Fri 12 Jan 2018 14:55 - 15:20 at Watercourt - Probability Chair(s): Lars Birkedal

We present a modular semantic account of Bayesian inference algorithms for probabilistic programming languages, as used in Bayesian 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. Finally, although not discussed in the paper, we provide an implementation of the concepts described in this paper as a Haskell library.

Fri 12 Jan

13:40 - 15:20: Research Papers - Probability at Watercourt
Chair(s): Lars BirkedalAarhus University
POPL-2018-papers151576080000013:40 - 14:05
Gilles BartheIMDEA Software Institute, Thomas EspitauUniversite Pierre et Marie Curie, Benjamin GregoireINRIA, Justin HsuUniversity College London, Pierre-Yves StrubEcole Polytechnique
POPL-2018-papers151576230000014:05 - 14:30
Aws AlbarghouthiUniversity of Wisconsin-Madison, Justin HsuUniversity College London
POPL-2018-papers151576380000014:30 - 14:55
Thomas EhrhardCNRS and University Paris Diderot, Michele PaganiUniversity Paris Diderot, Christine TassonUniversity Paris Diderot
POPL-2018-papers151576530000014:55 - 15:20
Adam ŚcibiorUniversity of Cambridge and MPI Tuebingen, Ohad KammarUniversity of Oxford, Matthijs VákárUniversity of Oxford, Sam StatonUniversity of Oxford, Hongseok YangUniversity of Oxford, Yufei CaiUniversity of Tuebingen, Klaus OstermannUniversity of Tuebingen, Sean K. MossUniversity of Cambridge, Chris HeunenUniversity of Edinburgh, Zoubin GhahramaniUniversity of Cambridge