This paper makes mathematically precise the idea that conditional probabilities are analogous to path liftings in geometry. The idea of lifting is modelled in terms of the category-theoretic concept of a lens, which can be interpreted as a consistent choice of arrow liftings. The category we study is the one of probability measures over a given standard Borel space, with morphisms given by the couplings, or transport plans. The geometrical picture is even more apparent once we equip the arrows of the category with weights, which one can interpret as "lengths" or "costs", forming a so-called weighted category, which unifies several concepts of category theory and metric geometry. Indeed, we show that the weighted version of a lens is tightly connected to the notion of submetry in geometry. Every weighted category gives rise to a pseudo-quasimetric space via optimization over the arrows. In particular, Wasserstein spaces can be obtained from the weighted categories of probability measures and their couplings, with the weight of a coupling given by its cost. In this case, conditionals allow one to form weighted lenses, which one can interpret as "lifting transport plans, while preserving their cost".

翻译：本文从数学上精确阐述了条件概率类似于几何中的路径提升这一思想。提升的概念通过范畴论中的透镜结构来建模，该结构可解释为箭头提升的一致选择。我们研究的范畴是给定标准博雷尔空间上的概率测度，其态射由耦合或传输计划给出。一旦我们赋予范畴中的箭头以权重，几何图景就更加明显：这些权重可被解释为"长度"或"成本"，从而形成所谓的加权范畴，它统一了范畴论和度量几何中的若干概念。事实上，我们证明了加权透镜与几何中子度量概念紧密相关。每个加权范畴通过对箭头进行优化而产生一个伪拟度量空间。特别地，Wasserstein空间可以从概率测度及其耦合的加权范畴中获得，其中耦合的权重由其成本给定。在这种情况下，条件概率使得我们可以构造加权透镜，这可以解释为"在保持成本的同时提升传输计划"。