nLab category-theoretic approaches to probability theory

Contents

Contents

Idea

There are a number of approaches to apply category theory to probability and related fields, such as statistics, information theory and dynamical systems.

On one hand, one can study the existing structures in traditional probability theory (such as probability spaces, integration, and so on) using a categorical lens. For instance, the Giry monad models the formation of spaces of probability measures and its iterations, used for example in the context of de Finetti's theorem.

On the other hand, one can try to express certain aspects of probability and statistics synthetically. One looks for structures and axioms which can be thought of as “fundamental” in probability and statistics, and which one can use to prove theorems, without having to use measure theory directly. One then proves that the usual measure-theoretic treatment is a model (or semantics) of this theory. This approach is often called synthetic probability theory, in analogy for example with synthetic differential geometry. One of the most recent approaches to synthetic probability theory is given by Markov categories.

The main end goals of categorical probability are

  • To generalize existing results in probability theory to more general settings, for example with less stringent conditions on countability, separability, etc.;
  • To find new results, which with the traditional methods would have been too complex to prove;
  • To make probability and related fields more accessible to practitioners, thanks to the fact that the formalism incorporates measure theory without requiring any deep knowledge of it.

Main structures of interest

Category theory was first developed to model particular structures in algebraic topology, and subsequently algebraic geometry, algebra, logic and computer science. Each one of these intended applications shaped a piece of the theory, adding to category theory the relevant structures of interest for each application.

The applications of category theory to probability are among the most recent, and are both bringing new categorical structures into the theory (such as Markov categories), as well as repurposing and reinterpreting existing ideas (such as monads).

Markov categories

Markov categories are a recent framework that models categories whose morphisms can be thought of as having randomness, such as stochastic maps and Markov kernels.

It has a graphical formalism which keeps track of the stochastic dependencies, and which can be used to prove theorems in probability purely graphically.

For more details, see Markov category.

Probability monads

Probability monads can be thought of as a way of adding a notion of “randomness” to an existing theory.

A monad often models the idea of “forming spaces of particular structures”, and in probability theory, one is interested in forming spaces of probability measures. Monads are particularly useful when this construction needs to be iterated, for example, when in de Finetti situations one needs to form probability measures over probability measures.

For more details, see probability monad.

Dagger categories

Dagger categories can be thought of as “undirected” categories, where morphisms can be seen as going either way as in an undirected graph.

In probability theory, joint distributions, or transport plans exhibit such a behavior, sometimes called Bayesian inversion. Several probabilistic ideas can be modelled in terms of dagger-categorical concepts, for example, conditional expectation.

For more details, see category of couplings.

Main results

Using category-theoretic methods, several results have been obtained in the past few years.

Firstly, some known concepts and results of probability theory have been given a category-theoretic description. (For example: expressing Kolmogorov's extension theorem as a cofiltered limit condition.) This allows to incorporate the existing theory of probability into the categorical framework, and is the basic starting point for further results. (For example, every time in traditional probability Kolmogorov’s extension theorem is invoked, one now knows that a certain universal property is being used.)

Secondly, some classical results of probability theory have been restated and reproven using category theory. Often this adds new insight into the problem, and allows, for example, to drop further unnecessary assumptions (see the next point). In addition, the category-theoretic formalism often trades higher complexity for higher abstraction. This way, while more abstract, the categorical proofs tend to be simpler than their measure-theoretic counterparts. (And so, they also allow to prove more difficult results more easily.)

Thirdly, and most importantly, new theorems have been proven, as well as generalizations and extensions of old theorems, especially from the discrete to the continuous case.

Here is a partial list, roughly in alphabetical order, roughly divided by subject.

(Work in progress, for more material see also the references below.)

