nLab Giry monad

Idea

The Giry monad (Giry 1980, following Lawvere 1962) is the monad on a category of suitable spaces which sends each suitable space $X$ to the space of suitable probability measures on $X$.

This is one of the main examples of probability monads, and hence one of the main structures used in categorical probability.

Definition

The Giry monad is defined on the category of measurable spaces, $\mathbf{Meas}$, 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

$$ev_U \colon G(X) \to [0,1]$$

sending a probability measure $P$ to $P(U)$, where $U$ ranges over all the measurable sets of $X$. The unit of the monad at $X$, traditionally denoted $\eta_X$, sends a point $x \in X$ to the Dirac measure at $x$, $\delta_x$, while the monad-multiplication is defined by the natural transformation

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

given by

$$\mu_X (Q)(U) =\textstyle{\int}_{q \in G(X)} ev_U(q) \,dQ \,.$$

This makes the endofunctor $G$ into a monad, $(G, \eta, \mu)$, 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, $\mathbf{Pol}$, which are separable metric spaces for which a complete metric exists. The morphisms of this category are continuous functions.

Write

$$P \colon \mathbf{Pol} \to \mathbf{Pol}$$

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

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

given by

$$\mu_X (M)(A) := \int_{P(X)} \tau(A) dM(\tau).$$

This makes the endofunctor $P$ 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 (=$\mathbf{Meas}_{G}$). It is one of the most important examples of a Markov category.

Algebras over the Giry Monad

Let $\mathbf{Alg}_G$ denote the category of algebras of the $G$-monad which has as objects those measurable spaces $X$ for which there exists a $G$-algebra $G(X) \xrightarrow{h} X$ which is an object in the Eilenberg-Moore category of the $G$-monad. The morphisms of $\mathbf{Alg}_G$ are those measurable functions $X \xrightarrow{m} Y$ such that $m$ constitutes an arrow in the Eilenberg-Moore category of the $G$-monad.

If $X$ is any measurable space then the space of probability measures $G{X}$ has a convex space structure defined pointwise: if $\{P_i\}_{i=1}^{n}$ is finite collection of probability measures on $X$ then, for every sequence $\{p_i\}_{i=1}^{n}$ with each $p_i \in [0,1]$ such that $\sum_{i=1}^{n} p_i = 1$, 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

$$\big(\sum_{i=1}^{n} p_i P_i\big)(U) = \sum_{i=1}^{n} p_i P_i(U).$$

Lemma

Given any $G$-algebra $G(X) \xrightarrow{h} X$ the base space $X$ has the structure of a convex space which makes the measurable function $h$ an affine (measurable) map. Moreover, morphisms of $G$-algebras are also affine maps.

Proof

Given $h$ define the convex space structure on $X$ by

$$\sum_{i=1}^{n} p_i x_i := h(\sum_{i=1}^{n} p_i \delta_{x_i}).$$

Because a $G$-algebra $h$ must satisfy $h \circ \mu_X = h \circ G{h}$ we have, for any finite sequence $Q_i \in GX$,

$$\begin{array}{lcl} (h \circ \mu_X)( \sum_{i=1}^{n} p_i \delta_{Q_i}) &=& (h \circ G{h})( \sum_{i=1}^{n} p_i \delta_{Q_i}) \\ h( \sum_{i=1}^{n} p_i Q_i) &=& h(\sum_{i=1}^{n} p_i \, \delta_{h(Q_i)}) \\ &=& \sum_{i=1}^{n} p_i h(Q_i) \end{array}$$

where the last line makes use of the definition of the convex structure on $X$. Thus, every $G$-algebra is affine.

