nLab Giry monad

Contents

Contents

Idea

The Giry monad (Giry 80, following Lawvere 62) is the monad on a category of suitable spaces which sends each suitable space XX to the space of suitable probability measures on XX.

It is one of the main examples of a probability monad, and hence one of the main structures used in categorical probability.

Definition

The Giry monad is defined on the category of measurable spaces, assigning to each measurable space XX the space G(X)G(X) of all probability measures on XX endowed with the σ \sigma -algebra generated by the set of all the evaluation maps

ev U:G(X)[0,1] ev_U \colon G(X) \to [0,1]

sending a probability measure PP to P(U)P(U), where UU ranges over all the measurable sets of XX. The unit of the monad sends a point xXx \in X to the Dirac measure at xx, δ x\delta_x, while the monad-multiplication is defined by the natural transformation

μ X:G(G(X))G(X) \mu_{X} \;\colon\; G\big(G(X)\big) \longrightarrow G(X)

given by

μ X(Q)(U) qG(X)ev U(q)dQ. \mu_X (Q)(U) \;\coloneqq\; \textstyle{\int}_{q \in G(X)} ev_U(q) \,dQ \,.

This makes the endofunctor GG 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, PolPol, which are separable metric spaces for which a complete metric exists. The morphisms of this category are continuous functions.

Write

P:PolPol P \colon Pol \to Pol

for the endofunctor which sends a space, XX, to the space of probability measures on the Borel subsets of XX. P(X)P(X) is equipped with the weakest topology which makes the integration map τ Xfdτ\tau \mapsto \int_{X}f d\tau continuous for any ff, a bounded, continuous, real function on XX.

There is a natural transformation

μ X:P(P(X))P(X) \mu_{X}: P(P(X)) \to P(X)

given by

μ X(M)(A):= P(X)τ(A)M(dτ). \mu_X (M)(A) := \int_{P(X)} \tau(A) M(d\tau).

This makes the endofunctor PP into a monad, and this is the Giry monad on Polish spaces.

Properties

Kleisli category

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.

Algebras over the Giry monad

We can’t say anything about the GG-algebras on the category of measurable spaces due to lack of structure and set-theoretical issues. However, the GG monad restricts to the full subcategory of standard Borel space where we can construct a factorization of the GG monad which allows us to understand how GG algebras arise via expectation maps.

If XX is any standard space then the space of probability measures GXG{X} is a superconvex space with the structure defined pointwise, i.e., if {P i} i=1 \{P_i\}_{i=1}^{\infty} is a finite collection of probability measures on XX then, for every sequence {p i} i=1 \{p_i\}_{i=1}^{\infty} with each p i[0,1]p_i \in [0,1] the countable affine sum i=1 p iP i\sum_{i=1}^{\infty} p_i P_i, is also a probability measure, defined at the measurable set UU in XX by

( i=1 p iP i)(U)= i=1 p iP i(U). \big(\sum_{i=1}^{\infty} p_i P_i\big)(U) = \sum_{i=1}^{\infty} p_i P_i(U).

Lemma

Given any GG-algebra h:GXXh: G{X} \rightarrow X the space XX has the structure of a superconvex space which makes the measurable function hh a countably affine (measurable) map.
Moreover, for any measurable space YY and GG-algebra k:GYYk:G{Y} \rightarrow Y on it, if f:(X,h)(Y,k)f: (X, h) \rightarrow (Y, k) is a map of GG-algebras then the measurable function ff is also countably affine.

Proof

Given hh define the superconvex space structure on XX by

i=1 p ix i:=h( i=1 p iδ x i). \sum_{i=1}^{\infty} p_i x_i := h(\sum_{i=1}^{\infty} p_i \delta_{x_i}).

Because a GG-algebra hh must satisfy hμ X=hGhh \circ \mu_X = h \circ G{h} we have

(hμ X)( i=1 p iδ Q i) = (hGh)( i=1 p iδ Q i) h( i=1 p iQ i) = h( i=1 p iδ h(Q i)) = i=1 p ih(Q i) \begin{array}{lcl} (h \circ \mu_X)( \sum_{i=1}^{\infty} p_i \delta_{Q_i}) &=& (h \circ G{h})( \sum_{i=1}^{\infty} p_i \delta_{Q_i}) \\ h( \sum_{i=1}^{\infty} p_i Q_i) &=& h(\sum_{i=1}^{\infty} p_i \, \delta_{h(Q_i)}) \\ &=& \sum_{i=1}^{\infty} p_i h(Q_i) \end{array}

where the last line makes use of the definition of the convex structure on XX. Thus every GG-algebra is countably affine.

