Contextual Equivalence for Probabilistic Programs with Continuous Random Variables and Scoring

Culpepper, Ryan; Cobb, Andrew

doi:10.1007/978-3-662-54434-1_14

Cited by 24 publications

(34 citation statements)

References 14 publications

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“…We defineμ(e, −) as a measure over arbitrary syntactic values-not just real numbers as with µ(e, K, −). This measure corresponds directly to the µ e of Culpepper and Cobb [2017] and [[e]] S of Borgström et al [2017]. The definition ofμ uses a generalization of eval from measurable sets of reals (A) to measurable sets of syntactic values (V ).…”

Section: 4mentioning

confidence: 99%

“…9 INFORMALLY RELATED WORK Our language and semantics are based on that of Culpepper and Cobb [2017], but unlike that language, which is simply-typed, ours is untyped and thus has recursion and nonterminating programs. Consequently, our logical relation must use step-indexing rather than type-indexing.…”

Section: From Let-style To Direct-stylementioning

confidence: 99%

“…The construction of our logical relation follows the tutorial of Pitts [2010] on the construction of biorthogonal, step-indexed [Ahmed 2006] logical relations. Instead of termination, we use the program measure as the observable behavior, following Culpepper and Cobb [2017]. But unlike that work, where the meaning of an expression is a measure over arbitrary syntactic values, we define the meaning of an expression and continuation together (representing a whole program) as a measure over the reals.…”

Section: From Let-style To Direct-stylementioning

confidence: 99%

“…But unlike that work, where the meaning of an expression is a measure over arbitrary syntactic values, we define the meaning of an expression and continuation together (representing a whole program) as a measure over the reals. This allows us to avoid the complication of defining a relation on measurable sets of syntactic values [Culpepper and Cobb 2017, the A relation].…”

Section: From Let-style To Direct-stylementioning

confidence: 99%

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Contextual equivalence for a probabilistic language with continuous random variables and recursion

Wand

Culpepper

Giannakopoulos

et al. 2018

Proc. ACM Program. Lang.

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We present a complete reasoning principle for contextual equivalence in an untyped probabilistic language. The language includes continuous (real-valued) random variables, conditionals, and scoring. It also includes recursion, since the standard call-by-value fixpoint combinator is expressible.We demonstrate the usability of our characterization by proving several equivalence schemas, including familiar facts from lambda calculus as well as results specific to probabilistic programming. In particular, we use it to prove that reordering the random draws in a probabilistic program preserves contextual equivalence. This allows us to show, for example, that (let x = e 1 in let = e 2 in e 0 ) = ctx (let = e 2 in let x = e 1 in e 0 ) (provided x does not occur free in e 2 and does not occur free in e 1 ) despite the fact that e 1 and e 2 may have sampling and scoring effects.

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Section: 4mentioning

confidence: 99%

Section: From Let-style To Direct-stylementioning

confidence: 99%

Section: From Let-style To Direct-stylementioning

confidence: 99%

Section: From Let-style To Direct-stylementioning

confidence: 99%

See 2 more Smart Citations

Contextual equivalence for a probabilistic language with continuous random variables and recursion

Wand

Culpepper

Giannakopoulos

et al. 2018

Proc. ACM Program. Lang.

Self Cite

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“…Several recent papers have started to look at this. For instance, Staton [2017] and Culpepper and Cobb [2017] have recently proposed equational methods for proving equivalences between higher-order probabilistic programs. Culpepper and Cobb [2017] Formal Verification of Higher-Order Probabilistic Programs 38:3…”

Section: Introductionmentioning

confidence: 99%

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

Sato

Aguirre

Barthe

et al. 2019

Proc. ACM Program. Lang.

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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.Pr z∼y [(π 1 (z)<.5∨π 2 (z)>.5)∧(π 1 (z)>.5)]Pr z∼y [π 1 (z)<.5∨π 2 (z)>.5] and this can be proved by applying the [Bayes] rule above, concluding the proof. We saw different components of PPVat work here: unary rules, subtyping, and a special rule for query. All these components can be assembled in more complex examples, as we show in Section 8.Monte Carlo Approximation. As a second example, we show how to use PPV to reason about other classical applications that do not use observations. We consider reasoning about expected value and variance of distributions. Concretely, we show convergence in probability of an implementation of the naive Monte Carlo approximation. This algorithm considers a distribution d and a real-valued function h, and tries to approximate the expected value of h(x) where x is sampled from d by sampling a number i of values from d and computing their mean.Consider the following implementation of Monte Carlo approximation:Our goal is to prove the convergence in probability of this algorithm, that is, the result can be made as accurate as desired by increasing the sample size (denoted by i above and n below). This is 1 We introduce the rule here to give some intuition, but this is also discussed in Section 6 after introducing PPV.

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Effectful Normal Form Bisimulation

Lago

Gavazzo

2019

Programming Languages and Systems

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Normal form bisimulation, also known as open bisimulation, is a coinductive technique for higher-order program equivalence in which programs are compared by looking at their essentially infinitary tree-like normal forms, i.e. at their Böhm or Lévy-Longo trees. The technique has been shown to be useful not only when proving metatheorems about λ-calculi and their semantics, but also when looking at concrete examples of terms. In this paper, we show that there is a way to generalise normal form bisimulation to calculi with algebraic effects,à la Plotkin and Power. We show that some mild conditions on monads and relators, which have already been shown to guarantee effectful applicative bisimilarity to be a congruence relation, are enough to prove that the obtained notion of bisimilarity, which we call effectful normal form bisimilarity, is a congruence relation, and thus sound for contextual equivalence. Additionally, contrary to applicative bisimilarity, normal form bisimilarity allows for enhancements of the bisimulation proof method, hence proving a powerful reasoning principle for effectful programming languages.

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Contextual Equivalence for Probabilistic Programs with Continuous Random Variables and Scoring

Cited by 24 publications

References 14 publications

Contextual equivalence for a probabilistic language with continuous random variables and recursion

Contextual equivalence for a probabilistic language with continuous random variables and recursion

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

Effectful Normal Form Bisimulation

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