To prove that any map of $G$-algebras $f: (X, h) \to (Y, k)$ is an affine map, we compute

$$\begin{array}{lcll} f(\sum_{i=1}^{n} p_i x_i) &=& f(h(\sum_{i=1}^{n} p_i \delta_{x_i})) & \\ &=& k(G(f)(\sum_{i=1}^{n} p_i \delta_{x_i})) & \\ &=& k(\sum_{i=1}^{n} p_i \delta_{f(x_i)}) & \\ &=& \sum_{i=1}^{n} p_i f(x_i) & \end{array}.$$

Lemma 3.1 shows that any object in $\mathbf{Alg}_{G}$ lies in both the categories $\mathbf{Meas}$ and $\mathbf{Cvx}$, where $\mathbf{Cvx}$ is the category of convex spaces. Similarly, any morphism in $\mathbf{Alg}_{G}$ is also a morphism in both $\mathbf{Meas}$ and $\mathbf{Cvx}$. An example of such an object is $\mathbb{R}_{\infty}$ which, as a measurable space, is the one-point compactification of the real-line with the Borel $\sigma$-algebra. As a convex space, $\mathbb{R}_{\infty}$ has the natural convex space structures on the real-line extended by the point $\infty$ with the property that $p r + (1-p) \infty = \infty$ for all $r \in \mathbb{R}$ and all $p \in [0,1)$.

Let $\mathbf{Meas} \cap \mathbf{Cvx}$ denote the category whose objects $X$ are convex spaces and, in addition, $X$ also posseses a measurable space structure such that all the algebraic operations of taking finite affine sums of elements, $\prod_{i=1}^n X \xrightarrow{\sum_{i=1}^n p_i \pi_i} X$, where $\pi_i$ is the coordinate projection function, is a measurable function. Note these maps are always an affine function because of the convex space structure on product spaces. Moreover, we require that each object in $\mathbf{Meas} \cap \mathbf{Cvx}$ have enough affine measurable functions $X \rightarrow \mathbb{R}_{\infty}$ to coseparate the points of $X$. The objects of $\mathbf{Meas} \cap \mathbf{Cvx}$ are called measurable convex spaces. The morphisms of $\mathbf{Meas} \cap \mathbf{Cvx}$ are affine measurable functions. Because $\mathbb{R}_{\infty}$ is a coseparator in $\mathbf{Cvx}$ it follows that $\mathbb{R}_{\infty}$ is a coseparator in $\mathbf{Meas} \cap \mathbf{Cvx}$.

For $X$ any measurable space the space $G(X)$ with the pointwise convex space structure and measurable structure is a measurable convex space because the evaluation maps $G(X) \xrightarrow{ev_U} \mathbb{R}_{\infty}$ coseparate any two distinct probability measures in $G(X)$ and the operation of taking affine sums is a measurable function because

Lemma

Let $p \in [0,1]$. The operation of taking an affine sum

$$G(X) \times G(X) \xrightarrow{ p \pi_1 + (1-p) \pi_2} G(X)$$

is a measurable function. This implies that all finite affine sum operators are measurable.

Proof

Let $U$ be a measurable set in $X$. By the pointwise convex space structure of $G(X)$ we have

$$ev_U \circ \big(p \pi_1 + (1-p) \pi_2\big) = \big(p \hat{\pi}_1 + (1-p) \hat{\pi}_2 \big) \circ (ev_U \times ev_U)$$

where $[0,1] \times [0,1] \xrightarrow{p \hat{\pi}_1 + (1-p) \hat{\pi}_2} [0,1]$ is the measurable affine sum on elements of the measurable convex space $[0,1]$, and $G(X) \times G(X) \xrightarrow{ev_U \times ev_U} [0,1] \times [0,1]$ is an affine measurable function because $ev_U$ is an affine measurable function. Since the right-hand side of the equation is measurable it follows, for every measurable set $W$ of $[0,1]$, that

$$(p \pi_1 + (1-p) \pi_2)^{-1}\big(ev_U^{-1}(W)\big) \in \Sigma_{G(X) \times G(X)}.$$

This last equation is true for all measurable sets $U$ in $X$, and since the affine measurable functions $ev_U$ generate the initial $\sigma$-algebra on $G(X)$ it follows that the function $(p \pi_i + (1-p) \pi_2)$ is measurable.

The second statement follows from the observation that we can use induction on the number of components in a product space, and, assuming $n$ components, that we are free to choose $n-1$ parameters $p_i \in [0,1]$ freely. Since every affine sum of $n$ elements is uniquely defined by $n-1$ parameters the result follows.