To prove the map of GG-algebras is a countably affine map we compute

f( i=1 p ix i) = f(h( i=1 p iδ x i)) = k(G(f)( i=1 p iδ x i) = k( i=1 p iδ f(x i)) = i=1 p if(x i) . \begin{array}{lcll} f(\sum_{i=1}^{\infty} p_i x_i) &=& f(h(\sum_{i=1}^{\infty} p_i \delta_{x_i})) & \\ &=& k(G(f)(\sum_{i=1}^{\infty} p_i \delta_{x_i}) & \\ &=& k(\sum_{i=1}^{\infty} p_i \delta_{f(x_i)}) & \\ &=& \sum_{i=1}^{\infty} p_i f(x_i) & \end{array}.

Let StdSCvx\mathbf{Std}\cap \mathbf{SCvx} be the category of standard spaces with a superconvex space structure with morphisms be countably affine measurable functions. Let V=[0,]V=[0,\infty] which is the one-point compactification of the interval [0,)[0,\infty). VV is a second-countable compact Hausdorff space so it is a Polish space, and hence VV with the Borel σ\sigma-algebra is a standard space. Let (StdSCvx) cos(\mathbf{Std} \cap \mathbf{SCvx})_{cos} denote the full subcategory of StdSCvx\mathbf{Std} \cap \mathbf{SCvx} consisting of those spaces which are coseparable by VV.

Given any standard space AA and any PG(A)P \in G(A) let Std(A,V)P˜V\mathbf{Std}(A, V) \xrightarrow{\tilde{P}} V denote the functional sending f AfdPf \mapsto \int_A f \, dP. Let V A=(StdSCvx) cos(A,V)V^A = (\mathbf{Std} \cap \mathbf{SCvx})_{cos}(A, V). Taking A=VA=V we obtain the space V VV^V of endomaps on VV. Recall that a VV-generalized point of AA is a functional P˜\tilde{P} satisfying, for all ϕV V\phi \in V^V and all mV Am \in V^A, the equation

ϕ(P˜(m))=P˜(ϕm). \phi \big( \tilde{P}(m) \big) = \tilde{P}(\phi \circ m).

The reader can verify than any such VV-generalized element P˜\tilde{P} is VV-linear, weakly averaging, and additive.

We say an object AA in (StdSCvx) cos(\mathbf{Std} \cap \mathbf{SCvx})_{cos} satisfies the fullness property if and only if for every PG(A)P \in G(A) the property

mV Am 1(P˜(m))={aA|P˜(m)=m(a)mV A} \displaystyle{ \bigcap_{m \in V^A}} m^{-1}\big(\tilde{P}(m)\big) = \{a \in A \, | \, \tilde{P}(m)=m(a) \quad \forall m \in V^A\} \ne \emptyset

holds.

Lemma

Every VV-generalized element P˜\tilde{P} of AA which satisfies the fullness property is a point, i.e, P˜=ev a\tilde{P} =ev_a for a unique element aAa \in A. (ev aev_a is the evaluation map at the point aa.)

Proof

Since the fullness property is satisfied there exist at least one element aAa \in A such that P˜(m)=m(a)\tilde{P}(m)=m(a). Since AA lies in (StdSCvx) cos(\mathbf{Std} \cap \mathbf{SCvx})_{cos} it is coseparated by VV, and hence there is at most one element aAa \in A satisfying, for all countably affine measurable maps AmVA \xrightarrow{m} V, the equation P˜(m)=m(a)\tilde{P}(m)=m(a).

Define Std SCvx\mathbf{Std}_{SCvx} to be the full subcategory of (StdSCvx) cos(\mathbf{Std} \cap \mathbf{SCvx})_{cos} consisting of those objects which satisfy the fullness property.

Note that VV is an object in Std SCvx\mathbf{Std}_{SCvx}. (This is exercise 8.23 in the text Sets for Mathematics by Lawvere and Rosebrugh.)

We now proceed to show that for every standard space XX that G(X)G(X) is an object in Std SCvx\mathbf{Std}_{SCvx}.

