{"id":606096,"date":"2019-08-29T06:37:05","date_gmt":"2019-08-29T13:37:05","guid":{"rendered":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/?post_type=msr-research-item&#038;p=606096"},"modified":"2023-02-21T03:47:24","modified_gmt":"2023-02-21T11:47:24","slug":"a-lambda-calculus-foundation-for-universal-probabilistic-programming","status":"publish","type":"msr-research-item","link":"https:\/\/new-cm-edgedigital.pages.dev\/en-us\/research\/publication\/a-lambda-calculus-foundation-for-universal-probabilistic-programming\/","title":{"rendered":"A lambda-calculus foundation for universal probabilistic programming"},"content":{"rendered":"<p>We develop the operational semantics of an untyped probabilistic \u03bb-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\u00a0\u03bb-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 <em>distribution-based semantics<\/em>. To move closer to inference techniques, we also define the <em>sampling-based semantics<\/em> 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 <em>Markov chain Monte Carlo<\/em> (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.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>We develop the operational semantics of an untyped probabilistic \u03bb-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\u00a0\u03bb-calculus to a continuous setting via creating a measure space on terms and [&hellip;]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"","msr_publisher_other":"","msr_booktitle":"","msr_chapter":"","msr_edition":"","msr_editors":"","msr_how_published":"","msr_isbn":"","msr_issue":"","msr_journal":"","msr_number":"","msr_organization":"ACM","msr_pages_string":"","msr_page_range_start":"33","msr_page_range_end":"46","msr_series":"","msr_volume":"","msr_copyright":"","msr_conference_name":"International Conference on Functional 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