We define a probabilistic programming language for Gaussian random variables with a first-class exact conditioning construct. We give operational, denotational and equational semantics for this language, establishing convenient properties like exchangeability of conditions. Conditioning on equality of continuous random variables is nontrivial, as the exact observation may have probability zero; this is Borel's paradox. Using categorical formulations of conditional probability, we show that the good properties of our language are not particular to Gaussians, but can be derived from universal properties, thus generalizing to wider settings. We define the Cond construction, which internalizes conditioning as a morphism, providing general compositional semantics for probabilistic programming with exact conditioning.

中文翻译：

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具有精确条件的概率程序的组合语义

我们为一流的精确条件构造定义了一种针对高斯随机变量的概率编程语言。我们为该语言提供了操作，指称和等式语义，并建立了诸如条件可交换性之类的便利属性。连续随机变量相等的条件并非无关紧要，因为精确的观察可能具有零概率；这是Borel的悖论。使用条件概率的分类表述，我们证明了我们语言的良好属性并不是高斯人所独有的，而是可以从通用属性派生而来的，因此可以推广到更广泛的环境中。我们定义了Cond构造，该构造将条件内部化为态射，为具有精确条件的概率编程提供了一般的组成语义。