A Gradual Probabilistic Lambda Calculus
This work addresses the challenge of balancing static and dynamic checking for developers in probabilistic programming, though it is incremental as it applies gradual typing to a new domain.
The authors tackled the problem of cumbersome or overly conservative static type checking in probabilistic programming languages by introducing GPLC, a gradual source probabilistic lambda calculus that allows programmers to gradually introduce or remove static type and probability annotations, and they established that GPLC is type safe and satisfies key gradual language criteria.
Probabilistic programming languages have recently gained a lot of attention, in particular due to their applications in domains such as machine learning and differential privacy. To establish invariants of interest, many such languages include some form of static checking in the form of type systems. However, adopting such a type discipline can be cumbersome or overly conservative. Gradual typing addresses this problem by supporting a smooth transition between static and dynamic checking, and has been successfully applied for languages with different constructs and type abstractions. Nevertheless, its benefits have never been explored in the context of probabilistic languages. In this work, we present and formalize GPLC, a gradual source probabilistic lambda calculus. GPLC includes a binary probabilistic choice operator and allows programmers to gradually introduce/remove static type -- and probability -- annotations. The static semantics of GPLC heavily relies on the notion of probabilistic couplings, as required for defining several relations, such as consistency, precision, and consistent transitivity. The dynamic semantics of GPLC is given via elaboration to the target language TPLC, which features a distribution-based semantics interpreting programs as probability distributions over final values. Regarding the language metatheory, we establish that TPLC -- and therefore also GPLC -- is type safe and satisfies two of the so-called refined criteria for gradual languages, namely, that it is a conservative extension of a fully static variant and that it satisfies the gradual guarantee, behaving smoothly with respect to type precision.