Probability and measure theory

(…)

Statistics

(…)

Information theory

(…)

Quantum probability and information theory

(…)

Probability in computer science

(…)

Structural results of categorical probability

(…)

See also

References

(in alphabetical order by author)

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  • Tobias Fritz, Andreas Klingler, Drew McNeely, Areeb Shah-Mohammed and Yuwen Wang, Hidden Markov models and the Bayes filter in categorical probability, 2024. (arXiv)

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  • Tobias Fritz and Antonio Lorenzin, Involutive Markov categories and the quantum de Finetti theorem, 2023. (arXiv)

  • Tobias Fritz and Paolo Perrone, Stochastic order on metric spaces and the ordered Kantorovich monad, Advances in Mathematics 366, 2020. (arXiv:1808.09898)

  • Tobias Fritz and Paolo Perrone, Monads, partial evaluations, and rewriting, Proceedings of MFPS, 2020. (arXiv)

  • Tobias Fritz and Paolo Perrone, A Probability Monad as the Colimit of Spaces of Finite Samples, Theory and Applications of Categories 34, 2019. (arXiv:1712.05363)

  • Tobias Fritz, Paolo Perrone, Bimonoidal Structure of Probability Monads, Proceedings of MFPS, 2018, (arXiv:1804.03527)

  • Tobias Fritz, Paolo Perrone, Sharwin Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, Mathematical Structures in Computer Science 31(8), 2021. (arXiv:1910.03752)

  • Tobias Fritz and Eigil Fjeldgren Rischel, Infinite products and zero-one laws in categorical probability, Compositionality 2(3) 2020. (arXiv:1912.02769)

  • Nicolas Gagne and Prakash Panangaden, A categorical characterization of relative entropy on standard Borel spaces, ENTCS 336, 2018. (arXiv)

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  • Chris Heunen, Ohad Kammar, Sam Staton and Hongseok Yang, A convenient category for higher-order probability theory, Proceedings of LICS 2017. (arXiv)

  • Bart Jacobs, Drawing from an Urn is Isometric, Proceedings of FoSSaCS, 2024. (pdf)

  • Bart Jacobs, Pearl’s and Jeffrey’s Update as Modes of Learning in Probabilistic Programming (arXiv:2309.07053)

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  • Bart Jacobs, Sufficient statistics and split idempotents in discrete probability theory, Proceedings of MFPS, 2023. (arXiv)

  • Bart Jacobs, Multinomial and Hypergeometric Distributions in Markov Categories, Proceedings of MFPS 2021. (arXiv)

  • Bart Jacobs, From Multisets over Distributions to Distributions over Multisets, Proceedings of LICS, 2021. (arXiv)

  • Bart Jacobs, From probability monads to commutative effectuses, Journal of Logical and Algebraic Methods in Programming 94, 2018.

    (doi:10.1016/j.jlamp.2016.11.006)

  • Bart Jacobs, Categorical aspects of parameter learning, 2018. (arXiv)

  • Bart Jacobs, A Channel-based Exact Inference Algorithm for Bayesian Networks, 2018. (arXiv)

  • Bart Jacobs, A Channel-Based Perspective on Conjugate Priors, 2017. (arXiv:1707.00269)

  • Bart Jacobs, Hyper Normalisation and Conditioning for Discrete Probability Distributions, LMCS 13, 2017. (arXiv)

  • Bart Jacobs, Aleks Kissinger, Fabio Zanasi, Causal Inference by String Diagram Surgery, (arXiv:1811.08338)

  • Bart Jacobs, Sam Staton, De Finetti’s construction as a categorical limit, Proceedings of CMCS, 2021. (arXiv:2003.01964)

  • Bart Jacobs and Dario Stein?, Overdrawing Urns using Categories of Signed Probabilities, Proceedings of Applied Category Theory, EPTCS 397, 2023. (arXiv)

  • Bart Jacobs, Fabio Zanasi, The Logical Essentials of Bayesian Reasoning, (arXiv:1804.01193)

  • Bart Jacobs, Fabio Zanasi, and Octavio Zapata, Bayesian Factorisation as Adjoint, abstract

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  • Dexter Kozen, Kim G. Larsen, Radu Mardare, Prakash Panangaden, Stone duality for Markov processes, Proceedings of LICS, 2013. (pdf)

  • Dexter Kozen, Radu Mardare, Prakash Panangaden, Stone duality for Markov processes, Proceedings of MFCS, 2013. (pdf)

  • Dexter Kozen, Alexandra Silva, Erik Voogd, Joint Distributions in Probabilistic Semantics, MFPS 2023. (arXiv)

  • Bill Lawvere, The category of probabilistic mappings, ms. 12 pages, 1962

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  • Tom Leinster, A short characterization of relative entropy, Journal of Mathematical Physics 60, 2019. (arXiv)

  • Tom Leinster, Entropy and Diversity, Cambridge University Press, 2021.

