Giry monad

The Giry monad, like every monad, consists of three structures:{{harvp|Giry|1982|page=69}}{{harvp|Riehl|2016}}{{harvp|Perrone|2024}}

• A functorial assignment, which in this case assigns to a measurable space $X$ a space of probability measures $PX$ over it;
• A natural map $\backslash delta:X\backslash to\; PX$ called the unit, which in this case assigns to each element of a space the Dirac measure over it;
• A natural map $\backslash mathcal\{E\}:PPX\backslash to\; PX$ called the multiplication, which in this case assigns to each probability measure over probability measures its expected value.

Let $(X,\; \backslash mathcal\{F\})$ be a measurable space.

Denote by $PX$ the set of probability measures over $(X,\; \backslash mathcal\{F\})$.

We equip the set $PX$ with a sigma-algebra as follows. First of all, for every measurable set $A\backslash in\; \backslash mathcal\{F\}$, define the map $\backslash varepsilon\_A:PX\backslash to\backslash mathbb\{R\}$ by $p\backslash longmapsto\; p(A)$.

We then define the sigma algebra $\backslash mathcal\{PF\}$ on $PX$ to be the smallest sigma-algebra which makes the maps $\backslash varepsilon\_A$ measurable, for all $A\backslash in\backslash mathcal\{F\}$ (where $\backslash mathbb\{R\}$ is assumed equipped with the Borel sigma-algebra).

Equivalently, $\backslash mathcal\{PF\}$ can be defined as the smallest sigma-algebra on $PX$ which makes the maps

:$p\backslash longmapsto\backslash int\_X\; f\; \backslash ,dp$

measurable for all bounded measurable $f:X\backslash to\backslash mathbb\{R\}$.{{harvp|Perrone|2024|page=238}}

The assignment $(X,\backslash mathcal\{F\})\backslash mapsto\; (PX,\backslash mathcal\; \{PF\})$ 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 $f:(X,\backslash mathcal\{F\})\backslash to(Y,\backslash mathcal\{G\})$, one assigns to $f$ the map $f\_*:(PX,\backslash mathcal\; \{PF\})\backslash to(PY,\backslash mathcal\; \{PG\})$ defined by

:$f\_*p\backslash ,(B)=p(f^\{-1\}(B))$

for all $p\backslash in\; PX$ and all measurable sets $B\backslash in\backslash mathcal\{G\}$.

Given a measurable space $(X,\backslash mathcal\{F\})$, the map $\backslash delta:(X,\backslash mathcal\{F\})\backslash to(PX,\backslash mathcal\{PF\})$ maps an element $x\backslash in\; X$ to the Dirac measure $\backslash delta\_x\backslash in\; PX$, defined on measurable subsets $A\backslash in\backslash mathcal\{F\}$ by

:$$

\delta_x(A) = 1_A(x) =

\begin{cases}

1 & \text{if }x\in A, \\

0 & \text{if }x\notin A.

\end{cases}

Let $\backslash mu\backslash in\; PPX$, i.e. a probability measure over the probability measures over $(X,\backslash mathcal\{F\})$. We define the probability measure $\backslash mathcal\{E\}\backslash mu\backslash in\; PX$ by

:$$

\mathcal{E}\mu(A) = \int_{PX} p(A)\,\mu(dp)

for all measurable $A\backslash in\backslash mathcal\{F\}$.

This gives a measurable, natural map $\backslash mathcal\{E\}:(PPX,\backslash mathcal\{PPF\})\backslash to(PX,\backslash mathcal\{PF\})$.

A mixture distribution, or more generally a compound distribution, can be seen as an application of the map $\backslash mathcal\{E\}$.

Let's see this for the case of a finite mixture. Let $p\_1,\backslash dots,p\_n$ be probability measures on $(X,\backslash mathcal\{F\})$, and consider the probability measure $q$ given by the mixture

:$$

q(A) = \sum_{i=1}^n w_i\,p_i(A)

for all measurable $A\backslash in\backslash mathcal\{F\}$, for some weights $w\_i\backslash ge\; 0$ satisfying $w\_1+\backslash dots+w\_n=1$.

We can view the mixture $q$ as the average $q=\backslash mathcal\{E\}\backslash mu$, where the measure on measures $\backslash mu\backslash in\; PPX$, which in this case is discrete, is given by

:$$

\mu = \sum_{i=1}^n w_i\,\delta_{p_i} .

More generally, the map $\backslash mathcal\{E\}:PPX\backslash to\; PX$ can be seen as the most general, non-parametric way to form arbitrary mixture or compound distributions.

The triple $(P,\backslash delta,\backslash mathcal\{E\})$ is called the Giry monad.

