ArXiv

Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference

2026-06-25 · 14:26 UTC ·Hanli Xu, Fengxiang He, Sarat Moka ·1 min read
Authors
Hanli Xu, Fengxiang He, Sarat Moka
Categories
stat.ML, cs.AI, cs.LG
arXiv
https://arxiv.org/abs/2606.27090v1
PDF
https://arxiv.org/pdf/2606.27090v1

Brief

Beyond Global Divergences develops a local-mass framework for Bayesian inference, introducing the Mass Index and regularised extended KL (RE-KL) to quantify polynomial/logarithmic decay of local mass and set-localised divergences (handling singular supports). The authors prove absolute, relative and directional inequalities comparing small-ball masses under forward/reverse KL, present controlled experiments, and release code.

Why it matters

Introduces two tools: Mass Index (records polynomial and logarithmic decay scales of local mass) and regularised extended KL (RE-KL), a set-localised divergence that admits singular components.

Key details

  • Mass Index shows how Bayesian updating alters local mass: power-log likelihood factors shift local-mass scales explicitly, while parameter-dependent supports or their smooth softenings change the local decay scale by varying the mass remaining near a parameter.
  • Using local RE-KL the authors prove absolute, relative, and directional inequalities for comparing local small-ball masses under the two KL directions; paper is 28 pages (3 figures, 2 tables), posted to arXiv:2606.27090v1 on 2026-06-25, code at https://github.com/Forsythia0604/Local-Mass-Framework.
Source evidence

Abstract

Global objectives, such as KL divergence and ELBO, are widely used in Bayesian inference for measuring distributional discrepancy. This paper studies their local-mass behaviour that is not directly captured by such objectives. We introduce and use two mathematical tools: (1) Mass Index for recording the polynomial and logarithmic decay scales of local mass, and (2) regularised extended KL (RE-KL), a set-localised divergence that can be formulated in the presence of singular components. Mass Indices help characterise how Bayesian updating changes local mass: (1) power-log likelihood factors shift it explicitly, and (2) parameter-dependent supports, or their smooth softenings, may change the local scale through the amount of mass that remains near the parameter value. Using local RE-KL, we prove absolute, relative, and directional inequalities for comparing local small-ball masses under the two KL directions. Together, these results provide a local theoretical account of local mass behaviour. Experiments provide controlled illustrations of the local behaviour. Code is available at https://github.com/Forsythia0604/Local-Mass-Framework.

Comment: 28 pages, 3 figures, 2 tables