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A Category for Unifying Gaussian Probability and Nondeterminism

Authors Dario Stein, Richard Samuelson

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ACM Subject Classification
  • Theory of computation → Probabilistic computation
  • Theory of computation → Categorical semantics
  • Mathematics of computing → Probability and statistics
Keywords
  • systems theory
  • hypergraph categories
  • Bayesian inference
  • category theory
  • Markov categories

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Abstract

We introduce categories of extended Gaussian maps and Gaussian relations which unify Gaussian probability distributions with relational nondeterminism in the form of linear relations. Both have crucial and well-understood applications in statistics, engineering, and control theory, but combining them in a single formalism is challenging. It enables us to rigorously describe a variety of phenomena like noisy physical laws, Willems' theory of open systems and uninformative priors in Bayesian statistics. The core idea is to formally admit vector subspaces D ⊆ X as generalized uniform probability distribution. Our formalism represents a first bridge between the literature on categorical systems theory (signal-flow diagrams, linear relations, hypergraph categories) and notions of probability theory.

Cite As Get BibTex

Dario Stein and Richard Samuelson. A Category for Unifying Gaussian Probability and Nondeterminism. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 13:1-13:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CALCO.2023.13

Author Details

Dario Stein
  • iHub, Radboud University Nijmegen, The Netherlands
Richard Samuelson
  • Humming Inc., Tacoma, WA, USA

Acknowledgements

It has been useful to discuss this work with many people. Particular thanks go to Tobias Fritz, Bart Jacobs, Dusko Pavlovic, Sam Staton and Alexander Terenin.

