Correctness of Sequential Monte Carlo Inference for Probabilistic Programming Languages

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.

Automatic Alignment in Higher-Order Probabilistic Programming Languages

“…For SMC, a standard inference approach is to resample at all likelihood updates [14,48]. This approach produces correct results asymptotically [24] but is highly problematic for certain models [39]. Such models require non-trivial and SMC-specific manual program rewrites to force good resampling locations and make SMC tractable.…”

“…Section 7 describes the evaluation and discusses its results. The paper also has an accompanying artifact that supports the evaluation [26]. Section 8 discusses related work and Section 9 concludes.…”

“…Section 3 introduces syntax and semantics for the calculus used to formalize the alignment analysis. An extended version of the paper is also available at arXiv [27]. We use the symbol † in the text to indicate that more information (e.g., proofs) is available in the extended version.…”

Compiling Universal Probabilistic Programming Languages with Efficient Parallel Sequential Monte Carlo Inference

“…Note that non-identical initial states in Algorithm 1 imply that different particles may traverse the blocks in B differently, and reach checkpoints at different times. Although this means that different particles can be at different blocks concurrently, the SMC algorithm is still correct [24]. This is an essential property of PCFGs that allows for the encoding of universal probabilistic programs in PCFG-based PPLs.…”

A Framework for Large Scale Particle Filters Validated with Data Assimilation for Weather Simulation

“…Particle filters are used for several purposes, like Data Assimilation (DA) [1], probabilistic programming [2,3,4], neural network optimization [5], localization and navigation [6]. Particle filters stands by their ability to work with nonlinear and/or non-Gaussian state space models in opposition to technics like Ensemble Kalman Filtering (EnKF).…”