Finite-depth higher-order abstract syntax trees for reasoning about probabilistic programs

We define a core calculus for the purpose of investigating reasoning principles of probabilistic programming languages. By using a variation of a technique called higher-order abstract syntax (HOAS), which is common in the implementation of domain-specific languages, the calculus captures the semantics of a stochastic language with observation while being agnostic to the details of its deterministic portions. By remaining agnostic to the non-stochastic portions of the language, this style of semantics enables the discovery of general reasoning principles for the principled manipulation of probabilistic program fragments by programmers, compilers, and analysis tools. This generality allows us to reason about probabilistic program fragments without the need to resort to the underlying measure theory in every instance, by instead enabling reasoning in terms of the core calculus in a way that we believe to be applicable to various surface-level languages.