References

  1. John C. Baez, Brandon Coya, and Franciscus Rebro. Props in network theory, 2018. URL: https://arxiv.org/abs/1707.08321.
  2. John C. Baez and Jason Erbele. Categories in control. Theory Appl. Categ., 30:836-881, 2015. URL: https://arxiv.org/abs/1405.6881.
  3. Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. A categorical semantics of signal flow graphs. In CONCUR 2014-Concurrency Theory: 25th International Conference, CONCUR 2014, Rome, Italy, September 2-5, 2014. Proceedings 25, pages 435-450. Springer, 2014. Google Scholar
  4. Filippo Bonchi, Pawel Sobocinski, and Fabio Zanasi. The calculus of signal flow diagrams I: linear relations on streams. Inform. Comput., 252, 2017. Google Scholar
  5. Filippo Bonchi, Paweł Sobociński, and Fabio Zanasi. Interacting Hopf algebras. Journal of Pure and Applied Algebra, 221(1):144-184, 2017. Google Scholar
  6. A. Carboni and R.F.C. Walters. Cartesian bicategories i. Journal of Pure and Applied Algebra, 49(1):11-32, 1987. URL: https://doi.org/10.1016/0022-4049(87)90121-6.
  7. Kenta Cho and B. Jacobs. Disintegration and Bayesian inversion via string diagrams. Mathematical Structures in Computer Science, 29:938-971, 2019. Google Scholar
  8. J Robin B Cockett and Stephen Lack. Restriction categories i: categories of partial maps. Theoretical computer science, 270(1-2):223-259, 2002. Google Scholar
  9. Brendan Fong. Decorated cospans. arXiv preprint arXiv:1502.00872, 2015. Google Scholar
  10. Brendan Fong. The Algebra of Open and Interconnected Systems. PhD thesis, University of Oxford, September 2016. Google Scholar
  11. Brendan Fong. Decorated corelations, 2017. URL: https://arxiv.org/abs/1703.09888.
  12. Brendan Fong, Paweł Sobociński, and Paolo Rapisarda. A categorical approach to open and interconnected dynamical systems. In Proceedings of the 31st annual ACM/IEEE symposium on Logic in Computer Science, pages 495-504, 2016. Google Scholar
  13. Brendan Fong and David I. Spivak. Hypergraph categories. ArXiv, abs/1806.08304, 2019. Google Scholar
  14. Brendan Fong and David I Spivak. An invitation to applied category theory: seven sketches in compositionality. Cambridge University Press, 2019. Google Scholar
  15. T. Fritz. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Adv. Math., 370, 2020. Google Scholar
  16. T. Fritz and Paolo Perrone. A probability monad as the colimit of spaces of finite samples. arXiv: Probability, 2017. Google Scholar
  17. Tobias Fritz, Tomáš Gonda, and Paolo Perrone. De Finetti’s theorem in categorical probability, 2021. URL: https://arxiv.org/abs/2105.02639.
  18. Tobias Fritz and Eigil Fjeldgren Rischel. Infinite products and zero-one laws in categorical probability. Compositionality, 2, August 2020. URL: https://doi.org/10.32408/compositionality-2-3.
  19. Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin. Bayesian Data Analysis. Chapman and Hall/CRC, 2nd ed. edition, 2004. Google Scholar
  20. Noah D Goodman, Joshua B. Tenenbaum, and The ProbMods Contributors. Probabilistic Models of Cognition. http://probmods.org, 2016. Accessed: 2021-3-26.
  21. Alexandre Goy and Daniela Petrişan. Combining probabilistic and non-deterministic choice via weak distributive laws. In Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '20, New York, NY, USA, 2020. Association for Computing Machinery. URL: https://doi.org/10.1145/3373718.3394795.
  22. Jakob Nebeling Hedegaard. Gaussian random fields - infinite, improper and intrinsic. Master’s thesis, Aalborg University, 2019. Google Scholar
  23. A.T. JAMES. The variance information manifold and the functions on it. In Multivariate Analysis–III, pages 157-169. Academic Press, 1973. URL: https://doi.org/10.1016/B978-0-12-426653-7.50016-8.
  24. Steffen Lauritzen and Frank Jensen. Stable local computation with conditional Gaussian distributions. Statistics and Computing, 11, November 1999. URL: https://doi.org/10.1023/A:1008935617754.
  25. Elena Di Lavore and Mario Román. Evidential decision theory via partial markov categories, 2023. URL: https://arxiv.org/abs/2301.12989.
  26. Joe Moeller and Christina Vasilakopoulou. Monoidal Grothendieck construction. arXiv preprint arXiv:1809.00727, 2018. Google Scholar
  27. SEAN MOSS and PAOLO PERRONE. A category-theoretic proof of the ergodic decomposition theorem. Ergodic Theory and Dynamical Systems, pages 1-27, 2023. URL: https://doi.org/10.1017/etds.2023.6.
  28. João Paixão, Lucas Rufino, and Paweł Sobociński. High-level axioms for graphical linear algebra. Science of Computer Programming, 218:102791, 2022. URL: https://doi.org/10.1016/j.scico.2022.102791.
  29. Sam Staton. Commutative semantics for probabilistic programming. In Hongseok Yang, editor, Programming Languages and Systems, pages 855-879, Berlin, Heidelberg, 2017. Springer Berlin Heidelberg. Google Scholar
  30. Dario Stein. GaussianInfer. https://github.com/damast93/GaussianInfer, 2021. Google Scholar
  31. Dario Stein. Structural Foundations for Probabilistic Programming Languages. PhD thesis, University of Oxford, 2021. Google Scholar
  32. Dario Stein. Decorated linear relations: Extending gaussian probability with uninformative priors, 2022. URL: https://arxiv.org/abs/2204.14024.
  33. Dario Stein and Sam Staton. Compositional semantics for probabilistic programs with exact conditioning (long version). In Proceedings of Thirty-Sixth Annual ACM/IEEE Conference on Logic in Computer Science (LICS 2021), 2021. Google Scholar
  34. Dario Stein and Sam Staton. Compositional semantics for probabilistic programs with exact conditioning (long version), 2021. URL: https://arxiv.org/abs/2101.11351.
  35. Alexander Terenin. Gaussian Processes and Statistical Decision-making in Non-Euclidean spaces. PhD thesis, Imperial College London, 2022. Google Scholar
  36. Jan-Willem van de Meent, Brooks Paige, Hongseok Yang, and Frank Wood. An introduction to probabilistic programming, 2018. URL: https://arxiv.org/abs/1809.10756.
  37. Jan C. Willems. Constrained probability. In 2012 IEEE International Symposium on Information Theory Proceedings, pages 1049-1053, 2012. URL: https://doi.org/10.1109/ISIT.2012.6283011.
  38. Jan C. Willems. Open stochastic systems. IEEE Transactions on Automatic Control, 58(2):406-421, 2013. URL: https://doi.org/10.1109/TAC.2012.2210836.
  39. Maaike Zwart and Dan Marsden. No-go theorems for distributive laws. In Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS '19. IEEE Press, 2019. Google Scholar
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