In probabilistic modelling, joint distributions are often of more interest than their marginals, but the standard composition of stochastic channels is defined by marginalization. Recently, the notion of 'copy-composition' was introduced in order to circumvent this problem and express the chain rule of the relative entropy fibrationally, but while that goal was achieved, copy-composition lacked a satisfactory origin story. Here, we supply such a story for two standard probabilistic tools: directed and undirected graphical models. We explain that (directed) Bayesian networks may be understood as "stochastic terms" of product type, in which context copy-composition amounts to a pull-push operation. Likewise, we show that (undirected) factor graphs compose by copy-composition. In each case, our construction yields a double fibration of decorated (co)spans. Along the way, we introduce a useful bifibration of measure kernels, to provide semantics for the notion of stochastic term, which allows us to generalize probabilistic modelling from product to dependent types.

翻译：在概率建模中，联合分布通常比其边缘分布更受关注，但随机通道的标准复合是通过边缘化定义的。最近引入的"复制复合"概念旨在规避此问题并以纤维化方式表达相对熵的链式法则，虽然该目标已实现，但复制复合缺乏令人满意的起源解释。本文为两种标准概率工具——有向和无向图模型——提供了这样的解释。我们阐明（有向）贝叶斯网络可被理解为乘积类型的"随机项"，在此语境下复制复合等价于拉回-前推操作。同样，我们证明（无向）因子图通过复制复合进行组合。在每种情况下，我们的构造都产生装饰（余）跨度的双重纤维化。在此过程中，我们引入了一个实用的测度核双纤维化，为随机项概念提供语义支持，这使得我们能够将概率建模从乘积类型推广到依赖类型。