In mathematics, the term categorical probability denotes a collection of
category-theoretic
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
approaches to
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
and related fields such as
statistics,
information theory
Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
and
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
.
The earliest ideas in the field were developed independently by
Lawvere
Francis William Lawvere (; born February 9, 1937) is a mathematician known for his work in category theory, topos theory and the philosophy of mathematics.
Biography
Lawvere studied continuum mechanics as an undergraduate with Clifford Truesdell ...
and by
Chentsov, where they defined a version of what we today call the
category of Markov kernels
In mathematics, the category of Markov kernels, often denoted Stoch, is the category whose objects are measurable spaces and whose morphisms are Markov kernels.
It is analogous to the category of sets and functions, but where the arrows can be ...
, and appeared in 1962 and 1965 respectively.
[N. N. Chentsov, The categories of mathematical statistics, Dokl. Akad. SSSR 164, 1965.]
Some of the most widely used structures in the theory are
*The
category of measurable spaces
In mathematics, the category of measurable spaces, often denoted Meas, is the category whose objects are measurable spaces and whose morphisms are measurable maps.
This is a category because the composition of two measurable maps is again measura ...
;
*
Markov categories Markov (Bulgarian, russian: Марков), Markova, and Markoff are common surnames used in Russia and Bulgaria. Notable people with the name include:
Academics
*Ivana Markova (born 1938), Czechoslovak-British emeritus professor of psychology at t ...
such as the
category of Markov kernels
In mathematics, the category of Markov kernels, often denoted Stoch, is the category whose objects are measurable spaces and whose morphisms are Markov kernels.
It is analogous to the category of sets and functions, but where the arrows can be ...
;
*
Probability monad
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...
s such as
Giry monad Giry may refer to:
People
* Arthur Giry (1848–1899), French historian
* Louis Giry (1596–1665), French lawyer, translator and writer
* Odet-Joseph Giry (1699–1761), French clergyman
* Sylvie Giry-Rousset (born 1965), French cross-country ski ...
.
References
*https://ncatlab.org/nlab/show/category-theoretic+approaches+to+probability+theory
*https://golem.ph.utexas.edu/category/2024/07/imprecise_probabilities_toward.html#more
Further reading
* https://ncatlab.org/nlab/show/Giry+monad#related_constructions
* https://golem.ph.utexas.edu/category/2024/08/introduction_to_categorical_pr.html#more
* Voevodsky's unfinished manuscript
Notes on categorical probability July 13, 2009.
External links
* https://golem.ph.utexas.edu/category/2020/06/categorical_probability_and_st.html
* https://golem.ph.utexas.edu/category/2020/06/statistics_for_category_theori.html
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Probability theory
Category theory