# nLab monads of probability, measures, and valuations

Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

In many categorical approaches to measure theory and probability, one considers a category of spaces, such as measurable spaces or topological spaces, and equips this category with a monad whose functor part assigns to each space $X$ a space $P X$ of measures, probability measures, or valuations on $X$, or a variation thereof.

For probability theory, this can be interpreted as adding to the points of a space $X$ new “random points”, modelled as probability measures or valuations. The old points, which we can think of as deterministic, are embedded in $P X$ via the unit of the monad $X\to P X$. Just as well, the Kleisli morphisms of $P$ can be seen as stochastic maps?. (Monads can be seen as ways of extending our spaces and functions to account for new phenomena, see for example extension system and monad in computer science.) Note that these probability measures are technically different from random elements?: they rather correspond to the laws? of the random elements.

Algebras of probability and measure monads can be interpreted as generalized convex spaces or conical spaces of a certain kind. For probability theory, in particular, the algebras of a probability monad can be seen as spaces equipped with a notion of expectation value of a random variable.

The details vary depending on the monad and on the category under consideration.

Many choices of categories and of monads are possible, depending on which aspects of measure theory or probability one wants to study. See the table below for more details.

The term “probability monad” was coined by Giry herself (see here).

## Functor, unit and multiplication

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## Algebras

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## Kleisli morphisms

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## Monoidal structure

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## Duality

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## Detailed list

Monad ($P$) Category Elements/points of $P X$ Extra structure of $P X$ $P$-Algebras References
distribution monad (a.k.a. finitary Giry monad, convex combination monad) Set convex combinations or finitely-supported probability measures (just a set) convex spaces Fritz '09, Jacobs '18
Giry monad Meas probability measures initial σ-algebra of evaluation maps Full characterization unknown. See also here. Lawvere '62, Giry '80
Giry monad Pol Borel probability measures initial topology of integration maps Full characterization unknown. See also here. Giry '80
Radon monad Comp Radon probability measures (or continuous valuations) weak topology w.r.t. continuous functions compact convex subsets of locally convex topological vector spaces Swirszcz '74, Keimel '08
ordered Radon monad CompOrd Radon probability measures (or continuous valuations) weak topology w.r.t. continuous functions, stochastic order compact convex subsets of ordered? locally convex topological vector spaces Swirszcz '74, Keimel '08
probabilistic powerdomain? dcpo, continuous domains? continuous valuations stochastic order abstract probabilistic domains? (continuous case) J-P '89
extended probabilistic powerdomain Top, stably compact spaces continuous valuations initial topology of evaluation maps, stochastic order Full characterization unknown. Dedicated section here Heckmann '96, A-J-K '04, GL-J '19, F-P-R '19
valuation monad on locales? Loc continuous valuations initial topology of evaluation maps Vickers '11
Kantorovich monad complete metric spaces Radon probability measures of finite first moment Kantorovich-Wasserstein metric closed convex subsets of Banach spaces van Breugel '05, F-P '19
ordered Kantorovich monad complete L-ordered? metric spaces Radon probability measures of finite first moment Kantorovich-Wasserstein metric, stochastic order closed convex subsets of ordered Banach spaces? F-P

(…to be expanded…)

## References

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

• Michèle Giry, A categorical approach to probability theory, Categorical aspects of topology and analysis (Ottawa, Ont., 1980), pp. 68–85, Lecture Notes in Math. 915 Springer 1982.

• T. Swirszcz, Monadic functors and convexity, Bulletin de l’Academie Polonais des Sciences 22, 1974 (pdf)

• Klaus Keimel?, The monad of probability measures over compact ordered spaces and its Eilenberg-Moore algebras, Topology and its Applications, 2008 (doi:10.1016/j.topol.2008.07.002)

• Reinhold Heckmann, Spaces of valuations, Papers on General Topology and Ap-plications, 1996 (doi:10.1111/j.1749-6632.1996.tb49168.x,pdf)

• Mauricio Alvarez-Manilla, Achin Jung, Klaus Keimel?, The probabilistic powerdomain for stably compact spaces, Theoretical Computer Science 328, 2004. Link here.

• C. Jones and Gordon. D. Plotkin?, A probabilistic powerdomain of evaluations, LICS 4, 1989. (doi:10.1109/LICS.1989.39173)

• Jean Goubault-Larrecq? and Xiaodong Jia, Algebras of the extended probabilistic powerdomain monad, ENTCS 345, 2019

• Tobias Fritz, Paolo Perrone and Sharwin Rezagholi, Probability, valuations, hyperspace: Three monads on Top and the support as a morphism, 2019. Link here.