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
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 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 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 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.
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.)
Carathéodory's extension theorem?: proven using probability monads (as codensity monads) in Van Belle’22.
Conditional expectations: first expressed categorically in Panangaden and CDPP’09, then in terms of probability monads (as codensity monads) in Van Belle’23 then using partial evaluations in Perrone’18, FP’20 and FGPR’23, and finally in categories of couplings in Ensarguet-Perrone’23 and Perrone-Van Belle’24.
De Finetti's theorem: stated, interpreted and proven in terms of Markov categories with probability monads. Main results in Fritz-Gonda-Perrone’21, see also Moss-Perrone’22 for further context. An analogous result was proven in the category of couplings, Ensarguet-Perrone’23. An additional, independent categorical approach is given in Jacobs-Staton’20.
Ergodic decomposition theorem: proven for deterministic dynamical systems using Markov categories in Moss-Perrone’23, and extended to the stochastic case using Markov categories and categories of couplings in Ensarguet-Perrone’23.
Stochastic versions of Gelfand duality: Furber-Jacobs’15 and Parzygnat’17.
Probabilistic graphical models: a categorical study of Bayesian networks Fong’12, a d-separation criterion for Markov categories Fritz-Klingler’22.
Idempotent completion of BorelStoch: a new measure-theoretic result, with several structural consequences, proven in FGLPS’23.
Kantorovich duality? in terms of probability monads: Perrone’18, Fritz-Perrone’19, Fritz-Perrone’20.
Kolmogorov extension theorem: described in terms of Markov categories in Fritz-Rischel’20, and in terms of probability monads (as codensity monads) in Van Belle’23.
Martingales: Described in terms of probability monads (as codensity monads) in Van Belle’23 and using partial evaluations in Perrone’18. A version of their convergence theorem using the category of couplings is given in Perrone-Van Belle’24.
Multinomial and hypergeometric distributions: described in terms of Markov categories in Jacobs’21.
Radon-Nikodym theorem: proven using probability monads (as codensity monads) in Van Belle’23.
Supports of probability measures: studied in terms of probability monads in Fritz-Perrone-Rezagholi’21 and in terms of Markov categories in FGLPS’23
Urn models, multinomials, and probability on sets, using category theory, including Markov categories: Jacobs’21a Jacobs’21b Jacobs’22, Jacobs-Stein’23, Jacobs’24.
Kolmogorov and Hewitt-savage’s zero-one laws: proven in terms of Markov categories in Fritz-Rischel’20, and in categories of couplings in Ensarguet-Perrone’23.
(…)
Bahadur’s theorem on ancillary statistics: proven in terms of Markov categories in Fritz’20
Basu’s theorem on minimal sufficient statistics?: proven in terms of Markov categories in Fritz’20.
Bayesian inverses: expressed categorically, with proofs about their core properties and their existence, in DDGS’18, Cho-Jacobs’19, Fritz’20, Parzygnat’24, and others.
Bayesian updating?: categorically Jacobs’23, Jacobs-Zanasi’18, and including using Markov categories, Di Lavore-Román’23.
Blackwell-Sherman-Stein theorem on statistical experiments: proven and generalized beyond the discrete case using Markov categories in FGPR’23.
Causal inference: using string diagrams JKZ’18, and using Markov categories Yin’22.
Filtering and smoothing for hidden Markov models: described in terms of Markov categories in Virgo’23, as well as in FKMSW’24.
Fisher-Neyman factorization theorem for sufficient statistics?: proven in terms of Markov categories in Fritz’20.
General results on sufficient statistics?: in terms of Markov categories Fritz’20, and in terms of idempotents Jacobs’23.
Improper priors for Gaussians, using Markov categories: Stein-Samuelson’23.
Machine learning: equivariance of neural networks using Markov categories in Cornish’24.
(…)
Characterizations of relative entropy in the discrete case: Baez-Fritz-Leinster’11, Baez-Fritz’14, Leinster’19, Fullwood-Parzygnat’21. (Quantum case in Parzygnat’22, see below.)
