A Cartesian Closed Category for Random Variables

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. We work with the Scott domain of random variables from a standard and fixed probability 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 a Scott continuous surjection onto the probabilistic power domain of the underlying Scott domain, which preserving canonical basis elements, establishing a new basic result in classical domain theory. If the underlying Scott domain is effectively given, then this map is also computable. We obtain a Cartesian closed category by enriching the category of Scott domains by a partial equivalence relation to capture the equivalence of random variables on these domains. The constructor of the domain of random variables on this category, with the two standard probability spaces, leads to four basic strong commutative monads, suitable for defining the semantics of probabilistic programming.