We extend the synthetic theories of discrete and Gaussian categorical probability by introducing a diagrammatic calculus for reasoning about hybrid probabilistic models in which continuous random variables, conditioned on discrete ones, follow a multivariate Gaussian distribution. This setting includes important classes of models such as Gaussian mixture models, where each Gaussian component is selected according to a discrete variable. We develop a string diagrammatic syntax for expressing and combining these models, give it a compositional semantics, and equip it with a sound and complete equational theory that characterises when two models represent the same distribution.

翻译：我们通过引入一种图示演算，扩展了离散与高斯范畴概率的合成理论，该演算用于推理一类混合概率模型，其中连续随机变量在给定离散变量的条件下服从多元高斯分布。该设定包含了诸如高斯混合模型等重要模型类别，其中每个高斯分量根据一个离散变量进行选择。我们为表达和组合这些模型开发了一种弦图语法，赋予其组合语义，并为其配备了一套可靠且完备的等式理论，用以刻画两个模型何时表示相同的分布。