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 this https URL.

Machine Learning (stat.ML); Artificial Intelligence (cs.AI); Machine Learning (cs.LG)

arXiv:2606.27090 [stat.ML]

(or arXiv:2606.27090v2 [stat.ML] for this version)

https://doi.org/10.48550/arXiv.2606.27090 Focus to learn more arXiv-issued DOI via DataCite

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.