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：我们在传统的Scott域框架内提出了一种新颖而简洁的构造，为概率编程提供语义，从而解决了该领域长期存在的开放性问题。与当前主要方法（在非标准或较为复杂的结构上使用概率测度或连续赋值）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Scott域的随机变量所构成的Scott域。将此类随机变量映射至其对应概率分布的映射，构成了一个可有效给出的、Scott连续的满射，该满射覆盖了底层Scott域的概率幂域，从而在经典域理论中建立了一个新的基本结果。通过对Scott域范畴进行富化以捕获这些域上随机变量的等价关系，我们得到了一个笛卡尔闭范畴。在此富化范畴上构建的随机变量域形成了一个强交换单子，适用于定义概率编程的语义。