in any open t interval fully determines the posterior expectation of interest. Harnessing this result, we propose novel regression methodology for approximation of posterior expectations based on tempering and the waste-free sequential Monte Carlo method of [Dau and Chopin, 2022], illustrating its effectiveness on a number for examples.

Enrique Cacicedo Germano

Title: Calibrating credible regions for Gibbs posterior distributions with covariance matrix approximation
Abstract: Modern machine learning applications involve optimizing an evaluation metric of interest that has an objective practical meaning. The traditional Bayesian setting relies on directly modeling the data-generating process, making model misspecification a relevant concern. Alternatively, a framework based on Gibbs posterior distributions that directly link data with quantities of interest via loss functions can provide robustness for inference and uncertainty quantification. In this study, we provide a simple solution for calibrating the learning rate in Gibbs posterior distributions. The algor...