标题： 构造性分解和条件模式 显示英文标题

标题： Constructive Disintegration and Conditional Modes

摘要： 贝叶斯统计中的核心操作是条件化，它由测度的分解概念来形式化。 然而，由于其定义的隐含性质，构造分解通常很困难。 机器学习中的一个常识性结果将分解的构造与概率密度函数在与给定观测一致的事件子集上的限制混为一谈。 我们提供了一套全面的数学工具，可用于构造分解，并将这些工具应用于找到微分流形上的分解密度。 利用我们的结果，我们提供了一个令人不安的简单例子，其中限制密度和分解密度存在显著差异。 受近似贝叶斯推断和贝叶斯反问题应用的启发，我们进一步研究了分解的众数。 我们证明了最近引入的“条件众数”概念通常并不等于通过分解得到的条件测度的众数，而是等于限制测度的众数。 我们还讨论了这两种测度之间差异的实际影响，倡导根据建模背景的不同，两种方法都有其适用性。 显示英文摘要

摘要： 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.