The Giry monad (Giry 80, following Lawvere 62) is the monad on a category of suitable spaces which sends each suitable space $X$ to the space of suitable probability measures on $X$.
It is one of the main examples of a probability monad, and hence one of the main structures used in categorical probability.
The Giry monad is defined on the category of measurable spaces, assigning to each measurable space $X$ the space $G(X)$ of all probability measures on $X$ endowed with the $\sigma$-algebra generated by the set of all the evaluation maps
sending a probability measure $P$ to $P(U)$, where $U$ ranges over all the measurable sets of $X$. The unit of the monad sends a point $x \in X$ to the Dirac measure at $x$, $\delta_x$, while the monad-multiplication is defined by the natural transformation
given by
This makes the endofunctor $G$ into a monad, and as such this is the Giry monad on measurable spaces, as originally defined by Lawvere 1962.
An alternative choice, convenient for analysis purposes, and introduced by Giry, is obtained by restricting the category of measurable spaces to the (full) subcategory which are those measurable spaces generated by Polish spaces, $Pol$, which are separable metric spaces for which a complete metric exists. The morphisms of this category are continuous functions.
Write
for the endofunctor which sends a space, $X$, to the space of probability measures on the Borel subsets of $X$. $P(X)$ is equipped with the weakest topology which makes the integration map $\tau \mapsto \int_{X}f d\tau$ continuous for any $f$, a bounded, continuous, real function on $X$.
There is a natural transformation
given by
This makes the endofunctor $P$ into a monad, and this is the Giry monad on Polish spaces.
The Kleisli morphisms of the Giry monad on Meas (and related subcategories) are Markov kernels. Therefore its Kleisli category is the category Stoch. It is one of the most important examples of a Markov category.
We can’t say anything about the $G$-algebras on the category of measurable spaces due to lack of structure and set-theoretical issues. However, we can analyze the $P$-algebras exploiting the continuous morphisms and countably dense subsets on the spaces $P{X}$, and give a class of non-free $P$-algebras.
If $X$ is any measurable space then the space of probability measures $P{X}$ is a convex space with the structure defined pointwise, i.e., if $\{P_i\}_{i=1}^{n}$ is a finite collection of probability measures on $X$ then, for every sequence $\{p_i\}_{i=1}^{n}$ with each $p_i \in [0,1]$ the affine sum $\sum_{i=1}^{n} p_i P_i$, is also a probability measure, defined at the measurable set $U$ in $X$ by
On the otherhand, given any $P$-algebra $h: P{X} \rightarrow X$ it follows that $X$ can also be endowed with a convex structure by defining, for every affine sum of $n$ elements of $X$,
To analyze $P$-algebras, via the Eilenberg-Moore representation as morphisms $h:P{X} \rightarrow X$, we require
Given any $P$-algebra $h: P{X} \rightarrow X$ the continuous function $h$ is an affine map.
Moreover, for any measurable space $Y$ and $P$-algebra $k:P{Y} \rightarrow Y$ on it, if $f: (X, h) \rightarrow (Y, k)$ is a map of $P$-algebras then the continuous function $f$ is also affine.
Because a $P$-algebra $h$ must satisfy $h \circ \mu_X = h \circ P{h}$ we have
where the last line makes use of the definition of the convex structure on $X$. Thus every $P$-algebra is affine.
To prove the map of $P$-algebras is an affine map we compute
Remark: The space $P{X}$ actually has a superconvex structure since, for $\lim_{n \rightarrow \infty} \sum_{i=1}^n p_i=1$ and each $p_i \in [0,1]$, the countable affine sum $\sum_{i=1}^{\infty} p_i P_i$ also defines a probability measure on $P{X}$. The preceding Lemma also then holds for the term “affine” replaced by “countably affine”.
(Uniqueness of $P$-algebras) Let $X$ be a Polish space and suppose $f,g:P{X} \rightarrow X$ are a pair of $P$-algebras. Then $f=g$.
The space $X$, being a separable metric space, has a countable dense subset $E$, and hence the set $D=\{\sum_{i=1}^n r_i \delta_{e_i} \, | \, e_i\in \E, r_i \in \mathbb{Q}\cap[0,1], \sum_{i=1}^n r_i=1, \text{ and } n \text{ finite}\}$ is dense in $P{X}$.(Parthasarathy, Probability measures on metric spaces,Theorem 6.3) Since the set $D$ is dense in $P{X}$ and $P{X}$ is a complete separable metric space it follows that the functions $f$ and $g$ are completely determined by their values on the dense subset $D$. We have $f(\sum_{i=1}^n r_i \delta_{e_i})= \sum_{i=1}^n r_i f(\delta_{e_i}) = \sum_{i=1}^n r_i e_i$, where the first equality follows by Lemma , and the second equality by the required condition on algebras, $h(\delta_e)=e$, and similiarly $g(\sum_{i=1}^n r_i \delta_{e_i}) = \sum_{i=1}^n r_i e_i.$ Since $f=g$ on the dense subset and both $f$ and $g$ are continuous maps it follows that $f=g$ on all of $P{X}$.
