Giry monad explained

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. 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

The Giry monad, like every monad, consists of three structures:

• A functorial assignment, which in this case assigns to a measurable space

X a space of probability measures PX over it;\delta:X\to PX called the unit, which in this case assigns to each element of a space the Dirac measure over it;l{E}:PPX\to PX called the multiplication, which in this case assigns to each probability measure over probability measures its expected value.

The space of probability measures

Let

(X, l{F}) be a measurable space. Denote by PX the set of probability measures over (X, l{F}) . We equip the set PX with a sigma-algebra as follows. First of all, for every measurable set A\in l{F} , define the map \varepsilon _{A:PX\toR} by p\longmapsto p(A) . We then define the sigma algebra l{PF} on PX to be the smallest sigma-algebra which makes the maps \varepsilon _{A} measurable, for all A\in l{F} (where R is assumed equipped with the Borel sigma-algebra).

Equivalently,

l{PF} can be defined as the smallest sigma-algebra on PX which makes the maps p\longmapsto\int _{X} f dp measurable for all bounded measurable f:X\to R .

The assignment

(X, l{F})\mapsto (PX, l {PF}) is part of an endofunctor on the category of measurable spaces, usually denoted again by P . Its action on morphisms, i.e. on measurable maps, is via the pushforward of measures. Namely, given a measurable map f:(X, l{F})\to(Y, l{G}) , one assigns to f the map f _{*:(PX,l{PF})\to(PY,l{PG})} defined by f ^{-1} _{*p(B) = p(f} (B)) for all p\in PX and all measurable sets B\in l{G} .

The Dirac delta map

Given a measurable space

(X, l{F}) , the map \delta:(X, l{F})\to(PX, l{PF}) maps an element x\in X to the Dirac measure \delta _{x\in} PX , defined on measurable subsets A\in l{F} by \delta _{x(A)} = 1 _{A(x)} = \begin{cases} 1 & if x\in A, \\ 0 & if x\notin A. \end{cases}

The expectation map

Let

\mu\in PPX , i.e. a probability measure over the probability measures over (X, l{F}) . We define the probability measure l{E}\mu\in PX byl{E}\mu(A) = \int _{PX} p(A)\mu(dp) for all measurable A\in l{F} .This gives a measurable, natural map l{E}:(PPX, l{PPF})\to(PX, l{PF}) .

Example: mixture distributions

A mixture distribution, or more generally a compound distribution, can be seen as an application of the map

l{E} . Let's see this for the case of a finite mixture. Let p _{1,...,p} _{n} be probability measures on (X, l{F}) , and consider the probability measure q given by the mixture q(A) = \sum ^{n} _{i =1} w _{ip} _{i(A)} for all measurable A\in l{F} , for some weights w _{i\ge} 0 satisfying w _{1+...+w} _{n=1} . We can view the mixture q as the average q = l{E}\mu , where the measure on measures \mu\in PPX , which in this case is discrete, is given by \mu = \sum ^{n} _{i =1} w _{i\delta} ^{p _{i}} . More generally, the map l{E}:PPX\to PX can be seen as the most general, non-parametric way to form arbitrary mixture or compound distributions.

The triple

(P,\delta, l{E}) is called the Giry monad.

Relationship with Markov kernels

l{PF} is that given measurable spaces (X, l{F}) and (Y, l{G}) , we have a bijective correspondence between measurable functions (X, l{F})\to(PY, l{PG}) and Markov kernels (X, l{F})\to(Y, l{G}) . This allows to view a Markov kernel, equivalently, as a measurably parametrized probability measure.

In more detail, given a measurable function

f:(X, l{F})\to(PY, l{PG}) , one can obtain the Markov kernel f ^{\flat:(X,l{F})\to(Y,l{G})} as follows,f ^{\flat(B|x)} = f(x)(B) for every x\in X and every measurable B\in l{G} (note that f(x)\in PY is a probability measure). Conversely, given a Markov kernel k:(X, l{F})\to(Y, l{G}) , one can form the measurable function k ^{\sharp:(X,l{F})\to(PY,l{PG})} mapping x\in X to the probability measure k ^{\sharp(x)\in} PY defined by k ^{\sharp(x)(B)} = k(B|x) for every measurable B\in l{G} . The two assignments are mutually inverse.Hom _{Meas} (X,PY) \cong Hom _{Stoch} (X,Y) 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.

Product distributions

Given measurable spaces

(PX, l{PF}) x (PY, l{PG})\to ( P(X x Y), l{P(F x G)} ) usually denoted by \nabla or by ⊗ .

The map

\nabla:PX x PY\to P(X x Y) is in general not an isomorphism, since there are probability measures on X x Y which are not product distributions, for example in case of correlation.However, the maps \nabla:PX x PY\to P(X x Y) and the isomorphism 1 \cong P 1 make the Giry monad a monoidal monad, and so in particular a commutative strong monad.

Further properties

• If a measurable space

(X, l{F}) is standard Borel, so is (PX, l{PF}) . Therefore the Giry monad restricts to the full subcategory of standard Borel spaces.

• The algebras for the Giry monad include compact convex subsets of Euclidean spaces, as well as the extended positive real line

[ 0 , inf ty] , with the algebra structure map given by taking expected values. For example, for [ 0 , inf ty] , the structure map e:P[ 0 , inf ty]\to [ 0 , inf ty] is given by p \longmapsto \int _{[0,infty)} xp(dx)

whenever

p is supported on [ 0 , inf ty) and has finite expected value, and e(p) =inf ty otherwise.

See also

• Mixture distribution
• Compound distribution
• de Finetti theorem
• Measurable space
• Markov kernel
• Monad (category theory)
• Monad (functional programming)
• Category of measurable spaces
• Category of Markov kernels
• Categorical probability

References

• Book: Giry, Michèle . Categorical Aspects of Topology and Analysis . A categorical approach to probability theory . Lecture Notes in Mathematics . 1982 . 915 . 68–85 . Springer. 10.1007/BFb0092872 . 978-3-540-11211-2 . https://link.springer.com/chapter/10.1007/BFb0092872.
• Doberkat . Ernst-Erich. Eilenberg-Moore algebras for stochastic relations. Information and Computation. 204. 12. 2006. 1756–1781. 10.1016/j.ic.2006.09.001.
• Avery . Tom. Codensity and the Giry monad. Journal of Pure and Applied Algebra. 220. 3. 1229–1251. 2016. 10.1016/j.jpaa.2015.08.017. 1410.4432. 119695729.
• Jacobs . Bart. From probability monads to commutative effectuses. Journal of Logical and Algebraic Methods in Programming. 94. 200–237. 2018. 10.1016/j.jlamp.2016.11.006.
• Fritz . Tobias. A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics. Advances in Mathematics. 370. 2020. 10.1016/j.aim.2020.107239. 1908.07021. 201103837.
• Moss . Sean. Perrone . Paolo. Probability monads with submonads of deterministic states. LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science. 2022. 10.1145/3531130.3533355. 2204.07003.
• Book: Riehl, Emily . Category Theory in Context. Chapter 5. Monads and their Algebras. 2016 . Dover. 978-0486809038 .
• Book: Perrone, Paolo . Starting Category Theory. Chapter 5. Monads and Comonads. 2024 . World Scientific. 10.1142/9789811286018_0005 . 978-981-12-8600-1. https://www.worldscientific.com/doi/10.1142/9789811286018_0005.

Further reading

• Monads of probability, measures, and valuations, in nLab.
• https://ncatlab.org/nlab/show/Giry+monad

External links

• What is a probability monad?, video tutorial.