Lemma

Let (X,σ(𝔽))(X, \sigma(\mathbb{F})) be a standard space. Then for QG 2(X)Q \in \G^2(X) we have

U𝔽ev U 1(Q˜(ev U))=μ X(Q)\displaystyle{ \bigcap_{U \in \mathbb{F}}} ev_U^{-1}\big( \tilde{Q}(ev_U) \big) = \mu_X(Q)

Proof

We have

ev U 1(Q˜(ev U))={RG(X)|R(U)=μ X(Q)[U]} ev_U^{-1}\big( \tilde{Q}(ev_U) \big) = \{R \in G(X) \, | \, R(U)=\mu_X(Q)[U] \}

so taking the intersection over all elements in the generating field (with a basis) yields the result using the well known result if XX is a standard space and P,RG(X)P,R \in G(X) satisfy P(U)=R(U)P(U)=R(U) for all U𝔽U \in \mathbb{F} then P=RP=R.

The proof of the next three lemmas, which are all straight forward, can be found in (Sturtz 25)

Lemma

Let (X,σ(𝔽))(X, \sigma(\mathbb{F})) be a standard space. Then every countably affine measurable function G(X)mVG(X) \xrightarrow{m} V is a countable affine sum of the form m= iλ iev U im = \sum_i \lambda_i ev_{U_i} where λ iV\lambda_i \in V and U i𝔽U_i \in \mathbb{F}.

Lemma

Let (X,σ(𝔽))(X, \sigma(\mathbb{F})) be a standard space. Then

G(X)mVm 1(Q˜(m))=μ X(Q)= U𝔽ev U 1(Q˜(ev U)). \displaystyle{ \bigcap_{G(X) \xrightarrow{m} V}}m^{-1}\big(\tilde{Q}(m)\big) =\mu_X(Q) = \displaystyle{\bigcap_{U \in \mathbb{F}}} ev_U^{-1}\big( \tilde{Q}(ev_U) \big).

Lemma

Let (X,σ(𝔽))(X, \sigma(\mathbb{F})) be a standard space. Then G(X)G(X) is an object in Std SCvx\mathbf{Std}_{SCvx}.

This lemma shows there exists a functor StdG^Std SCvx\mathbf{Std} \xrightarrow{\hat{G}} \mathbf{Std}_{SCvx} which is the Giry monad functor with codomain Std SCvx\mathbf{Std}_{SCvx}, and coupled with the partial forgetful functor Std SCvx𝒰 SCvxStd\mathbf{Std}_{SCvx} \xrightarrow{\mathcal{U}_{SCvx}} \mathbf{Std} which forgets the superconvex space structure, we obtain a factorization of the GG monad. (This will be an adjoint factorization once we prove some more facts.)

Theorem

Let V\mathbf{V} denote the full subcategory of Std SCvx\mathbf{Std}_{SCvx} consisting of the single object VV. The functor defined (on objects) by

Std SCvx op 𝒴 Set V A Std SCvx(A,) \begin{array}{ccc} \mathbf{Std}_{SCvx}^{op} & \xrightarrow{\mathcal{Y}} & \mathbf{Set}^{\mathbf{V}} \\ A & \mapsto & \mathbf{Std}_{SCvx}(A, \bullet) \end{array}

is a full and faithful functor.

Proof

In the category Std SCvx\mathbf{Std}_{SCvx} every countably affine measurable function is determined by its value on points 1aA1 \xrightarrow{a} A. Hence to prove the fully faithful property it suffices to prove those properties on points.

Since AA lies in Std SCvx\mathbf{Std}_{SCvx}, which has VV as a coseparator, the faithful property holds. The fullness property follows from Lemma .

Let δ a\delta_a denote the Dirac measure at aa.

Corollary

If AA is an object in Std SCvx\mathbf{Std}_{SCvx} then there exists a unique countably affine measurable function G(A)ϵ AAG(A) \xrightarrow{\epsilon_A} A such that ϵ A(δ a)=a\epsilon_A(\delta_a)=a for all aAa \in A.

Proof

Let VιStd SCvx\mathbf{V} \xrightarrow{\iota} \mathbf{Std}_{SCvx} denote the inclusion functor. Let AιA\downarrow \iota denote the slice category of arrows AmVA \xrightarrow{m} V, and let AιπVA \downarrow \iota \xrightarrow{\pi} \mathbf{V} denote the projection functor. For 𝒟 A=AιπVιStd SCvx\mathcal{D}_A = A \downarrow \iota \xrightarrow{\pi} \mathbf{V} \xrightarrow{\iota} \mathbf{Std}_{SCvx} the theorem is equivalent to saying A=lim𝒟 AA = \lim \mathcal{D}_A with the projection map at component mm being mm.

