Formal verification of higher-order probabilistic programs: reasoning about approximation, convergence, Bayesian inference, and optimization

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on foundational semantics issues. In this paper, we take a step further by developing a suite of logics, collectively named PPV for proving properties of programs written in an expressive probabilistic higher-order language with continuous sampling operations and primitives for conditioning distributions. Our logics mimic the comfortable reasoning style of informal proofs using carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show expressiveness by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) ⊤⊤-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs.