从局部质量视角重新审视贝叶斯推断中的分布差异问题
Beyond Global Divergences: A Local-Mass Perspective on Bayesian Inference
- 引入局部质量指数与正则化扩展KL，刻画参数附近质量分布的衰减特性
- 揭示似然函数幂律与参数支撑变化对局部质量的影响机制
- 为小球区域质量比较提供方向性不等式，适合理论研究者深入理解贝叶斯更新
全局目标如KL散度和ELBO在贝叶斯推断中广泛用于度量分布差异，但其未能捕捉局部质量行为。本文提出两种数学工具：(1) 局部质量指数，用于记录局部质量的多项式与对数衰减尺度；(2) 正则化扩展KL（RE-KL），一种可处理奇异成分的局部位散度。局部质量指数揭示了：(1) 幂律似然因子会直接改变局部质量分布，(2) 参数依赖支撑或其平滑形式可通过保留于参数附近的质量总量改变局部尺度。利用局部RE-KL，我们证明了在两个KL方向下局部小球质量的绝对、相对及方向性不等式。这些结果共同构建了局部质量行为的理论框架。实验提供了对局部行为的受控可视化验证。代码已公开于 https://github.com/Forsythia0604/Local-Mass-Framework。
原文摘要 · Abstract (English)
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.
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