Robustly Parameterised Higher-Order Probabilistic Models

Authors Fredrik Dahlqvist, Vincent Danos, Ilias Garnier



PDF
Thumbnail PDF

File

LIPIcs.CONCUR.2016.23.pdf
  • Filesize: 0.55 MB
  • 15 pages

Document Identifiers

Author Details

Fredrik Dahlqvist
Vincent Danos
Ilias Garnier

Cite AsGet BibTex

Fredrik Dahlqvist, Vincent Danos, and Ilias Garnier. Robustly Parameterised Higher-Order Probabilistic Models. In 27th International Conference on Concurrency Theory (CONCUR 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 59, pp. 23:1-23:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.CONCUR.2016.23

Abstract

We present a method for constructing robustly parameterised families of higher-order probabilistic models. Parameter spaces and models are represented by certain classes of functors in the category of Polish spaces. Maps from parameter spaces to models (parameterisations) are continuous and natural transformations between such functors. Naturality ensures that parameterised models are invariant by change of granularity -- ie that parameterisations are intrinsic. Continuity ensures that models are robust with respect to their parameterisation. Our method allows one to build models from a set of basic functors among which the Giry probabilistic functor, spaces of cadlag trajectories (in continuous and discrete time), multisets and compact powersets. These functors can be combined by guarded composition, product and coproduct. Parameter spaces range over the polynomial closure of Giry-like functors. Thus we obtain a class of robust parameterised models which includes the Dirichlet process, various point processes (random sequences with values in Polish spaces) and other classical objects of probability theory. By extending techniques developed in prior work, we show how to reduce the questions of existence, uniqueness, naturality, and continuity of a parameterised model to combinatorial questions only involving finite spaces.
Keywords
  • Probability
  • category theory
  • Giry monad

Metrics

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

References

  1. I. Beylin and P. Dybjer. Extracting a proof of coherence for monoidal categories from a proof of normalization for monoids. In Types for Proofs and Programs, pages 47-61. Springer Berlin Heidelberg, 1995. Google Scholar
  2. P. Billingsley. Convergence of Probability Measures. Wiley, 1968. Google Scholar
  3. N. Bourbaki. Elements de mathématique. Topologie Générale. Springer, 1971. Google Scholar
  4. F. Dahlqvist, V. Danos, and I. Garnier. Giry and the machine. Feb 2016. To appear in the proceedings of MFPS XXXII. Google Scholar
  5. V. Danos and I. Garnier. Dirichlet is natural. Electronic Notes in Theoretical Computer Science, 319:137 - 164, 2015. The 31st Conference on the Mathematical Foundations of Programming Semantics (MFPS XXXI). Google Scholar
  6. J. Desharnais, V. Gupta, R. Jagadeesan, and P. Panangaden. Metrics for labelled Markov processes. Theoretical Computer Science, 318(3):323-354, 2004. Google Scholar
  7. S.N. Ethier and T.G. Kurtz. Markov processes: characterization and convergence. Probability and mathematical statistics. Wiley, 1986. Google Scholar
  8. B. A. Frigyik, A. Kapila, and M.R. Gupta. Introduction to the Dirichlet distribution and related processes. Technical report, University of Washington., 2010. Google Scholar
  9. M. Giry. A categorical approach to probability theory. In Categorical Aspects of Topology and Analysis, number 915 in Lecture Notes In Math., pages 68-85. Springer-Verlag, 1981. Google Scholar
  10. J. Goubault-Larrecq and K. Keimel. Choquet-Kendall-Matheron theorems for non-Hausdorff spaces. Mathematical Structures in Computer Science, 21(03):511-561, 2011. Google Scholar
  11. P.T. Johnstone. Stone Spaces. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1986. Google Scholar
  12. O. Kallenberg. Probabilistic Symmetries and Invariance Principles. Number v. 10 in Applied probability. Springer, 2005. Google Scholar
  13. A. S. Kechris. Classical descriptive set theory, volume 156 of Graduate Text in Mathematics. Springer, 1995. Google Scholar
  14. K. Keimel and G. Plotkin. Predicate transformers for extended probability and non-determinism. Mathematical Structures in Computer Science, 19(03):501-539, 2009. Google Scholar
  15. S. Mac Lane. Categories for the Working Mathematician. Graduate Texts in Mathematics. Springer, 1998. Google Scholar
  16. R. Mardare, P. Panangaden, and G. Plotkin. Quantitative algebraic reasoning. To appear in LICS 2016, 2016. Google Scholar
  17. K.R. Parthasarathy. Probability Measures on Metric Spaces. AMS Chelsea Publishing Series. Academic Press, 1972. Google Scholar
  18. R. Segala and N. Lynch. Probabilistic simulations for probabilistic processes. Nordic Journal of Computing, 2(2):250-273, 1995. Google Scholar
  19. R. Tix, K. Keimel, and G. Plotkin. Semantic domains for combining probability and non-determinism. Electronic Notes in Th. Comp. Sc., 222:3-99, 2009. Google Scholar
  20. F. van Breugel and J. Worrell. A behavioural pseudometric for probabilistic transition systems. Theoretical Computer Science, 331(1):115-142, 2005. 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