One of the properties of the sigma-algebra $\backslash mathcal\{PF\}$ is that given measurable spaces $(X,\backslash mathcal\{F\})$ and $(Y,\backslash mathcal\{G\})$, we have a bijective correspondence between measurable functions $(X,\backslash mathcal\{F\})\backslash to(PY,\backslash mathcal\{PG\})$ and Markov kernels $(X,\backslash mathcal\{F\})\backslash to(Y,\backslash mathcal\{G\})$. This allows to view a Markov kernel, equivalently, as a measurably parametrized probability measure.{{harvp|Giry|1982|page=71}}

In more detail, given a measurable function $f:(X,\backslash mathcal\{F\})\backslash to(PY,\backslash mathcal\{PG\})$, one can obtain the Markov kernel $f^\backslash flat:(X,\backslash mathcal\{F\})\backslash to(Y,\backslash mathcal\{G\})$ as follows,

:$$

f^\flat(B|x) = f(x)(B)

for every $x\backslash in\; X$ and every measurable $B\backslash in\backslash mathcal\{G\}$ (note that $f(x)\backslash in\; PY$ is a probability measure).

Conversely, given a Markov kernel $k:(X,\backslash mathcal\{F\})\backslash to(Y,\backslash mathcal\{G\})$, one can form the measurable function $k^\backslash sharp:(X,\backslash mathcal\{F\})\backslash to(PY,\backslash mathcal\{PG\})$ mapping $x\backslash in\; X$ to the probability measure $k^\backslash sharp(x)\backslash in\; PY$ defined by

:$$

k^\sharp(x)(B) = k(B|x)

for every measurable $B\backslash in\backslash mathcal\{G\}$.

The two assignments are mutually inverse.

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

:$$

\mathrm{Hom}_\mathrm{Meas} (X,PY) \cong \mathrm{Hom}_\mathrm{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.

Given measurable spaces $(X,\backslash mathcal\{F\})$ and $(Y,\backslash mathcal\{G\})$, one can form the measurable space $(PX,\backslash mathcal\{PX\})\backslash times\; (PY,\backslash mathcal\{PY\})=(X\backslash times\; Y,\; \backslash mathcal\{F\}\backslash times\backslash mathcal\{G\})$ with the product sigma-algebra, which is the product in the category of measurable spaces.

Given probability measures $p\backslash in\; PX$ and $q\backslash in\; PY$, one can form the product measure $p\backslash otimes\; q$ on $(X\backslash times\; Y,\; \backslash mathcal\{F\}\backslash times\backslash mathcal\{G\})$. This gives a natural, measurable map

:$$

(PX,\mathcal{PF})\times (PY,\mathcal{PG})\to \big(P(X\times Y), \mathcal{P(F\times G)}\big)

usually denoted by $\backslash nabla$ or by $\backslash otimes$.

The map $\backslash nabla:PX\backslash times\; PY\backslash to\; P(X\backslash times\; Y)$ is in general not an isomorphism, since there are probability measures on $X\backslash times\; Y$ which are not product distributions, for example in case of correlation.

However, the maps $\backslash nabla:PX\backslash times\; PY\backslash to\; P(X\backslash times\; Y)$ and the isomorphism $1\backslash cong\; P1$ make the Giry monad a monoidal monad, and so in particular a commutative strong monad.

• If a measurable space $(X,\backslash mathcal\{F\})$ is standard Borel, so is $(PX,\backslash mathcal\{PF\})$. 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 $[0,\backslash infty]$, with the algebra structure map given by taking expected values.{{harvp|Doberkat|2006|pages=1772-1776}} For example, for $[0,\backslash infty]$, the structure map $e:P[0,\backslash infty]\backslash to\; [0,\backslash infty]$ is given by

:$$

p \longmapsto \int_{[0,\infty)} x\,p(dx)

:whenever $p$ is supported on $[0,\backslash infty)$ and has finite expected value, and $e(p)=\backslash infty$ otherwise.

• 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

{{reflist}}

{{refbegin}}

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|chapter = A categorical approach to probability theory

|series = Lecture Notes in Mathematics

|date = 1982

|volume = 915

|pages = 68–85

|publisher = Springer

|doi = 10.1007/BFb0092872

|isbn = 978-3-540-11211-2

|chapter-url = https://link.springer.com/chapter/10.1007/BFb0092872

}}

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• {{cite journal

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}}

• {{cite journal

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• {{cite journal

| last = Fritz | first = Tobias

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• {{cite conference

| last1 = Moss | first1 = Sean

| last2 = Perrone | first2 = Paolo

| title = Probability monads with submonads of deterministic states

| book-title = LICS '22: Proceedings of the 37th Annual ACM/IEEE Symposium on Logic in Computer Science

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• {{cite book

|last = Riehl |first = Emily

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}}

• {{cite book

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|date = 2024

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|doi = 10.1142/9789811286018_0005

|isbn = 978-981-12-8600-1

|chapter-url = https://www.worldscientific.com/doi/10.1142/9789811286018_0005

}}

{{refend}}

• [https://ncatlab.org/nlab/show/monads+of+probability%2C+measures%2C+and+valuations Monads of probability, measures, and valuations], in nLab.
• https://ncatlab.org/nlab/show/Giry+monad

• [https://www.youtube.com/watch?v=3Da88Tgw_rM What is a probability monad?], video tutorial.

Category:Measure theory

Category:Probability theory

Category:Category theory