research-article

A type theory for probability density functions

Authors:

Ashish Agarwal,

Alexander GrayAuthors Info & Claims

ACM SIGPLAN Notices, Volume 47, Issue 1

Pages 545 - 556

https://doi.org/10.1145/2103621.2103721

Published: 25 January 2012 Publication History

Abstract

There has been great interest in creating probabilistic programming languages to simplify the coding of statistical tasks; however, there still does not exist a formal language that simultaneously provides (1) continuous probability distributions, (2) the ability to naturally express custom probabilistic models, and (3) probability density functions (PDFs). This collection of features is necessary for mechanizing fundamental statistical techniques. We formalize the first probabilistic language that exhibits these features, and it serves as a foundational framework for extending the ideas to more general languages. Particularly novel are our type system for absolutely continuous (AC) distributions (those which permit PDFs) and our PDF calculation procedure, which calculates PDF s for a large class of AC distributions. Our formalization paves the way toward the rigorous encoding of powerful statistical reformulations.

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Cited By

Zhou PSun XZhang LLi M(2024)Enhanced predictive PDF control of stochastic distribution systems with neural network compensation and its applicationJournal of Process Control10.1016/j.jprocont.2024.103328143(103328)Online publication date: Nov-2024
https://doi.org/10.1016/j.jprocont.2024.103328
Chen MKatoen JKlinkenberg LWinkler T(2022)Does a Program Yield the Right Distribution?Computer Aided Verification10.1007/978-3-031-13185-1_5(79-101)Online publication date: 7-Aug-2022
https://dl.acm.org/doi/10.1007/978-3-031-13185-1_5
Lee WYu HRival XYang H(2019)Towards verified stochastic variational inference for probabilistic programsProceedings of the ACM on Programming Languages10.1145/33710844:POPL(1-33)Online publication date: 20-Dec-2019
https://dl.acm.org/doi/10.1145/3371084
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Index Terms

A type theory for probability density functions
1. Mathematics of computing
  1. Probability and statistics
    1. Statistical paradigms
      1. Statistical graphics
2. Theory of computation
  1. Semantics and reasoning
    1. Program reasoning
      1. Program analysis
    2. Program semantics

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Information & Contributors

Information

Published In

cover image ACM SIGPLAN Notices

ACM SIGPLAN Notices Volume 47, Issue 1

POPL '12

January 2012

569 pages

ISSN:0362-1340

EISSN:1558-1160

DOI:10.1145/2103621

Issue’s Table of Contents

POPL '12: Proceedings of the 39th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
January 2012
602 pages
ISBN:9781450310833
DOI:10.1145/2103656
General Chair:
John Field
Google, USA
,
Program Chair:
Michael Hicks
University of Maryland, College Park, USA

Copyright © 2012 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 25 January 2012

Published in SIGPLAN Volume 47, Issue 1

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Cited By

Zhou PSun XZhang LLi M(2024)Enhanced predictive PDF control of stochastic distribution systems with neural network compensation and its applicationJournal of Process Control10.1016/j.jprocont.2024.103328143(103328)Online publication date: Nov-2024
https://doi.org/10.1016/j.jprocont.2024.103328
Chen MKatoen JKlinkenberg LWinkler T(2022)Does a Program Yield the Right Distribution?Computer Aided Verification10.1007/978-3-031-13185-1_5(79-101)Online publication date: 7-Aug-2022
https://dl.acm.org/doi/10.1007/978-3-031-13185-1_5
Lee WYu HRival XYang H(2019)Towards verified stochastic variational inference for probabilistic programsProceedings of the ACM on Programming Languages10.1145/33710844:POPL(1-33)Online publication date: 20-Dec-2019
https://dl.acm.org/doi/10.1145/3371084
Bornholt JMytkowicz TMcKinley KBalasubramonian RDavis AAdve S(2014) Uncertain< T > Proceedings of the 19th international conference on Architectural support for programming languages and operating systems10.1145/2541940.2541958(51-66)Online publication date: 24-Feb-2014
https://doi.org/10.1145/2541940.2541958
Bhat SBorgström JGordon ARusso C(2013)Deriving probability density functions from probabilistic functional programsProceedings of the 19th international conference on Tools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-642-36742-7_35(508-522)Online publication date: 16-Mar-2013
https://dl.acm.org/doi/10.1007/978-3-642-36742-7_35
Klinkenberg LBlumenthal CChen MHaase DKatoen J(2024)Exact Bayesian Inference for Loopy Probabilistic Programs using Generating FunctionsProceedings of the ACM on Programming Languages10.1145/36498448:OOPSLA1(923-953)Online publication date: 29-Apr-2024
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Wang DHoffmann JReps TFreund SYahav E(2021)Sound probabilistic inference via guide typesProceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation10.1145/3453483.3454077(788-803)Online publication date: 19-Jun-2021
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Mak COng CPaquet HWagner D(2021)Densities of Almost Surely Terminating Probabilistic Programs are Differentiable Almost EverywhereProgramming Languages and Systems10.1007/978-3-030-72019-3_16(432-461)Online publication date: 23-Mar-2021
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Wang DKahn DHoffmann J(2020)Raising expectations: automating expected cost analysis with typesProceedings of the ACM on Programming Languages10.1145/34089924:ICFP(1-31)Online publication date: 3-Aug-2020
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Narayanan PShan C(2020)Symbolic Disintegration with a Variety of Base MeasuresACM Transactions on Programming Languages and Systems10.1145/337420842:2(1-60)Online publication date: 19-May-2020
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