A Gradual Probabilistic Lambda Calculus

2026-04-06

AI Summary

Static type systems in probabilistic programming are often overly conservative, limiting flexibility. This work proposes GPLC—the first probabilistic λ-calculus that supports gradual introduction or removal of both type and probabilistic annotations, seamlessly integrating static and dynamic checking. Leveraging probabilistic coupling theory, the authors formally characterize consistency and precision relationships, and design a corresponding type system together with a semantic refinement mechanism that elaborates programs into a target language, TPLC. GPLC guarantees type safety while satisfying two core criteria of gradual typing: conservative extension over its fully static variant and the gradual guarantee. This represents the first successful application of gradual typing to the domain of probabilistic programming.

Abstract

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.

Problem

• probabilistic programming
• gradual typing
• type systems
• static checking
• dynamic checking

Innovation

• gradual typing
• probabilistic lambda calculus
• probabilistic couplings
• type safety
• gradual guarantee