Consider the cone over 𝒟 A\mathcal{D}_A with vertex G(A)G(A) and natural transformation components 𝔼 (m)=𝔼 (id V)G(m)\mathbb{E}_{\bullet}(m) = \mathbb{E}_{\bullet}(id_V) \circ \G(m).

Since A=lim𝒟 AA=\lim \mathcal{D}_A there exists a unique Std SCvx\mathbf{Std}_{SCvx}-morphism G(A)ϵ AAG(A) \xrightarrow{\epsilon_A} A such that mϵ A=𝔼 (m)m \circ \epsilon_A = \mathbb{E}_{\bullet}(m) for all countably affine maps AmVA \xrightarrow{m} V. It follows that on δ aG(A)\delta_a \in G(A) that, for all AmVA \xrightarrow{m} V in Std SCvx\mathbf{Std}_{SCvx} that m(ϵ A(δ a))=m(a)m(\epsilon_A(\delta_a)) = m(a). Since VV is a coseparator in Std SCvx\mathbf{Std}_{SCvx} it follows ϵ A(δ a)=a\epsilon_A(\delta_a)=a.

A more appropriate notation for the unique morphism ϵ A\epsilon_A is 𝔼 (id A)\mathbb{E}_{\bullet}(id_A) which, in the special case of AA lying in an \mathbb{R}-vector space coincides with the usual interpretation. For an arbitrary space AA the function G(A)𝔼 (id A)AG(A) \xrightarrow{\mathbb{E}_{\bullet}(id_A)} A is the unique morphism such that, for every PG(A)P \in G(A), 𝔼 P(id A)A\mathbb{E}_P(id_A) \in A is the unique point such that m(𝔼 P(id A))= Am(a)dPm(\mathbb{E}_{P}(id_A)) = \int_A m(a) \, dP for all countably affine measurable functions AmVA \xrightarrow{m} V.

Lemma

The function ϵ A\epsilon_A is a GG-algebra.

Proof

The property ϵ A(δ a)=a\epsilon_A(\delta_a)=a follows from the preceding corollary. To prove the property ϵ Aμ A=ϵ AG(ϵ A)\epsilon_A \circ \mu_A = \epsilon_A \circ G(\epsilon_A) compose both sides of that equation by a countably affine measurable map AmVA \xrightarrow{m} V. If we spell both sides of that equation out, using the property Amd(μ X(Q))= PG(A)𝔼 P(m)dQ\int_A m \, d(\mu_X(Q)) = \int_{P \in \G(A)} \mathbb{E}_{P}(m) dQ, the equation holds valid. The result of the lemma follows from the property that VV coseparates, i.e, the set of morphisms AmVA \xrightarrow{m} V are jointly monic on AA.

Note that what we have shown is that Std SCvx\mathbf{Std}_{SCvx} is a subcategory of Alg G\mathbf{Alg}_{G}. Proving the reverse inclusion (or finding a counterexample) is an interesting open problem.

Concerning the PP algebras the above method of using VV generalize points can be used to obtain similar results. (Doberkat 03) gives a different representation for the algebras of PP, although, like the Eilenberg-Moore characterization, the representation is descriptive but not constructive. His representation for the algebras is based upon the idea that we want continuous maps h:P(X)Xh:P(X) \rightarrow X such that the ‘fibres’ are convex and closed, and such that δ x\delta_x, the Delta distribution on xx, is in the fibre over xx. And there’s another condition which requires a compact subset of P(X)P(X) to be sent to a compact subset of XX.

As an example of PP-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 nX \subset \mathbb{R}^n as a bounded, closed, and convex subset of the Euclidean space n\mathbb{R}^n. A vector x nx_{\star} \in \mathbb{R}^n is called a barycenter of the probability measure τP(X)\tau \in P(X) iff, for all linear functionals gg on n\mathbb{R}^n, the property g(x )= Xg(x)dτg(x_{\star}) = \int_X g(x) d\tau holds. Since every linear functional on n\mathbb{R}^n is given by g()=x,g(\cdot)=\langle x, \cdot \rangle for a unique x nx \in \mathbb{R}^n, the defining property of a barycenter (for convex subsets of n\mathbb{R}^n) given above is equivalent to saying that, for all x nx \in \mathbb{R}^n, the property x,x = yXx,ydτ\langle x, x_{\star} \rangle = \int_{y \in X} \langle x, y\rangle d\tau holds.

