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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