Given any measurable space $X$ and any $P \in G(X)$ the expected value of an $\mathbb{R}_{\infty}$-valued measurable function is given by the functional

$$\begin{array}{rcc} hom_{\mathbf{Meas}}(X, \mathbb{R}_{\infty}) &\xrightarrow{\hat{\mathbb{E}}_{P}}& \mathbb{R}_{\infty} \\ f & \mapsto & \int_X f \, dP \end{array}.$$

which is (1) weakly averaging: $\mathbb{E}_{P}(\overline{c}) = c$ for every constant function $X \xrightarrow{\overline{c}} \mathbb{R}_{\infty}$ with value $c \in\mathbb{R}_{\infty}$, and (2) scalar equivariant: $\mathbb{E}_P(\lambda \cdot f) = \lambda \cdot \mathbb{E}_P(f)$ for all $\lambda \in \mathbb{R}_{\infty}$. If we let $\mathbb{R}_{\infty}^{hom_{\mathbf{Meas}}(X,\mathbb{R}_{\infty})}|_{wa+se}$ denote the set of all weakly averaging and scalar equivariant functionals from $hom_{\mathbf{Meas}}(X, \mathbb{R}_{\infty})$ to $\mathbb{R}_{\infty}$ then we have a bijective correspondence between $G(X)$ and $\mathbb{R}_{\infty}^{hom_{\mathbf{Meas}}(X,\mathbb{R}_{\infty})}|_{wa+se}$. The correspondence is $P \mapsto \hat{\mathbb{E}}_{P}$ and $J \mapsto \hat{J}$ where the probability measure $\hat{J}$ is defined on a measurable set $U$ by $\hat{J}(U) = J(\chi_U)$.

Let $\mathbb{R}_{\infty}^X = hom_{\mathbf{Meas} \cap \mathbf{Cvx}}(X, \mathbb{R}_{\infty})$. Taking $X=\mathbb{R}_{\infty}$ we obtain the space $\mathbb{R}_{\infty}^{\mathbb{R}_{\infty}}$ of affine measurable endomaps on $\mathbb{R}_{\infty}$.

An $\mathbb{R}_{\infty}$-generalized point of an object $X$ in $\mathbf{Meas} \cap \mathbf{Cvx}$ is a functional $\mathbb{R}_{\infty}^X \xrightarrow{J} \mathbb{R}_{\infty}$ satisfying, for all $\phi \in \mathbb{R}_{\infty}^{\mathbb{R}_{\infty}}$ and all $f \in \mathbb{R}_{\infty}^X$, the equation

$$\phi \big( J(f) \big) = J(\phi \circ f)$$

which implies that $J$ is (1) weakly averaging, and (2) scalar equivariant. (Generalized points are defined in Definition 8.19 of Sets for Mathematics, and several basic properties are discussed therein.)

Note that if $P \in G(X)$ then the functional $\mathbb{R}_{\infty}^X \xrightarrow{\mathbb{E}_P} \mathbb{R}_{\infty}$, which is the restriction of the functional $\hat{ \mathbb{E}}_P$ to affine measurable functions, is an $\mathbb{R}_{\infty}$-generalized point of $X$ since, for all $f \in \mathbb{R}_{\infty}^X$ and all $\phi \in \mathbb{R}_{\infty}^{ \mathbb{R}_{\infty}}$, $\mathbb{E}_P(\phi \circ f) = \phi( \mathbb{E}_P(f) )$. Since we are restricting the operators $\hat{\mathbb{E}}_{P}$ to operate only on affine measurable functions, yielding the operators $\mathbb{E}_P$, we have

Lemma

If $J$ is an $\mathbb{R}_{\infty}$-generalized element of $X$ then there exists a $P \in G(X)$ such that $J=\mathbb{E}_P$.

