揭示条件概率密度构造中的关键差异，澄清了机器学习中常见误解。
Constructive Disintegration and Conditional Modes
- 提出一套数学工具，用于在微分流形上构造条件概率测度。
- 发现受限密度与真实条件密度可能严重不符，导致推理偏差。
- 指出条件极值点与真实条件分布极值不一致，需根据场景选择方法。
条件化是贝叶斯统计的核心操作，形式化为测度的分解（disintegration）。然而由于定义隐含，构造分解常具挑战性。机器学习中一种流行观点将分解构造等同于将概率密度函数限制在与观测一致的子集上。本文提供了一套完整的数学工具，用于在微分流形上构造分解，并给出了一个令人震惊的简单例子：受限密度与分解密度显著偏离。针对近似贝叶斯推断和贝叶斯反问题的应用，我们进一步研究了分解的模式（conditional modes）。结果表明，近期提出的“条件模式”并不等于通过分解得到的条件测度的模式，而是受限测度的模式。我们讨论了两种测度间的差异在实践中的影响，主张根据建模需求选择合适的方法。
原文摘要 · Abstract (English)
Conditioning, the central operation in Bayesian statistics, is formalised by the notion of disintegration of measures. However, due to the implicit nature of their definition, constructing disintegrations is often difficult. A folklore result in machine learning conflates the construction of a disintegration with the restriction of probability density functions onto the subset of events that are consistent with a given observation. We provide a comprehensive set of mathematical tools which can be used to construct disintegrations and apply these to find densities of disintegrations on differentiable manifolds. Using our results, we provide a disturbingly simple example in which the restricted density and the disintegration density drastically disagree. Motivated by applications in approximate Bayesian inference and Bayesian inverse problems, we further study the modes of disintegrations. We show that the recently introduced notion of a "conditional mode" does not coincide in general with the modes of the conditional measure obtained through disintegration, but rather the modes of the restricted measure. We also discuss the implications of the discrepancy between the two measures in practice, advocating for the utility of both approaches depending on the modelling context.
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