We present a novel, yet rather simple construction within the traditional framework of Scott domains to provide semantics to probabilistic programming, thus obtaining a solution to a long-standing open problem in this area. Unlike current main approaches that employ some probability measures or continuous valuations on non-standard or rather complex structures, we use the Scott domain of random variables from a standard sample space -- the unit interval or the Cantor space -- to any given Scott domain. The map taking any such random variable to its corresponding probability distribution provides an effectively given, Scott continuous surjection onto the probabilistic power domain of the underlying Scott domain, establishing a new basic result in classical domain theory. We obtain a Cartesian closed category by enriching the category of Scott domains to capture the equivalence of random variables on these domains. The construction of the domain of random variables on this enriched category forms a strong commutative monad, which is suitable for defining the semantics of probabilistic programming.

翻译：本文在斯科特域的传统框架内提出了一种新颖且相当简单的构造，为概率编程提供语义，从而解决了该领域一个长期存在的开放问题。与当前主流方法在非标准或复杂结构上使用概率测度或连续估值不同，我们采用从标准样本空间（单位区间或康托尔空间）到任意给定斯科特域的随机变量斯科特域。将任何此类随机变量映射到其对应概率分布的映射，提供了对底层斯科特域的概率幂域的有效给定斯科特连续满射，这在经典域理论中建立了一个新的基本结果。通过丰富斯科特域的范畴以捕捉这些域上随机变量的等价性，我们获得了一个笛卡尔闭范畴。在此丰富范畴上随机变量域的构造形成了一个强交换单子，适用于定义概率编程的语义。