Proof

Let $X$ be an object in $\mathbf{Meas} \cap \mathbf{Cvx}$. We have the inclusion function $hom_{\mathbf{Meas} \cap \mathbf{Cvx}}(X, \mathbb{R}_{\infty}) \xrightarrow{\iota} hom_{\mathbf{Meas}}(X, \mathbb{R}_{\infty})$ which induces the restriction mapping

$$\mathbb{R}_{\infty}^{hom_{\mathbf{Meas}}(X,\mathbb{R}_{\infty})}|_{wa+se} \xrightarrow{\mathbb{R}_{\infty}^{\iota}} \mathbb{R}_{\infty}^{hom_{\mathbf{Meas} \cap \mathbf{Cvx}}(X, \mathbb{R}_{\infty})}|_{\mathbb{R}_{\infty}-generalized pt}$$

which is a surjective function. (The conditions on both functional spaces are identical: the elements are weakly averaging and scalar equivariant.) This specifies an equivalence relation on the set $\mathbb{R}_{\infty}^{hom_{\mathbf{Meas}}(X,\mathbb{R}_{\infty})}|_{wa+se}$ defined by $\hat{\mathbb{E}}_P \cong \hat{\mathbb{E}}_Q$ if and only if the restriction of those functionals are equal on the set of all affine measurable functions $X \rightarrow \mathbb{R}_{\infty}$. Thus every $\mathbb{R}_{\infty}$-generalized point of $X$ comes from a functional $\hat{\mathbb{E}}_P$, which in turn arises from the probability measure $P$ on $X$.

We say an object $X$ in $\mathbf{Meas} \cap \mathbf{Cvx}$ satisfies the fullness property if and only if for every $P \in G(X)$ the property

$$\displaystyle{ \bigcap_{f \in \mathbb{R}_{\infty}^X}} f^{-1}\big(\mathbb{E}_{P}(f)\big) = \{x \in X \, | \, \mathbb{E}_P(f)=f(x) \quad \forall f \in \mathbb{R}_{\infty}^X\} \ne \emptyset$$

holds.

Define $\mathbf{Cvx}_{Meas}$ to be the full subcategory of $\mathbf{Meas} \cap \mathbf{Cvx}$ consisting of those objects which satisfy the fullness property.
The space $\mathbb{R}_{\infty}$ is an object in $\mathbf{Cvx}_{Meas}$ because, for every $P \in G(\mathbb{R}_{\infty})$, we have $\mathbb{E}_P(id_{\mathbb{R}_{\infty}}) \in \cap_{f \in \mathbb{R}_{\infty}^{\mathbb{R}_{\infty} }} f^{-1}(\mathbb{E}_{P}(f))$. (See exercise 8.23 in Sets for Mathematics.) Trivially, the space $\mathbf{1}$ is also an object in $\mathbf{Cvx}_{Meas}$.

Theorem

The restricted Yoneda embedding functor defined (on objects) by

$$\begin{array}{ccc} \mathbf{Cvx}_{Meas}^{op} & \xrightarrow{\mathcal{Y}} & \mathbf{Set}^{\mathcal{R}} \\ X & \mapsto & hom_{\mathbf{Cvx}_{Meas}}(X, \bullet) \end{array}$$

is a full and faithful functor.

Proof

In the category $\mathbf{Cvx}_{Meas}$ every affine measurable function $X \xrightarrow{g} Y$ is determined by its value on points $\mathbf{1} \xrightarrow{x} X$. Hence to prove the fully faithful property it suffices to prove those properties on points.

Faithful: Note $\mathcal{Y}(x)$ is the evaluation map $\mathbb{R}_{\infty}^X \xrightarrow{ev_x} \mathbb{R}_{\infty}$. Let $\mathbf{1} \xrightarrow{x_i} X$, for $i=1,2$ be two points of $X$. If $f(x_1) = \mathcal{Y}(x_1)f = \mathcal{Y}(x_2)f = f(x_2)$ for all $f \in \mathbb{R}_{\infty}^X$, then since $X$ has enough affine measurable maps to $\mathbb{R}_{\infty}$ to coseparate points it follows that $x_1=x_2$ and $\mathcal{Y}$ is faithful functor.

