这篇论文从局部质量角度拆解贝叶斯推理，用Mass Index和RE-KL两个新工具分析了KL散度忽略的局部行为，做变分推断的朋友可以看看。
该论文引入质量指数（Mass Index）记录贝叶斯推断中局部质量的幂和对数衰减尺度，并提出正则化扩展KL散度（RE-KL）以处理奇异成分。作者证明在KL散度的两个方向下，局部小球质量的绝对、相对和方向性不等式。实验提供了受控的局部行为示意图。代码已开源。
Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference
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.