  • Radu Mardare, Prakash Panangaden, Gordon Plotkin, Free complete Wasserstein algebras, LMCS 14, 2018. (pdf)

  • Radu Mardare, Prakash Panangaden, Gordon Plotkin, Quantitative algebraic reasoning, Proceedings of LICS, 2016. (pdf)

  • Cristina Matache, Sean Moss, Sam Staton, Ariadne Si Suo, Denotational semantics for languages for inference: semirings, monads, and tensors, Proceedings of LAFI, 2023. (arXiv)

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  • Sean Moss and Paolo Perrone, A category-theoretic proof of the ergodic decomposition theorem, Ergodic Theory and Dynamical Systems, 2023. (arXiv:2207.07353)

  • Sean Moss, Paolo Perrone, Probability monads with submonads of deterministic states, LICS 2022. (arXiv:2204.07003)

  • Prakash Panangaden, The category of Markov kernels, ENTCS, 1999. (full text)

  • Prakash Panangaden, Labelled Markov processes, Imperial College Press, 2009.

  • Prakash Panangaden, A categorical view of conditional expectation, (slides)

  • Arthur J. Parzygnat, A functorial characterization of von Neumann entropy, Cahiers de Topologie et de Géométrie Différentielle Catégorique LXIII(1), 2022. (arXiv)

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  • Arthur J. Parzygnat, Discrete probabilistic and algebraic dynamics: a stochastic commutative Gelfand-Naimark Theorem, 2017. (arXiv)

  • Arthur J. Parzygnat, Francesco Buscemi, Axioms for retrodiction: achieving time-reversal symmetry with a prior, Quantum 7(1013), 2023. arXiv

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  • Arthur J. Parzygnat and Benjamin P. Russo, A noncommutative Bayes theorem, Linear Algebra Applications 644, 2022. (arXiv)

  • Paolo Perrone, Markov Categories and Entropy, IEEE Transactions on Information Theory 70(3), 2024. (arXiv)

  • Paolo Perrone, Lifting couplings in Wasserstein spaces, 2021. (arXiv)

  • Paolo Perrone, Categorical Probability and Stochastic Dominance in Metric Spaces, (thesis)

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  • Sam Staton and Ned Summers, Quantum de Finetti Theorems as Categorical Limits, and Limits of State Spaces of C-algebras_, Proceedings of Quantum Physics and Logic, 2022. (arXiv)

  • Dario Stein and Richard Samuelson, A Category for unifying Gaussian Probability and Nondeterminism, Proceedings of CALCO, LIPIcs 10, 2023. (arXiv)

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  • Ruben Van Belle, A categorical treatment of the Radon-Nikodym theorem and martingales, 2023. (arXiv)

  • Ruben Van Belle, A categorical proof of the Carathéodory extension theorem, 2022. (arXiv)

  • Ruben Van Belle, Probability monads as codensity monads, Theory and Applications of Categories, 38(21), 2022. (arXiv)

  • Steve Vickers, A monad of valuation locales, 2011. Link here.

  • Nathaniel Virgo, Unifilar Machines and the Adjoint Structure of Bayesian Filtering, Proceedings of ACT, 2023. (arXiv)

  • Yimu Yin, Jiji Zhang ,_Markov categories, causal theories, and the do-calculus_, 2022. (arXiv)

(…more to add…)

category: probability

Last revised on April 4, 2024 at 19:04:06. See the history of this page for a list of all contributions to it.