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

2026-06-26 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, extended

Summary

This research introduces a local-mass framework for Bayesian inference, addressing how probability mass behaves near specific parameter values, a detail often missed by global objectives like KL divergence and ELBO. It presents two key mathematical tools: the Mass Index, which quantifies local small-ball behavior through polynomial and logarithmic decay scales, and regularised extended KL (RE-KL), a set-localised divergence capable of handling singular components. The study proves that Bayesian updating preserves local mass scales under regular likelihoods. However, non-regular likelihoods, such as power-log factors or parameter-dependent supports, can explicitly shift these local scales. Crucially, the paper demonstrates that global KL convergence does not guarantee the preservation of local mass behavior. Experiments provide controlled illustrations of these theoretical findings, covering regular mass, depletion, logarithmic corrections, and the directional nature of local RE-KL.

Key takeaway

For AI scientists evaluating or designing Bayesian models, it's critical to recognize that global divergence metrics like KL or ELBO are insufficient for understanding local mass concentration. You should explicitly analyze local small-ball probabilities using tools like the Mass Index and RE-KL to ensure your models accurately capture critical parameter behaviors, particularly near sparse or singular regions. Ignoring this risks misinterpreting model uncertainty or approximation quality.

Key insights

Global divergence metrics fail to capture local probability mass behavior in Bayesian inference.

Principles

• Bayesian updating can shift local mass scales.
• Softening hard constraints may alter local mass.
• KL divergence directionality impacts local mass comparison.

Method

The framework uses Mass Index to quantify local small-ball probability decay and regularised extended KL (RE-KL) as a set-localised divergence to compare local masses, even for singular measures.

In practice

• Use Mass Index to distinguish priors with similar power but different log behavior.
• Be aware that softening hard constraints can change local mass concentration.
• Consider RE-KL directionality when comparing variational approximations.

Topics

• Bayesian Inference
• Local Mass Analysis
• Kullback-Leibler Divergence
• Variational Approximation
• Mass Index
• RE-KL Divergence

Code references

• Forsythia0604/Local-Mass-Framework

Best for: Research Scientist, AI Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.