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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 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.
The Giry monad, like every monad, consists of three structures: [6] [7] [8]
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]
Given a measurable space , the map maps an element to the Dirac measure , defined on measurable subsets by [6]
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]
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]
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]
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]
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Submission declined on 16 April 2024 by
ToadetteEdit (
talk). This submission is not adequately supported by
reliable sources. Reliable sources are required so that information can be
verified. If you need help with referencing, please see
Referencing for beginners and
Citing sources.
Where to get help
How to improve a draft
You can also browse Wikipedia:Featured articles and Wikipedia:Good articles to find examples of Wikipedia's best writing on topics similar to your proposed article. Improving your odds of a speedy review To improve your odds of a faster review, tag your draft with relevant WikiProject tags using the button below. This will let reviewers know a new draft has been submitted in their area of interest. For instance, if you wrote about a female astronomer, you would want to add the Biography, Astronomy, and Women scientists tags. Editor resources
This draft has been resubmitted and is currently awaiting re-review. | ![]() |
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 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.
The Giry monad, like every monad, consists of three structures: [6] [7] [8]
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]
Given a measurable space , the map maps an element to the Dirac measure , defined on measurable subsets by [6]
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]
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]
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]
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]