Full: If $J \in Nat( hom(X, \cdot), hom(\mathbf{1}, \cdot) )$ then, at the single component of $\mathcal{R}$, $J$ is a functional $\mathbb{R}_{\infty}^X \rightarrow \mathbb{R}_{\infty}$ which is weakly averaging and scalar covariant. In other words, $J$ is an $\mathbb{R}_{\infty}$-generalized point of $X$. Now to complete the proof we employ Lemma 3.3: $J = \mathbb{E}_P$ for some $P \in \G(X)$. Then, because $X \in_{ob} \mathbf{Cvx}_{Meas}$, it satisfies the fullness property and we have, for all $f \in \mathbb{R}_{\infty}^X$, the property that $J(f) = \mathbb{E}_P(f) \in Im(f)$. So there exists an element $x_f \in X_f = \{x \in X \, | \, J(f) = f(x) \}$ such that $J(f)=f(x_f)$. But the fullness property says $\cap_f X_f \ne \emptyset$. Thus there exist an $x \in X$ such that $J(f) = f(x)$ for all $f \in \mathbb{R}_{\infty}^X$ from which it follows that $\mathcal{Y}(x) = ev_x$ and we conclude that $\mathcal{Y}$ is a full functor.

Corollary

Let $\mathcal{R}$ denote the full subcategory of $\mathbf{Cvx}_{Meas}$ consisting of the single object $\mathbb{R}_{\infty}$, and let $\iota: \R \hookrightarrow \mathbf{Cvx}_{Meas}$ denote the inclusion functor. The right Kan extension of $\iota$ along $\iota$ is the identity functor on $\mathbf{Cvx}_{Meas}$ along with the identity natural transformation $id: id_{\mathbf{Cvx}_{Meas}} \circ \iota \Rightarrow \iota$. In other words, $\iota$ is a codense functor.

Proof

We can compute the right Kan extension of $\iota$ along $\iota$ pointwise, for each $X \in_{ob} \mathbf{Cvx}_{Meas}$, as $\lim \mathcal{D}_X$ where

$$\mathcal{D}_X = X \downarrow \iota \xrightarrow{\pi} \mathcal{R} \xrightarrow{\iota} \mathbf{Cvx}_{Meas}$$

where $X\downarrow \iota$ denotes the slice category whose objects consist of affine measurable functions $X \xrightarrow{f} \mathbb{R}_{\infty}$, and $X \downarrow \iota \xrightarrow{\pi} \mathcal{R}$ denotes the projection functor.

We claim that $\lim \mathcal{D}_X = (X, \{f\}_{f \in \mathbb{R}_{\infty}^X})$. To obtain a contradiction, suppose to the contrary that $X$ is not the limit of $\mathcal{D}_X$. Then there exists a cone $\mathbf{1} \xrightarrow{\beta} \mathcal{D}_X$ such that there exists no $x \in X$ such that

(1)$$\{\beta_f\}_{f \in \mathbb{R}_{\infty}^X} = \{f(x)\}_{f \in \mathbb{R}_{\infty}^X} = \{ ev_x(f) \}_{f \in \mathbb{R}_{\infty}^X}.$$

But the functional

$$\begin{array}{ccc} \mathbb{R}_{\infty}^X & \xrightarrow{\beta_{\bullet}} & \mathbb{R}_{\infty} \\ f & \mapsto & \beta_f \end{array}$$

is an $\mathbb{R}_{\infty}$-generalized point of $X$ because $\beta$ is a cone over the diagram $\mathcal{D}_X$. That it to say it satisfies $\beta_{\phi \circ f} = \phi \circ \beta_f$ for all $\phi \in \mathbb{R}_{\infty}^{\mathbb{R}_{\infty}}$ and all $f \in \mathbb{R}_{\infty}^X$. But the hypothesis that there exists no $x \in X$ such that equation (1) holds leads to the contradiction that there does exists an $\mathbb{R}_{\infty}$-generalized point of $X$ given by the functional $\beta_{\bullet}$ which is not an evaluation map $ev_x$ for some $x \in X$. This contradicts the result in Theorem 3.4 that the restricted Yoneda embedding $\mathcal{Y}$ is a full functor. (When $X=\lim \mathcal{D}_X$ it follows that the Kan extension is also the identity function on morphisms.) Hence we conclude $(X, \{f\}_{f \in \mathbb{R}_{\infty}^X})= \lim \mathcal{D}_X$. This proves that the right Kan extension of $\iota$ along $\iota$ consists of the identity functor on $\mathbf{Cvx}_{Meas}$ from which it readily follows that the natural transformation $id_{\mathbf{Cvx}_{Meas}} \circ \iota \rightarrow \iota$ is the identity natural transformation.