Let $X$ be a Polish space and suppose $h:P{X} \rightarrow X$ is a $P$-algebra. Then $h$ is a free $P$-algebra if and only if there exists a $\mathbf{Cvx}$-isomorphism $\phi: X \rightarrow P{Y}$ for some Polish space $Y$.
Suppose there exists a $\mathbf{Cvx}$-isomorphism $\phi: X \rightarrow P{Y}$ then $\hat{h} = \phi \circ h \circ P(\phi^{-1}): P(P{Y}) \rightarrow P{Y}$ is also a $P$-algebra. Since the multiplication natural transformation $\mu: P \circ P \Rightarrow P$ gives the free $P$-algebra $\mu_Y: P(P{Y}) \rightarrow P{Y}$ it follows by the uniqueness lemma that $\hat{h}=\mu_Y$. Because $\phi$ is a $\mathbf{Cvx}$-isomorphism it then follows that $h$ is a free $P$-algebra.
Conversely, if $h$ is a free $P$-algebra then, since all free $P$-algebras are given by $\mu_Y: P(P{Y}) \rightarrow P{Y}$ for every space $Y$, there must exists a particular space $Y$ such that $X = P{Y}$. Hence the identity map yields a $\mathbf{Cvx}$-isomorphism $id: X \rightarrow P{Y}$.
The space $P{\mathbf{2}} \times P{\mathbf{2}}$ is not $\mathbf{Cvx}$-isomorphic to $P{X}$ for any Polish space $X$.
Every element of $P{\mathbf{2}} \times P{\mathbf{2}}$ is characterized precisely by two independent real numbers $r,s \in [0,1]$. The only space $X$ for which every element $Q \in P{X}$ is characterized by two parameters is $P{\mathbf{3}}$. However those two parameters are not independent since choosing the first parameter as one forces the second parameter to be zero. An equivalent way to state this is to say that the coequalizer of the pair of points $(0,1)$ and $(1,1)$ of the space $[0,1] \times [0,1]$ is, up to isomorphism, $P(\mathbf{3})$. The congruence relation $R$ is the relationship on $[0,1]\times [0,1]$ with the equivalence classes specified by
is $\mathbf{Cvx}$-isomorphic to $P(\mathbf{3})$. (Draw the picture.)
(Marginalization $P$-algebra) Let $\pi_i: P(\mathbf{2})\times P(\mathbf{2}) \rightarrow P(\mathbf{2})$, for $i=1,2$ denote the two coordinate projection maps of the product space $P(\mathbf{2}) \times P(\mathbf{2})$. The function $m:P( P{\mathbf{2}} \times P{\mathbf{2}}) \rightarrow P{\mathbf{2}} \times P{\mathbf{2}}$ specified by marginalization, $Q \mapsto \big( (\mu_2\circ P{\pi_1})Q, (\mu_2 \circ P{\pi_2})Q\big)$, is a $P$-algebra.
The function $m$ is continuous because each of the component maps are continuous. The condition $m \circ \eta_{P{\mathbf{2}} \times P{\mathbf{2}}} = id_{P{\mathbf{2}} \times P{\mathbf{2}}}$ follows because for each element $(Q,R) \in P{\mathbf{2}} \times P{\mathbf{2}}$ we have $\mu_2(G{\pi_1} \delta_{(Q,R)}) = \mu_2(\delta_{\pi_1(Q,R)}) = \mu_2(\delta_P)=Q$ and similarly $\mu_2(P{\pi_2} \delta_{(Q,R)}) =R$.
To prove the required condition $P{m} \circ m= \mu_{P{2} \times P{2}} \circ m$ first note that $m$ is an affine function because each component map $\mu_2 \circ P{\pi_i}$, for $i=1,2$, is a composite of affine maps. Moreover, since $P{\mathbf{2}} \times P{\mathbf{2}}$ is a separable metric space it has a countable dense subset of elements $(u_k, v_k)$ so that the set of elements consisting of affine sums of Dirac measures of the form $\sum_{j=1}^p s_j \delta_{(u_j,v_j)}$ for $p$ a finite number is a dense subset of $P(P{\mathbf{2}} \times P{\mathbf{2}})$. In turn it follows that affine sums of elements of the form $\sum_{i=1}^n r_i \delta_{\sum_{j=1}^p s_j^i \delta_{(u_j^i,v_j^i)}}$ specify a dense subset of $P(P(P(\mathbf{2}) \times P(\mathbf{2})))$. A direct computation on the dense elements, using the fact that all the maps are affine, then yields $P{m} \circ m= \mu_{P{\mathbf{2}} \times P{\mathbf{2}}} \circ m$ on all the elements of the dense set. Since the maps are all continuous it therefore follows that $P{m} \circ m= \mu_{P{\mathbf{2}} \times P{\mathbf{2}}} \circ m$ on all elements of $P(P(P{\mathbf{2}} \times P{\mathbf{2}}))$. This completes the proof that $m$ is a $P$-algebra.
The category of $P$-algebras is not equivalent to the Kleisi category of the $P$-monad for Polish spaces. In other words, there are non-free $P$-algebras.
By Lemma we know that the marginalization map $m:P( P{\mathbf{2}} \times P{\mathbf{2}}) \rightarrow P{\mathbf{2}} \times P{\mathbf{2}}$ is a $P$-algebra. By Lemma we know that $P{\mathbf{2}} \times P{\mathbf{2}}$ is not $\mathbf{Cvx}$-isomorphic to $P{Y}$ for any space $Y$. Hence by Lemma it follows that $m$ is not a free $P$-algebra, and hence the category of $P$-algebras is not equivalent to the Kleisi category for the $P$-monad.
(Doberkat 03) gives a different representation for the algebras of $P$. His representation for the algebras is based upon the idea that we want continuous maps $h:P(X) \rightarrow X$ such that the ‘fibres’ are convex and closed, and such that $\delta_x$, the Delta distribution on $x$, is in the fibre over $x$. And there’s another condition which requires a compact subset of $P(X)$ to be sent to a compact subset of $X$.
As an example of $P$-algebras, represented via convex spaces, Doberkat gives the example of closed and bounded convex subsets of some Euclidean space, and shows that the construction of a barycenter yields an algebra. We summarize that construction here.
Fix $X \subset \mathbb{R}^n$ as a bounded, closed, and convex subset of the Euclidean space $\mathbb{R}^n$. A vector $x_{\star} \in \mathbb{R}^n$ is called a barycenter of the probability measure $\tau \in P(X)$ iff, for all linear functionals $g$ on $\mathbb{R}^n$, the property $g(x_{\star}) = \int_X g(x) d\tau$ holds. Since every linear functional on $\mathbb{R}^n$ is given by $g(\cdot)=\langle x, \cdot \rangle$ for a unique $x \in \mathbb{R}^n$, the defining property of a barycenter (for convex subsets of $\mathbb{R}^n$) given above is equivalent to saying that, for all $x \in \mathbb{R}^n$, the property $\langle x, x_{\star} \rangle = \int_{y \in X} \langle x, y\rangle d\tau$ holds.
The barycenter of $\tau \in P(X)$ exists, it is uniquely determined, and it is an element of $X$.
Let $h(\tau)$ be the barycenter of $\tau \in P(X)$. Then $(X,h)$ is an algebra for the $P$-monad.
The example given by the unit square $X= P(\mathbf{2}) \times P(\mathbf{2})=[0,1] \times [0,1] \subset \mathbb{R}^2$ fits into this theory nicely. The barycenter map is given by the marginalization map $m:P( P(\mathbf{2}) \times P(\mathbf{2})) \rightarrow P(\mathbf{2}) \times P(\mathbf{2})$. However when the space $X$ cannot be characterized by a finite number of parameters, the above theory using barycenters directly cannot be applied, even though the marginalization map $P( P(X) \times P(Y)) \rightarrow P(X) \times P(Y)$ is still a $P$-algebra. (It is still a barycenter map but not within the framework of $\mathbb{R}^n$.) These barycenter maps are the components of a natural transformation characterizing the counit of an adjunction of the $P$-monad.
More information concerning the use of barycenter maps in finding algebras can be found at Radon monad.
Finally, we note that Doberkat points out that for discrete Polish space $X$ that $X$ is disconnected, and hence there can be no continuous map $P{X} \rightarrow \mathbf{X}$. Hence $X$, irrelavant of any convex structure we endow it with, cannot be an algebra.
See also monads of probability, measures, and valuations.
