We develop the operational semantics of an untyped probabilistic λ-calculus with continuous distributions, and both hard and soft constraints, as a foundation for universal probabilistic programming languages such as CHURCH, ANGLICAN, and VENTURE. Our first contribution is to adapt the classic operational semantics of λ-calculus to a continuous setting via creating a measure space on terms and defining step-indexed approximations. We prove equivalence of big-step and small-step formulations of this distribution-based semantics. To move closer to inference techniques, we also define the sampling-based semantics of a term as a function from a trace of random samples to a value. We show that the distribution induced by integration over the space of traces equals the distribution-based semantics. Our second contribution is to formalize the implementation technique of trace Markov chain Monte Carlo (MCMC) for our calculus and to show its correctness. A key step is defining sufficient conditions for the distribution induced by trace MCMC to converge to the distribution-based semantics. To the best of our knowledge, this is the first rigorous correctness proof for trace MCMC for a higher-order functional language, or for a language with soft constraints.
|Title of host publication||Proceedings of the 21st ACM SIGPLAN International Conference on Functional Programming|
|Place of Publication||Nara, Japan|
|Number of pages||14|
|Publication status||Published - 4 Sep 2016|
|Event||21st ACM SIGPLAN International Conference on Functional Programming - Nara, Japan|
Duration: 18 Sep 2016 → 24 Sep 2016
|Name||ACM SIGPLAN Notices|
|Conference||21st ACM SIGPLAN International Conference on Functional Programming|
|Abbreviated title||ICFP 2016|
|Period||18/09/16 → 24/09/16|
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- School of Informatics - Chair in Computer Security
- Laboratory for Foundations of Computer Science
- Foundations of Computation
Person: Academic: Research Active