Corollary

If $X$ is an object in $\mathbf{Cvx}_{Meas}$ then there exists a unique affine measurable function $G(X) \xrightarrow{\epsilon_X} X$ such that $\epsilon_X(\delta_x)=x$ for all $x \in X$.

Proof

From the proof of the preceding corollary $X = \lim \mathcal{D}_X$ with the projection map at component $f$ being $f$.

We can construct a cone over $\mathcal{D}_X$ with vertex $G(X)$ and natural transformation components $\mathbb{E}_{\bullet}(f) = \mathbb{E}_{\bullet}(id_{\mathbb{R}_{\infty}}) \circ G(f)$.

Since $X=\lim \mathcal{D}_X$ there exists a unique $\mathbf{Cvx}_{Meas}$-morphism $G(X) \xrightarrow{\epsilon_X} X$ such that $f \circ \epsilon_X = \mathbb{E}_{\bullet}(f)$ for all affine maps $X \xrightarrow{f} \mathbb{R}_{\infty}$. It follows that for each Dirac measure $\delta_x \in G(X)$ that, for all $X \xrightarrow{f} \mathbb{R}_{\infty}$ in $\mathbf{Cvx}_{Meas}$ that $f(\epsilon_X(\delta_x)) = f(x)$. Since $\mathbb{R}_{\infty}$ is a coseparator in $\mathbf{Cvx}_{Meas}$ it follows $\epsilon_X(\delta_x)=x$.

The defining property of $\epsilon_X$ is that it is the unique affine measurable function such that, for all $f \in \mathbb{R}_{\infty}^X$, the property

$$f \circ \epsilon_X = \mathbb{E}_{\bullet}(id_X) \circ G(f) = \mathbb{E}_{\bullet}(f)$$

holds. In the special case of $X = \mathbb{R}_{\infty}$ or for $X$ a closed convex subset of $\mathbb{R}^n$ it follows that for each fixed $P \in G(X)$ that $\epsilon_X(P) = \mathbb{E}_{P}(id_X)$. Henceforth we use the notation $\mathbb{E}_{\bullet}(id_X)$ for the unique affine measurable function $G(X) \rightarrow X$ such that, for each fixed $P \in G(X)$, the property

$$f \big( \mathbb{E}_P(id_X)\big) = \mathbb{E}_P(f) \quad \quad \forall f \in \mathbb{R}_{\infty}^X$$

holds. In other words, by definition, $\mathbb{E}_P(id_X)$ is the unique point in $X$ such that the preceding equation holds.

Lemma

If $X \in_{ob} \mathbf{Cvx}_{Meas}$ then the function $G(X) \xrightarrow{\mathbb{E}_{\bullet}(id_X)} X$ is a G-algebra.

Proof

We need to show the following two properties: (1) for all $x \in X$ we have $\mathbb{E}_{\delta_x}(id_X) = x$, and (2)$\mathbb{E}_{\bullet}(id_X) \circ \mu_X = \mathbb{E}_{\bullet}(id_X) \circ G(\mathbb{E}_{\bullet}(id_X))$.

The first property follows from the preceding corollary and remark concerning notation.

To prove the second property let $Q \in G^2(X)$ and compose both sides of the required condition by an affine measurable map $X \xrightarrow{f} \mathbb{R}_{\infty}$, and use the property that $f \circ \mathbb{E}_{\bullet}(id_X) = \mathbb{E}_{\bullet}(f)$. The left-hand side of the required condition, after composition on the left hand side with $f$, is given by

$$\mathbb{E}_{\bullet}(f) \big( \mu_X(Q) \big)= \mathbb{E}_{\mu_X(Q)}(f) = \int_X f \, d\mu_X(Q) = \int_{P \in GX} \big( \int_X f \, dP\big)dQ= \int_{P \in G(X)} \mathbb{E}_P(f)dQ$$

On the other hand, the right hand side of the required condition in (2), after composition with $f$, yields

$$\mathbb{E}_{\bullet}(f)\big( G(\mathbb{E}_{\bullet}(id_X))Q \big) = \mathbb{E}_{G( \mathbb{E}_{\bullet}(id_X))Q}(f) = \mathbb{E}_Q(f \circ \mathbb{E}_{\bullet}(id_X)) = \mathbb{E}_Q(\mathbb{E}_{\bullet}(f)) = \int_{P \in G(X)} \mathbb{E}_P(f) \, dQ$$

which coincides with the left hand side of the required condition. The result of the lemma now follows from the property that $\mathbb{R}_{\infty}$ coseparates, i.e, the set of affine measurable functions $X \xrightarrow{f} \mathbb{R}_{\infty}$ are jointly monic on $X$.

