New paper introduces Mass Index and RE-KL for local Bayesian inference

A new arXiv paper by Hanli Xu, Fengxiang He, and Sarat Moka (arXiv:2606.27090) challenges the standard reliance on global divergences like KL divergence and ELBO in Bayesian inference. The authors argue these global objectives overlook important local-mass behavior—how probability mass concentrates or decays in small neighborhoods around parameter values. To fill this gap, they introduce two mathematical tools: the Mass Index, which records polynomial and logarithmic decay scales of local mass, and regularized extended KL (RE-KL), a set-localized divergence that works even with singular components. The paper, 28 pages with 3 figures and 2 tables, provides a rigorous theoretical framework.

The authors show that Mass Indices characterize how Bayesian updating shifts local mass via (1) power-log likelihood factors and (2) parameter-dependent supports (or smooth softenings) that alter local scale. Using local RE-KL, they prove absolute, relative, and directional inequalities comparing small-ball masses under both KL directions. Controlled experiments illustrate the local dynamics. Practically, this work could improve variational inference, MCMC diagnostics, and uncertainty quantification by revealing where models focus probability—critical for anomaly detection, rare event modeling, and high-stakes decision-making. Code is available on GitHub.