Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference
Abstract
Global objectives, such as KL divergence and ELBO, are widely used in Bayesian inference for measuring distributional discrepancy. This paper studies distributional ``local-mass behaviours'' that are not directly captured by such global 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, and show that the directional comparison remains visible in the variational posteriors of Bayesian neural networks up to ResNet-50 on ImageNet. Code is available at https://github.com/Forsythia0604/Local-Mass-Framework.
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BibTeX
@misc{xu2026beyond,
title = {{Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference}},
author = {Hanli Xu and Fengxiang He and Sarat Moka},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.27090},
eprint = {2606.27090},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.27090},
}
RIS
TY - UNPB
TI - Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference
AU - Hanli Xu
AU - Fengxiang He
AU - Sarat Moka
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2606.27090
UR - https://arxiv.org/abs/2606.27090
AB - Global objectives, such as KL divergence and ELBO, are widely used in Bayesian inference for measuring distributional discrepancy. This paper studies distributional ``local-mass behaviours'' that are not directly captured by such global 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, and show that the directional comparison remains visible in the variational posteriors of Bayesian neural networks up to ResNet-50 on ImageNet. Code is available at https://github.com/Forsythia0604/Local-Mass-Framework.
ER -