A Domain-Theoretic Approach to Statistical Programming Languages

Abstract

We give a domain-theoretic semantics to a statistical programming language, using the plain old category of dcpos, in contrast to some more sophisticated recent proposals. Remarkably, our monad of minimal valuations is commutative, which allows for program transformations that permute the order of independent random draws, as one would expect. A similar property is not known for Jones and Plotkin' s monad of continuous valuations. Instead of working with true real numbers, we work with exact real arithmetic, providing a bridge towards possible implementations. (Implementations by themselves are not addressed here.) Rather remarkably, we show that restricting ourselves to minimal valuations does not restrict us much: all measures on the real line can be modeled by minimal valuations on the domain IR ⊥ of exact real arithmetic. We give three operational semantics for our language, and we show that they are all adequate with respect to the denotational semantics. We also explore quite a few examples in order to demonstrate that our semantics computes exactly as one would expect, and in order to debunk the myth that a semantics based on continuous maps would not be expressive enough to encode measures with non-compact support using only measures with compact support, or to encode measures via non-continuous density functions, for instance.

Central Submonads and Notions of Computation

“…Multiple experts in the field believe that it is not commutative 6 . More recently, progress has been made in identifying commutative submonads of V on the category DCPO [Goubault-Larrecq et al 2021;Jia et al 2021a,b]. The difficulty in (dis)proving the commutativity of V boils down to (dis)proving the following Fubini-style equation…”

“…In the above equation, the notion of integration is given by the valuation integral (see [Jones 1990] for more information). Several different submonads of V are constructed in [Goubault-Larrecq et al 2021;Jia et al 2021a] by using topological methods that allow the authors to restrict the space of valuations to particularly nice ones where the above Fubini-style equation may be shown to hold.…”

“…Identifying commutative submonads of a given strong monad has been done elsewhere, e.g., [Goubault-Larrecq et al 2021;Vákár et al 2019]. The cited works do not produce the central submonad of the ambient one, but rather a different commutative submonad.…”