Marcin Szymczak | Searcharxiv

arXiv2020

Weakest Preexpectation Semantics for Bayesian Inference

We present a semantics of a probabilistic while-language with soft conditioning and continuous distributions which handles programs diverging with positive probability. To this end, we extend the probabilistic guarded command language (pGCL) with draws from continuous distributions and a score operator. The main contribution is an extension of the standard weakest preexpectation semantics to support these constructs. As a sanity check of our semantics, we define an alternative trace-based semantics of the language, and show that the two semantics are equivalent. Various examples illustrate the applicability of the semantics.

cs.PL↗

arXiv2017

A Lambda-Calculus Foundation for Universal Probabilistic Programming

We develop the operational semantics of an untyped probabilistic lambda-calculus with continuous distributions, as a foundation for universal probabilistic programming languages such as Church, Anglican, and Venture. Our first contribution is to adapt the classic operational semantics of lambda-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a term as a function from a trace of random samples to a value. We show that the distribution induced by integrating over all traces equals the distribution-based semantics. Our second contribution is to formalize the implementation technique of trace Markov chain Monte Carlo (MCMC) for our calculus and to show its correctness. A key step is defining sufficient conditions for the distribution induced by trace MCMC to converge to the distribution-based semantics. To the best of our knowledge, this is the first rigorous correctness proof for trace MCMC for a higher-order functional language.

cs.PL↗