论文标题
阁楼中的狮子 - Borel-Kolmogorov悖论的分辨率
The lion in the attic -- A resolution of the Borel--Kolmogorov paradox
论文作者
论文摘要
Borel - Kolmogorov的调理悖论在零概率零事件上，自100年前发现以来，学生和研究人员都着迷于学生和研究人员。经典条件仅相对于阳性概率事件有效。如果我们在此类集合上忽略了此约束和条件，例如，对于连续分布的随机变量$ y $，类型$ \ {y = y \} $的事件，几乎可以选择任何概率度量作为此类集合的条件度量。在概率为零的子集上，在调理的环境中，有许多描述和解释。但是，大多数治疗方法无法提供有关如何避免这种情况的明确说明。我们建议通过定义使用Hausdorff度量的条件度量的版本来缩小这一差距。这使得典型的选择仅取决于空间的几何形状，从而消除了任何歧义。我们描述了在Borel-Kolmogorov Paradox的背景下产生的一系列可能的措施，并将与规范度量相吻合的措施分类。本手稿的目的是为单一条件概率提供手册：我们给出一个明确的解释，在该解释中，出现（不）出现歧义，以及如何通过规范的选择一劳永逸地摆脱这种歧义。
The Borel--Kolmogorov paradox of conditioning with respect to events of prior probability zero has fascinated students and researchers since its discovery more than 100 years ago. Classical conditioning is only valid with respect to events of positive probability. If we ignore this constraint and condition on such sets, for example events of type $\{Y=y\}$ for a continuously distributed random variable $Y$, almost any probability measure can be chosen as the conditional measure on such sets. There have been numerous descriptions and explanations of the paradox' appearance in the setting of conditioning on a subset of probability zero. However, most treatments don't supply explicit instructions on how to avoid it. We propose to close this gap by defining a version of conditional measure which utilizes the Hausdorff measure. This makes the choice canonical in the sense that it only depends on the geometry of the space, thus removing any ambiguity. We describe the set of possible measures arising in the context of the Borel--Kolmogorov paradox and classify those coinciding with the canonical measure. The objective of this manuscript is to provide a manual for singular conditional probability: We give an explicit explanation in which settings ambiguity arises (and where not) and how to get rid of this ambiguity once and for all by a canonical choice.