Lemma

The barycenter of τP(X)\tau \in P(X) exists, it is uniquely determined, and it is an element of XX.

Theorem

Let h(τ)h(\tau) be the barycenter of τP(X)\tau \in P(X). Then (X,h)(X,h) is an algebra for the PP-monad.

The example given by the unit square X=P(2)×P(2)=[0,1]×[0,1] 2X= 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(2)×P(2))P(2)×P(2)m:P( P(\mathbf{2}) \times P(\mathbf{2})) \rightarrow P(\mathbf{2}) \times P(\mathbf{2}). However when the space XX 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)×P(Y))P(X)×P(Y)P( P(X) \times P(Y)) \rightarrow P(X) \times P(Y) is still a PP-algebra. (It is still a barycenter map but not within the framework of n\mathbb{R}^n.) These barycenter maps are the components of a natural transformation characterizing the counit of an adjunction of the PP-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 XX that XX is disconnected, and hence there can be no continuous map PXXP{X} \rightarrow \mathbf{X}. Hence XX, irrelavant of any convex structure we endow it with, cannot be an algebra.

See also monads of probability, measures, and valuations.

Voevodsky’s work

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:

See Miami Talk abstract

…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.

Panangaden’s monad

Prakash Panangaden in Probabilistic Relations defines the category SRelSRel (stochastic relations) to have as objects sets equipped with a σ\sigma-field. Morphisms are conditional probability densities or stochastic kernels. So, a morphism from (X,Σ X)( X, \Sigma_X) to (Y,Σ Y)( Y, \Sigma_Y) is a function h:X×Σ Y[0,1]h: X \times \Sigma_Y \to [0, 1] such that

  1. BΣ Y.λxX.h(x,B)\forall B \in \Sigma_Y . \lambda x \in X . h(x, B) is a bounded measurable function,
  2. xX.λBΣ Y.h(x,B)\forall x \in X . \lambda B \in \Sigma_Y . h(x, B) is a subprobability measure on Σ Y\Sigma_Y.

If kk is a morphism from YY to ZZ, then khk \cdot h from XX to ZZ is defined as (kh)(x,C)= Yk(y,C)h(x,dy)(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, SRelSRel is the Kleisli category of the functor over the category of measurable spaces and measurable functions which sends a measurable space, XX, to the measurable space of subprobability measures on XX. 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 XX to a coproduct X+YX + Y. This you can iterate to form a process which looks to see where in YY you eventually end up. This relates to SRelSRel being traced.

There is a monad on MeasureSpacesMeasureSpaces, 1+:MeasMeas1 + -: Meas \to Meas. A probability measure on 1+X1 + X is a subprobability measure on XX. Panangaden’s monad is a composite of Giry’s and 1+1 + -.

History

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.

References

The idea originates with

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

  • O. de la Tullaye, L’intégration considérée comme l’algèbre d’un triple. Rapport de Stage de D.E.A. manuscrit 1971.

The article

shows, in effect, that the Giry monad restricted to countable measurable spaces (with the discrete σ\sigma-algebra) yields the restricted Giry functor G|:Meas cMeasG|: \mathbf{Meas}_c \rightarrow \mathbf{Meas} which has the codensity monad GG. 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 NN of all natural numbers with the powerset σ\sigma-algebra is codense in PolPol - every continuous function PXXP{X} \rightarrow X is completely determined by its values on the countable dense subset of PXP{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, obtained by double dualizing into [0,][0,\infty], which then yields the characterization of GG-algebras summarized above, which is from the article

  • Kirk Sturtz, A factorization of the Giry monad on standard Borel spaces using [0,][0,\infty]-generalized points, [[arXiv:2409.14861]]

Some corrections from an earlier version of the Categorical Probability Theory 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.

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:

  • Fong: Causal Theories - A Categorical Perspective on Bayesian Networks, (2013) arXiv:1301.6201

See also:

Discussion of the Giry monad extended to simplicial sets and used to characterize quantum contextuality via simplicial homotopy theory:

exposition:

category: probability

Last revised on June 15, 2025 at 08:51:03. See the history of this page for a list of all contributions to it.