@incollection{yang_commutative_2017,
address = {Berlin, Heidelberg},
title = {Commutative {Semantics} for {Probabilistic} {Programming}},
volume = {10201},
isbn = {978-3-662-54433-4 978-3-662-54434-1},
url = {http://link.springer.com/10.1007/978-3-662-54434-1_32},
abstract = {We show that a measure-based denotational semantics for probabilistic programming is commutative. The idea underlying probabilistic programming languages (Anglican, Church, Hakaru, ...) is that programs express statistical models as a combination of prior distributions and likelihood of observations. The product of prior and likelihood is an unnormalized posterior distribution, and the inference problem is to ﬁnd the normalizing constant. One common semantic perspective is thus that a probabilistic program is understood as an unnormalized posterior measure, in the sense of measure theory, and the normalizing constant is the measure of the entire semantic domain.},
language = {en},
urldate = {2019-11-23},
booktitle = {Programming {Languages} and {Systems}},
publisher = {Springer Berlin Heidelberg},
author = {Staton, Sam},
editor = {Yang, Hongseok},
year = {2017},
doi = {10.1007/978-3-662-54434-1_32},
note = {ZSCC: NoCitationData[s0] },
keywords = {Bayesianism, Probabilistic programming, Programming language theory, Semantics},
pages = {855--879}
}
@article{heunen_convenient_2017,
title = {A {Convenient} {Category} for {Higher}-{Order} {Probability} {Theory}},
url = {http://arxiv.org/abs/1701.02547},
abstract = {Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.},
urldate = {2019-10-10},
journal = {arXiv:1701.02547 [cs, math]},
author = {Heunen, Chris and Kammar, Ohad and Staton, Sam and Yang, Hongseok},
month = jan,
year = {2017},
note = {arXiv: 1701.02547}
}
@article{staton_semantics_2016,
title = {Semantics for probabilistic programming: higher-order functions, continuous distributions, and soft constraints},
shorttitle = {Semantics for probabilistic programming},
url = {http://arxiv.org/abs/1601.04943},
doi = {10/ggdf97},
abstract = {We study the semantic foundation of expressive probabilistic programming languages, that support higher-order functions, continuous distributions, and soft constraints (such as Anglican, Church, and Venture). We define a metalanguage (an idealised version of Anglican) for probabilistic computation with the above features, develop both operational and denotational semantics, and prove soundness, adequacy, and termination. They involve measure theory, stochastic labelled transition systems, and functor categories, but admit intuitive computational readings, one of which views sampled random variables as dynamically allocated read-only variables. We apply our semantics to validate nontrivial equations underlying the correctness of certain compiler optimisations and inference algorithms such as sequential Monte Carlo simulation. The language enables defining probability distributions on higher-order functions, and we study their properties.},
urldate = {2019-11-23},
journal = {Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16},
author = {Staton, Sam and Yang, Hongseok and Heunen, Chris and Kammar, Ohad and Wood, Frank},
year = {2016},
note = {ZSCC: 0000071
arXiv: 1601.04943},
keywords = {Bayesianism, Probabilistic programming, Programming language theory, Semantics},
pages = {525--534}
}
@article{scibior_denotational_2017,
title = {Denotational validation of higher-order {Bayesian} inference},
volume = {2},
issn = {24751421},
url = {http://arxiv.org/abs/1711.03219},
doi = {10.1145/3158148},
abstract = {We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in data science and machine learning. Sophisticated inference algorithms are often explained in terms of composition of smaller parts. However, neither their theoretical justification nor their implementation reflects this modularity. We show how to conceptualise and analyse such inference algorithms as manipulating intermediate representations of probabilistic programs using higher-order functions and inductive types, and their denotational semantics. Semantic accounts of continuous distributions use measurable spaces. However, our use of higher-order functions presents a substantial technical difficulty: it is impossible to define a measurable space structure over the collection of measurable functions between arbitrary measurable spaces that is compatible with standard operations on those functions, such as function application. We overcome this difficulty using quasi-Borel spaces, a recently proposed mathematical structure that supports both function spaces and continuous distributions. We define a class of semantic structures for representing probabilistic programs, and semantic validity criteria for transformations of these representations in terms of distribution preservation. We develop a collection of building blocks for composing representations. We use these building blocks to validate common inference algorithms such as Sequential Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between the semantic manipulation and its traditional measure theoretic origins, we use Kock's synthetic measure theory. We demonstrate its usefulness by proving a quasi-Borel counterpart to the Metropolis-Hastings-Green theorem.},
number = {POPL},
urldate = {2019-10-10},
journal = {Proceedings of the ACM on Programming Languages},
author = {Ścibior, Adam and Kammar, Ohad and Vákár, Matthijs and Staton, Sam and Yang, Hongseok and Cai, Yufei and Ostermann, Klaus and Moss, Sean K. and Heunen, Chris and Ghahramani, Zoubin},
month = dec,
year = {2017},
note = {arXiv: 1711.03219},
pages = {1--29}
}