Vladimir Voevodsky has also worked on a category theoretic treatment of probability theory, and gave few talks on this at IHES, Miami, in Moscow etc. Voevodsky had in mind applications in mathematical biology?, for example, population genetics:
…a categorical study of probability theory where “categorical” is understood in the sense of category theory. Originally, I developed this approach to probability to get a better understanding of the constructions which I had to deal with in population genetics. Later it evolved into something which seems to be also interesting from a purely mathematical point of view. On the elementary level it gives a category which is useful for the work with probabilistic constructions involving complicated combinations of stochastic processes of different types. On a more advanced level, applying in this context the old idea of a functor as a generalized object one gets a better view of the relationship between probability and the theory of (pre-)ordered topological vector spaces.
A talk in Moscow (20 Niv 2008, in Russian) can be viewed here, wmv 223.6 Mb. Abstract:
In early 60-ies Bill Lawvere defined a category whose objects are measurable spaces and morphisms are Markov kernels. I will try to show how this category allows one to think about many of the notions of probability theory in categorical terms and to connect probabilistic objects to objects of other types through various functors.
Voevodsky’s unfinished notes on categorical probability theory have been released posthumously.
Prakash Panangaden in Probabilistic Relations defines the category $SRel$ (stochastic relations) to have as objects sets equipped with a $\sigma$-field. Morphisms are conditional probability densities or stochastic kernels. So, a morphism from $( X, \Sigma_X)$ to $( Y, \Sigma_Y)$ is a function $h: X \times \Sigma_Y \to [0, 1]$ such that
If $k$ is a morphism from $Y$ to $Z$, then $k \cdot h$ from $X$ to $Z$ is defined as $(k \cdot h)(x, C) = \int_Y k(y, C)h(x, d y)$.
Panangaden’s definition differs from Giry’s in the second clause where subprobability measures are allowed, rather than ordinary probability measures.
Panangaden emphasises that the mechanism is similar to the way that the category of relations can be constructed from the power set functor. Just as the category of relations is the Kleisli category of the powerset functor over the category of sets Set, $SRel$ is the Kleisli category of the functor over the category of measurable spaces and measurable functions which sends a measurable space, $X$, to the measurable space of subprobability measures on $X$. This functor gives rise to a monad.
What is gained by the move from probability measures to subprobability measures? One motivation seems to be to model probabilistic processes from $X$ to a coproduct $X + Y$. This you can iterate to form a process which looks to see where in $Y$ you eventually end up. This relates to $SRel$ being traced.
There is a monad on $MeasureSpaces$, $1 + -: Meas \to Meas$. A probability measure on $1 + X$ is a subprobability measure on $X$. Panangaden’s monad is a composite of Giry’s and $1 + -$.
measure, probability measure, pushforward measure, convex mixture
Radon monad, distribution monad, extended probabilistic powerdomain
The adjunction underlying the Giry monad was originally developed by Lawvere in 1962, prior to the full recognition of the relationship between monads and adjunctions. Although P. Huber had already shown in 1961 that every adjoint pair gives rise to a monad, it wasn’t until 1965 that the constructions of Eilenberg-Moore, and Kleisli, made the essential equivalence of both concepts manifest.
Lawvere’s construction was written up as an appendix to a proposal to the Arms Control and Disarmament Agency, set up by President Kennedy as part of the State Department to handle planning and execution of certain treaties with the Soviet Union. This appendix was intended to provide a reasonable framework for arms control verification protocols (Lawvere 20).
At that time, Lawvere was working for a “think tank” in California, and the purpose of the proposal was to provide a means for verifying compliance with limitations on nuclear weapons. In the 1980’s, Michèle Giry was collaborating with another French mathematician at that time who was also working with the French intelligence agency, and she was able to obtain a copy of the appendix. Giry then developed and extended some of the ideas in the appendix (Giry 80)
Gian-Carlo Rota had also (somehow) obtained a copy of the appendix, which ended up in the library at The American Institute of Mathematics, and only became publicly available in 2012.
From Lawvere 20:
I’d like to say that the idea of the category of probabilistic mappings, the document corresponding to that was not part of a seminar, as some of the circulations say, essentially it was the document submitted to the arms control and disarmament agency after suitable checking that the Pentagon didn’t disagree with it. Because of the fact that for arms control agencies as a side responsibility the forming of arms control agreements and part of these agreements must involve agreed upon protocols of verification. So the idea of that paper did not provide such protocols, but it purported to provide reasonable framework within which such protocol can be formulated.
