Compositional Semantics for Probabilistic Programs with Exact Conditioning

Stein, Dario; Staton, Sam

doi:10.1109/lics52264.2021.9470552

Cited by 13 publications

(10 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We gave a construction to combine probability and nondeterminism on vector spaces, which is generally not possible in a seamless way [11,24]. The original motivation for this work comes from programming language theory and the semantics of probabilistic programming [22]. While in statistics literature, noisy observations are seen as the primary operation of interest, our focus on singular covariances is natural from a logical or programming perspective: Copying and conditioning are fundamental operations, however they do lead to highly singular distributions.…”

Section: Outlook and Discussionmentioning

confidence: 99%

“…We believe that GaussEx solves the outstanding characterization of Cond(Gauss) in [22]. The category Cond(GaussEx) can express both logical (solving linear equations) and probabilistic computation (conditioning Gaussians).…”

Section: Outlook and Discussionmentioning

confidence: 99%

“…Conditioning on equality: In the context of Gaussian probability, we can condition two random variables U, V to be exactly equal [22] by introducing an auxiliary variable Z = U − V for their difference, and computing the conditional distribution (U, V )|Z = 0. For example, if U, V ∼ N (0, 1) are independent standard normal variables, then the conditional distribution U |Z = 0 is N (0, 0.5).…”

Section: Conditioningmentioning

confidence: 99%

“…The categorical framework has a canonical counterpart in the compositional semantics of probabilistic programs [19,21]. In [22], we treat exact conditioning of Gaussian random variables from a programming-languages viewpoint. The study of uninformative priors is very natural in this context, because it asks for a unit for the conditioning operation.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Decorated Linear Relations: Extending Gaussian Probability with Uninformative Priors

Stein¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce extended Gaussian distributions as a precise and principled way of combining Gaussian probability uninformative priors, which indicate complete absence of information. To give an extended Gaussian distribution on a finite-dimensional vector space X is to give a subspace D, along which no information is known, together with a Gaussian distribution on the quotient X/D. We show that the class of extended Gaussians remains closed under taking conditional distributions. We then introduce decorated linear maps and relations as a general framework to combine probability with nondeterminism on vector spaces, which includes extended Gaussians as a special case. This enables us to apply methods from categorical logic to probability, and make connections to the semantics of probabilistic programs with exact conditioning.

show abstract

Section: Outlook and Discussionmentioning

confidence: 99%

Section: Outlook and Discussionmentioning

confidence: 99%

Section: Conditioningmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Decorated Linear Relations: Extending Gaussian Probability with Uninformative Priors

Stein¹

2022

Preprint

Self Cite

View full text Add to dashboard Cite

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

View full text Add to dashboard Cite

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.

show abstract