Characterizations of relative entropy on standard Borel spaces: Gagne-Panangaden’18.
Entropy in terms of operads: Bradley’21.
Entropy from the point of view of topos theory: Constantin-Döring’20.
Entropy, relative entropy and data processing inequalities?: expressed in terms of enriched Markov categories in Perrone’24.
A book on the relationship between metrics, entropy and diversity in ecology, with a category-theoretic mindset: Leinster’21.
(…)
Quantum versions of Bayesian inverses: Coecke-Spekkens’12, Parzygnat-Russo’22, Parzygnat-Buscemi’23, Fullwood-Parzygnat’23, Parzygnat’24.
Quantum versions of conditioning and disintegrations: Parzygnat’21, Parzygnat-Russo’23.
Quantum versions of de Finetti's theorem: Staton-Summers’22 and Fritz-Lorenzin’23.
Categorical descriptions of dilations: Parzygnat’19 and GHL’21.
Categorical characterization of von Neumann entropy: Parzygnat’22.
Quantum versions of Markov categories: Parzygnat’20 and Fritz-Lorenzin’23.
(…)
Bayesian updating in probabilistic programming: Jacobs’23.
A cartesian closed category with a probability monad: HKSY’17, SSSW’21.
Stochastic -calculus: AKM’21.
Probability monads for probabilistic computation: Jones-Plotkin’89, Jacobs’18, DKPS’2023.
Probabilistic programming? and random graphs?: AFKKMRSY’24.
Stochastic memoization via monads: Kaddar-Staton’23.
(…)
Foundational results on Markov categories: Golubtsov’99, Cho-Jacobs’19, Fritz’20.
Constructions and analysis of categories of Markov kernels and of transport plans: Lawvere’62, Chentsov’65, Giry’82, Panangaden’99, DDGS’18, Perrone’21, Kozen-Silva-Voogd’23.
Constructions of free Markov and CD categories: Fritz-Liang’23.
Notions of conditional independence as universal properties: Simpson’18, Simpson’24.
Study of joint and marginal distributions for probability monads: Fritz-Perrone’18.
Markov categories for partial maps: Di Lavore-Román’23.
Connections between Markov categories and categories of couplings: Fritz’20 and Ensarguet-Perrone’23.
Connections between Markov categories and probability monads: Golubtsov’02, FGPR’23, Moss-Perrone’22, FGGPS’23.
Connections between positive Markov categories and semicartesian monoidal categories: FGGPS’23.
Construction of point-free probability monads (on locales): Vickers’11.
Results expressing probability monads as codensity monads: Avery’16 and more recently Van Belle’22 with concrete applications to measure theory.
Constructions of probability monads on metric spaces, and connections with metric geometry and Banach spaces: AJK’04, Fritz-Perrone’19, Fritz-Perrone’20.
Constructions of probability monads on topological spaces, and connections with functional analysis: Swirszcz’74, Giry’82, Jones-Plotkin’89, Heckmann’96, Keimel’08, GLJ’19, Fritz-Perrone-Rezagholi’21, Kristel-Peterseim’24.
A monad encoding probabilistic point processes? combining probability and nondeterminism: Dash-Staton’20.
Random variables in terms of sheaves: Simpson’17, Simpson’24.
(…)
(in alphabetical order by author)
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Rob Cornish, Stochastic Neural Network Symmetrization in Markov Categories, 2024. (arXiv)
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Arthur J. Parzygnat, Reversing information flow: retrodiction in semicartesian categories, 2024. (arXiv)
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Tobias Fritz, Tomáš Gonda, Paolo Perrone, Eigil Fjeldgren Rischel, Representable Markov categories and comparison of statistical experiments in categorical probability, Theoretical Computer Science 961, 2023. (arXiv:2010.07416)
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(…more to add…)
Last revised on October 16, 2024 at 16:28:50. See the history of this page for a list of all contributions to it.