Paradoxes of probabilistic programming: and how to condition on events of measure zero with infinitesimal probabilities

Jacobs, Jules

doi:10.1145/3434339

Cited by 7 publications

(3 citation statements)

References 12 publications

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Section: Related Workmentioning

confidence: 99%

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Correctness of Sequential Monte Carlo Inference for Probabilistic Programming Languages

Lundén¹,

Borgström²,

Broman³

2021

Preprint

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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.

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Section: Related Workmentioning

confidence: 99%

“…Other work related to ours include Jacobs [17], Vákár et al [39], and Staton et al [35]. Jacobs [17] discusses problems with models in which observe (related to weight) statements occur conditionally. While our results show that SMC inference for such models is correct, the models themselves may not be useful.…”

Section: Related Workmentioning

confidence: 99%

Correctness of Sequential Monte Carlo Inference for Probabilistic Programming Languages

Lundén¹,

Borgström²,

Broman³

2021

Preprint

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show abstract

“…Unlike traditional foundations for probability in measurable spaces, they are wellsuited to higher-order data. • While naive handling of conditional probabilities can lead to paradoxes [20], it was shown in [33] that in more restrictive models of probability 'exact conditioning' can be given a consistent meaning. Fritz's Markov categories [6] were used to formulate the result.…”

Section: Introductionmentioning

confidence: 99%

Probability monads with submonads of deterministic states - Extended version

Moss¹,

Perrone²

2022

Preprint

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Probability theory can be studied synthetically as the computational effect embodied by a commutative monad. In the recently proposed Markov categories, one works with an abstraction of the Kleisli category and then defines deterministic morphisms equationally in terms of copying and discarding. The resulting difference between 'pure' and 'deterministic' leads us to investigate the 'sober' objects for a probability monad, for which the two concepts coincide. We propose natural conditions on a probability monad which allow us to identify the sober objects and define an idempotent sobrification functor. Our framework applies to many examples of interest, including the Giry monad on measurable spaces, and allows us to sharpen a previously given version of de Finetti's theorem for Markov categories. This is an extended version of the paper accepted for the Logic In Computer Science (LICS) conference 2022. In this document we include more mathematical details, including all the proofs, of the statements and constructions given in the published version.About citing this work. All the definitions, propositions, and theorems appearing in the published version also appear here, with the same numbering as in the published version. There is one result here, Lemma 3.18, not present in the published version. The numbering of particular equations is however inevitably different between the two versions. Because of this, if future readers need to refer to any of the equations contained here, we recommend them to refer to the corresponding definition or theorem instead.

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Correctness of Sequential Monte Carlo Inference for Probabilistic Programming Languages

Lundén

Borgström

Broman

2021

Programming Languages and Systems

View full text Add to dashboard Cite

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.

show abstract

Paradoxes of probabilistic programming: and how to condition on events of measure zero with infinitesimal probabilities

Cited by 7 publications

References 12 publications

Correctness of Sequential Monte Carlo Inference for Probabilistic Programming Languages

Correctness of Sequential Monte Carlo Inference for Probabilistic Programming Languages

Probability monads with submonads of deterministic states - Extended version

Correctness of Sequential Monte Carlo Inference for Probabilistic Programming Languages

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