A Cartesian Closed Category for Random Variables

Abbas Edalat

Abstract:

We present a novel, yet rather simple construction within the tradi-
tional framework of Scott domains to provide semantics to proba-
bilistic programming, thus obtaining a solution to a long-standing
open problem in this area. We work with the Scott domain of ran-
dom 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 proba-
bility 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 re-
sult 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 do-
mains by a partial equivalence relation to capture the equivalence
of random variables on these domains. The constructor of the do-
main 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.