Abstract
Probabilistic programming is an approach to reasoning under uncertainty by encoding inference problems as programs. In order to solve these inference problems, probabilistic programming languages (PPLs) employ different inference algorithms, such as sequential Monte Carlo (SMC), Markov chain Monte Carlo (MCMC), or variational methods. Existing research on such algorithms mainly concerns their implementation and efficiency, rather than the correctness of the algorithms themselves when applied in the context of expressive PPLs. To remedy this, we give a correctness proof for SMC methods in the context of an expressive PPL calculus, representative of popular PPLs such as WebPPL, Anglican, and Birch. Previous work have studied correctness of MCMC using an operational semantics, and correctness of SMC and MCMC in a denotational setting without term recursion. However, for SMC inference—one of the most commonly used algorithms in PPLs as of today—no formal correctness proof exists in an operational setting. In particular, an open question is if the resample locations in a probabilistic program affects the correctness of SMC. We solve this fundamental problem, and make four novel contributions: (i) we extend an untyped PPL lambda calculus and operational semantics to include explicit resample terms, expressing synchronization points in SMC inference; (ii) we prove, for the first time, that subject to mild restrictions, any placement of the explicit resample terms is valid for a generic form of SMC inference; (iii) as a result of (ii), our calculus benefits from classic results from the SMC literature: a law of large numbers and an unbiased estimate of the model evidence; and (iv) we formalize the bootstrap particle filter for the calculus and discuss how our results can be further extended to other SMC algorithms.
Highlights
Probabilistic programming is a programming paradigm for probabilistic models, encompassing a wide range of programming languages, libraries, and platforms [5,13,14,25,32,37,38]
(i) We extend the calculus from Borgstrom et al [3] to include explicit resample terms, expressing explicit synchronization points for performing resampling in sequential Monte Carlo (SMC)
This is a powerful result, since it implies that when implementing SMC for probabilistic programming languages (PPLs), any method for selecting resampling locations in a program is correct under mild conditions (Theorem 1 or Theorem 2) that are most often, if not always, fulfilled in practice
Summary
Probabilistic programming is a programming paradigm for probabilistic models, encompassing a wide range of programming languages, libraries, and platforms [5,13,14,25,32,37,38]. (i) We extend the calculus from Borgstrom et al [3] to include explicit resample terms, expressing explicit synchronization points for performing resampling in SMC With this extension, we define a semantics which limits the number of evaluated resample terms, laying the foundation for the remaining contributions. This includes both a generic SMC algorithm, and two standard correctness results from the SMC literature: a law of large numbers [6], and the unbiasedness of the likelihood estimate [26]. Their approach is both general and compositional in the different inference transformations, while we abstract over parts of the SMC algorithm This allows us, in particular, to relate directly to standard SMC correctness results. The lemmas proved in the extended version are explicitly marked with †
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