The idea originates with
W. Lawvere, The category of probabilistic mappings, ms. 12 pages, 1962 (Lawvere Probability 1962)
(notice that the statement of origin on p.1 is wrong)
and was picked up and published in:
Michèle Giry, A categorical approach to probability theory, Categorical aspects of topology and analysis (Ottawa, Ont., 1980), pp. 68–85, Lecture Notes in Math. 915 Springer 1982 (doi:10.1007/BFb0092872)
(there are allegedly a few minor analytically incorrect points and gaps in proofs, observed by later authors).
Historical comments on the appearance of Lawvere 62 are made in
According to E. Burroni (2009), the Giry monad appears also in
The article
shows, in effect, that the Giry monad restricted to countable measurable spaces (with the discrete $\sigma$-algebra) yields the restricted Giry functor $G|: \mathbf{Meas}_c \rightarrow \mathbf{Meas}$ which has the codensity monad $G$. This suggest that the natural numbers $\mathbb{N}$ are ‘’sufficient’‘ in some sense. Indeed, the full subcategory of Polish spaces consisting of the single object $N$ of all natural numbers with the powerset $\sigma$-algebra is codense in $Pol$ for the reasons delineated in the proof above - every continuous function $P{X} \rightarrow X$ is completely determined by its values on the countable dense subset of $P{X}$.
The article
views probability measures via double dualization, restricted to weakly averaging affine maps which preserves limits. (A more satisfactory description of probability measures arises from recognizing the need for viewing them as weakly-averaging countably affine maps, rather than just finite affine maps, as discussed in that article.)
Some corrections from an earlier version of that article, were pointed out in
Apart from these papers, there are similar developments in
Franck van Breugel, The metric monad for probabilistic nondeterminism, features both the Lawvere/Giry monad and Panangaden’s monad.
Ernst-Erich Doberkat, Characterizing the Eilenberg-Moore algebras for a monad of stochastic relations (pdf)
Ernst-Erich Doberkat, Kleisli morphisms and randomized congruences, Journal of Pure and Applied Algebra Volume 211, Issue 3, December 2007, Pages 638-664 https://doi.org/10.1016/j.jpaa.2007.03.003
N. N. Cencov, Statistical decisions rules and optimal Inference, Translations of Math. Monographs 53, Amer. Math. Society 1982
(blog comment) Cencov’s “category of statistical decisions” coincides with Giry’s (Lawvere’s) category. I ($\leftarrow$ somebody) have the sense that Cencov discovered this category independently of Lawvere although years later.
category cafe related to Giry monad: category theoretic probability, coalgebraic modal logic
Samson Abramsky et al. Nuclear and trace ideals in tensored ∗-Categories,arxiv:math/9805102, on the representation of probability theory through monads, which looks to work Giry’s monad into a context even more closely resembling the category of relations.
There is also relation with work of Jacobs et al.
Robert Furber, Bart Jacobs, Towards a categorical account of conditional probability, arxiv:1306.0831
Bart Jacobs, Probabilities, distribution monads and convex categories, Theoretical
Computer Science 412(28) (2011) pp.3323–3336. https://doi.org/10.1016/j.tcs.2011.04.005, (preprint)
J. Culbertson and K. Sturtz use the Giry monad in their categorical approach to Bayesian reasoning and inference (both articles contain further references to the categorical approach to probability theory):
Jared Culbertson and Kirk Sturtz, A categorical foundation for Bayesian probability, Applied Cat. Struc. 2013 (preprint as arXiv:1205.1488)
Jared Culbertson and Kirk Sturtz, Bayesian machine learning via category theory, 2013 (arxiv:1312.1445)
Elisabeth Burroni, Lois distributives. Applications aux automates stochastiques, TAC 22, 2009 pp.199-221 (journal page)
where she derives stochastic automata as algebras for a suitable distributive law on the monoid and Giry monads.
B. Fong has a section on the Giry monad in his paper on Bayesian networks:
See also:
Discussion of the Giry monad extended to simplicial sets and used to characterize quantum contextuality via simplicial homotopy theory:
Cihan Okay, Sam Roberts, Stephen D. Bartlett, Robert Raussendorf, Topological proofs of contextuality in quantum mechanics, Quantum Information and Computation 17 (2017) 1135-1166 [arXiv:1701.01888, doi:10.26421/QIC17.13-14-5]
Cihan Okay, Aziz Kharoof, Selman Ipek, Simplicial quantum contextuality, Quantum 7 (2023) 1009 [arXiv:2204.06648, doi:10.22331/q-2023-05-22-1009]
Aziz Kharoof, Cihan Okay, Homotopical characterization of strongly contextual simplicial distributions on cone spaces [arXiv:2311.14111]
exposition:
Last revised on August 9, 2024 at 03:38:07. See the history of this page for a list of all contributions to it.