Document Open Access Logo

The Beta-Bernoulli process and algebraic effects

Authors Sam Staton, Dario Stein, Hongseok Yang, Nathanael L. Ackerman, Cameron E. Freer, Daniel M. Roy

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2018.141.pdf
  • Filesize: 0.51 MB
  • 15 pages

Document Identifiers

Author Details

Sam Staton
  • Univ. Oxford
Dario Stein
  • Univ. Oxford
Hongseok Yang
  • KAIST
Nathanael L. Ackerman
  • Harvard Univ.
Cameron E. Freer
  • Borelian
Daniel M. Roy
  • Univ. Toronto

Cite AsGet BibTex

Sam Staton, Dario Stein, Hongseok Yang, Nathanael L. Ackerman, Cameron E. Freer, and Daniel M. Roy. The Beta-Bernoulli process and algebraic effects. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 141:1-141:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.141

Abstract

In this paper we use the framework of algebraic effects from programming language theory to analyze the Beta-Bernoulli process, a standard building block in Bayesian models. Our analysis reveals the importance of abstract data types, and two types of program equations, called commutativity and discardability. We develop an equational theory of terms that use the Beta-Bernoulli process, and show that the theory is complete with respect to the measure-theoretic semantics, and also in the syntactic sense of Post. Our analysis has a potential for being generalized to other stochastic processes relevant to Bayesian modelling, yielding new understanding of these processes from the perspective of programming.

Subject Classification

ACM Subject Classification
  • Theory of computation → Probabilistic computation
Keywords
  • Beta-Bernoulli process
  • Algebraic effects
  • Probabilistic programming
  • Exchangeability

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Nathanael L. Ackerman, Cameron E. Freer, and Daniel M. Roy. Exchangeable random primitives. Workshop on Probabilistic Programming Semantics (PPS 2016), 2016. URL: http://pps2016.soic.indiana.edu/files/2015/12/xrp.pdf.
  2. Danel Ahman and Sam Staton. Normalization by evaluation and algebraic effects. In Proc. MFPS 2013, volume 298 of Electron. Notes Theor. Comput. Sci, pages 51-69, 2013. Google Scholar
  3. M. Alves Diniz, L. E. Salasar, and R. Bassi Stern. Positive polynomials on closed boxes. arXiv e-print 1610.01437, 2016. Google Scholar
  4. Ales Bizjak and Lars Birkedal. Step-indexed logical relations for probability. In Proc. FOSSACS 2015, pages 279-294, 2015. Google Scholar
  5. Carsten Führmann. Varieties of effects. In Proc. FOSSACS 2002, pages 144-159, 2002. Google Scholar
  6. Robert Furber and Bart Jacobs. From Kleisli categories to commutative C*-algebras: Probabilistic Gelfand duality. Log. Methods Comput. Sci., 11(2), 2015. URL: http://dx.doi.org/10.2168/LMCS-11(2:5)2015.
  7. Bart Jacobs. From probability monads to commutative effectuses. J. Log. Algebr. Methods Program., 94:200-237, 2018. Google Scholar
  8. Patricia Johann, Alex Simpson, and Janis Voigtländer. A generic operational metatheory for algebraic effects. In Proc. LICS 2010, pages 209-218, 2010. Google Scholar
  9. Ohad Kammar and Gordon D. Plotkin. Algebraic foundations for effect-dependent optimisations. In Proc. POPL 2012, pages 349-360, 2012. Google Scholar
  10. Alexander Kechris. Classical Descriptive Set Theory. Springer, 1995. Google Scholar
  11. Oleg Kiselyov and Chung-Chieh Shan. Probabilistic programming using first-class stores and first-class continuations. In Proc. 2010 ACM SIGPLAN Workshop on ML, 2010. Google Scholar
  12. Anders Kock. Monads on symmetric monoidal closed categories. Arch. Math., 21:1-10, 1970. Google Scholar
  13. Anders Kock. Commutative monads as a theory of distributions. Theory Appl. Categ., 26(4):97-131, 2012. Google Scholar
  14. Dexter Kozen. Semantics of probabilistic programs. J. Comput. System Sci., 22(3):328-350, 1981. Google Scholar
  15. F. E. J. Linton. Autonomous equational categories. J. Math. Mech., 15:637-642, 1966. Google Scholar
  16. Vikash Mansinghka, Daniel Selsam, and Yura Perov. Venture: a higher-order probabilistic programming platform with programmable inference. arXiv e-print 1404.0099, 2014. Google Scholar
  17. Walter D. Neumann. On the quasivariety of convex subsets of affine space. Arch. Math., 21:11-16, 1970. Google Scholar
  18. Gordon Plotkin and John Power. Algebraic operations and generic effects. Appl. Categ. Structures, 11(1):69-94, 2003. Google Scholar
  19. David Pollard. A user’s guide to measure theoretic probability. Cambridge University Press, 2001. Google Scholar
  20. Victoria Powers and Bruce Reznick. Polynomials that are positive on an interval. Trans. Amer. Math. Soc., 352(10):4677-4692, 2000. Google Scholar
  21. Matija Pretnar. The Logic and Handling of Algebraic Effects. PhD thesis, Univ. Edinburgh, 2010. Google Scholar
  22. Anna B. Romanowska and Jonathan D. H. Smith. Modes. World Scientific, 2002. Google Scholar
  23. Davide Sangiorgi and Valeria Vignudelli. Environmental bisimulations for probabilistic higher-order languages. In Proc. POPL 2016, 2016. Google Scholar
  24. Mark J. Schervish. Theory of statistics. Springer, 1995. Google Scholar
  25. Sam Staton. An algebraic presentation of predicate logic. In Proc. FOSSACS 2013, pages 401-417, 2013. Google Scholar
  26. Sam Staton. Instances of computational effects: An algebraic perspective. In Proc. LICS 2013, 2013. Google Scholar
  27. Sam Staton, Dario Stein, Hongseok Yang, Nathanael L. Ackerman, Cameron E. Freer, and Daniel M. Roy. The Beta-Bernoulli process and algebraic effects. arXiv e-print 1802.09598, 2018. Google Scholar
  28. Sam Staton, Hongseok Yang, Nathanael Ackerman, Cameron Freer, and Daniel M. Roy. Exchangeable random processes and data abstraction. Workshop on Probabilistic Programming Semantics (PPS 2017), 2017. URL: https://pps2017.soic.indiana.edu/files/2017/01/staton-yang-ackerman-freer-roy.pdf.
  29. M. H. Stone. Postulates for the barycentric calculus. Ann. Mat. Pura Appl. (4), 29:25-30, 1949. Google Scholar
  30. Yee Whye Teh, Michael I. Jordan, Matthew J. Beal, and David M. Blei. Hierarchical Dirichlet processes. J. Amer. Statist. Assoc., 101(476):1566-1581, 2006. Google Scholar
  31. John von Neumann. Various techniques used in connection with random digits. Nat. Bur. Stand. Appl. Math. Series, 12:36-38, 1951. Google Scholar
  32. Mitchell Wand, Theophilos Giannakopoulos, Andrew Cobb, and Ryan Culpepper. Contextual equivalence for a probabilistic language with continuous random variables and recursion. Workshop on Probabilistic Programming Semantics (PPS 2018), 2018. URL: https://pps2018.soic.indiana.edu/files/2018/01/paper.pdf.
  33. Jeff Wu. Reduced traces and JITing in Church. Master’s thesis, Mass. Inst. of Tech., 2013. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact