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Giry monad

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In mathematics, the Giry monad is a construction that assigns to a measurable space a space of probability measures over it, equipped with a canonical sigma-algebra.[1][2][3][4][5] It is one of the main examples of a probability monad.

It is implicitly used in probability theory whenever one considers probability measures which depend measurably on a parameter (giving rise to Markov kernels), or when one has probability measures over probability measures (such as in de Finetti's theorem).

Like many iterable constructions, it has the category-theoretic structure of a monad, on the category of measurable spaces.

Construction

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The Giry monad, like every monad, consists of three structures:[6][7][8]

  • A functorial assignment, which in this case assigns to a measurable space a space of probability measures over it;
  • A natural map called the unit, which in this case assigns to each element of a space the Dirac measure over it;
  • A natural map called the multiplication, which in this case assigns to each probability measure over probability measures its expected value.

The space of probability measures

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Let be a measurable space. Denote by the set of probability measures over . We equip the set with a sigma-algebra as follows. First of all, for every measurable set , define the map by . We then define the sigma algebra on to be the smallest sigma-algebra which makes the maps measurable, for all (where is assumed equipped with the Borel sigma-algebra). [6]

Equivalently, can be defined as the smallest sigma-algebra on which makes the maps

measurable for all bounded measurable .[9]

The assignment is part of an endofunctor on the category of measurable spaces, usually denoted again by . Its action on morphisms, i.e. on measurable maps, is via the pushforward of measures. Namely, given a measurable map , one assigns to the map defined by

for all and all measurable sets . [6]

The Dirac delta map

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Given a measurable space , the map maps an element to the Dirac measure , defined on measurable subsets by[6]

The expectation map

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Let , i.e. a probability measure over the probability measures over . We define the probability measure by

for all measurable . This gives a measurable, natural map .[6]

Example: mixture distributions

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A mixture distribution, or more generally a compound distribution, can be seen as an application of the map . Let's see this for the case of a finite mixture. Let be probability measures on , and consider the probability measure given by the mixture

for all measurable , for some weights satisfying . We can view the mixture as the average , where the measure on measures , which in this case is discrete, is given by

More generally, the map can be seen as the most general, non-parametric way to form arbitrary mixture or compound distributions.

The triple is called the Giry monad.[1][2][3][4][5]

Relationship with Markov kernels

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One of the properties of the sigma-algebra is that given measurable spaces and , we have a bijective correspondence between measurable functions and Markov kernels . This allows to view a Markov kernel, equivalently, as a measurably parametrized probability measure.[10]

In more detail, given a measurable function , one can obtain the Markov kernel as follows,

for every and every measurable (note that is a probability measure). Conversely, given a Markov kernel , one can form the measurable function mapping to the probability measure defined by

for every measurable . The two assignments are mutually inverse.

From the point of view of category theory, we can interpret this correspondence as an adjunction

between the category of measurable spaces and the category of Markov kernels. In particular, the category of Markov kernels can be seen as the Kleisli category of the Giry monad.[3][4][5]

Product distributions

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Given measurable spaces and , one can form the measurable space with the product sigma-algebra, which is the product in the category of measurable spaces. Given probability measures and , one can form the product measure on . This gives a natural, measurable map

usually denoted by or by .[4]

The map is in general not an isomorphism, since there are probability measures on which are not product distributions, for example in case of correlation. However, the maps and the isomorphism make the Giry monad a monoidal monad, and so in particular a commutative strong monad.[4]

Further properties

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  • If a measurable space is standard Borel, so is . Therefore the Giry monad restricts to the full subcategory of standard Borel spaces.[1][4]
  • The algebras for the Giry monad include compact convex subsets of Euclidean spaces, as well as the extended positive real line , with the algebra structure map given by taking expected values.[11] For example, for , the structure map is given by
whenever is supported on and has finite expected value, and otherwise.

See also

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Citations

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  1. ^ a b c Giry (1982)
  2. ^ a b Avery (2016), pp. 1231–1234
  3. ^ a b c Jacobs (2018), pp. 205–106
  4. ^ a b c d e f Fritz (2020), pp. 19–23
  5. ^ a b c Moss & Perrone (2022), pp. 3–4
  6. ^ a b c d e Giry (1982), p. 69
  7. ^ Riehl (2016)
  8. ^ Perrone (2024)
  9. ^ Perrone (2024), p. 238
  10. ^ Giry (1982), p. 71
  11. ^ Doberkat (2006), pp. 1772–1776

References

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  • Giry, Michèle (1982). "A categorical approach to probability theory". Categorical Aspects of Topology and Analysis. Lecture Notes in Mathematics. Vol. 915. Springer. pp. 68–85. doi:10.1007/BFb0092872. ISBN 978-3-540-11211-2.

Further reading

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