Lemma

Let $X \in_{ob} \mathbf{Cvx}_{Meas}$. Every affine measurable function $X \xrightarrow{f} \mathbb{R}_{\infty}$ yields a morphism of $G$-algebras.

Proof

We have already noted, for every $P \in G(X)$, that $\mathbb{E}_P(id_X) \in X$ is the unique point in $X$ such that $f(\mathbb{E}_{P}(id_X)) = \mathbb{E}_P(f)$ for all affine measurable functions $X \xrightarrow{f} \mathbb{R}_{\infty}$. That equation is equivalent to the statement

$$f \circ \mathbb{E}_{\bullet}(id_X) = \mathbb{E}_{\bullet}(id_{\mathbb{R}_{\infty}}) \circ G(f).$$

But since both $X$ and $\mathbb{R}_{\infty}$ are objects in $\mathbf{Cvx}_{Meas}$ it follows by Lemma 3.6 that both $\mathbb{E}_{\bullet}(id_X)$ and $\mathbb{E}_{\bullet}(id_{\mathbb{R}_{\infty}})$ are $G$-algebras. Hence $f$ is a morphism of those algebras.

Lemma

For every measurable space $X$ the space $G(X)$ is an object in $\mathbf{Cvx}_{Meas}$. For every measurable function $X \xrightarrow{f} Y$ the pushforward map $G(f)$ is an affine measurable function.

Proof

By Lemma 3.2 and the paragraph preceding that lemma we know that $G(X)$ is a measurable convex space. We need to show it also satisfies the fullness property.

Note that $\mu_X$ is an affine measurable function and that $\mu_X = \mathbb{E}_{\bullet}(id_{G(X)})$ because, for every measurable set $U$ in $X$, we have

$$(ev_U \circ \mu_X)(Q) = \mu_X(Q)[U] = \int_{P \in G(X)} ev_U(P) \, dQ = \mathbb{E}_Q(ev_U) = \big(ev_U \circ \mathbb{E}_{\bullet}(id_{G(X)})\big)(Q).$$

Since these evaluation maps are jointly monic on $G(X)$ it follows that $\mu_X = \mathbb{E}_{\bullet}(id_{G(X)})$. Using this result, along with Lemma 3.8, we obtain the result that for $Q \in G^2(X)$ and all affine maps $G(X) \xrightarrow{f} \mathbb{R}_{\infty}$ the element $\mu_X(Q) \in \cap_{f \in \mathbb{R}_{\infty}^{G(X)}} f^{-1}(\mathbb{E}_Q(f))$ which shows the fullness property is satisfied for $G(X)$. Hence $G(X) \in_{ob} \mathbf{Cvx}_{Meas}$.

The fact that $G(f)$ is an affine function follows immediately by the pointwise convex space structure of $G(X)$ and $G(Y)$. The fact that it is a measurable function follows because $G$ is an endofunctor on $\mathbf{Meas}$.

By the preceding lemma we obtain the ‘’free functor’‘ $\mathbf{Meas} \xrightarrow{\mathcal{F}} \mathbf{Cvx}_{Meas}$, which is the Giry monad (functor) $G$ viewed as a functor into $\mathbf{Cvx}_{Meas}$. There is also a ‘’forgetful functor’‘ $\mathbf{Cvx}_{Meas} \xrightarrow{\mathcal{U}} \mathbf{Meas}$ which forgets the convex space structure. (We are using the terms free functor and forgetful functor before we actually show they have adjoints simply because we need to name them.)

Lemma

Let $X \in_{ob} \mathbf{Cvx}_{Meas}$. The affine measurable functions $\mathbb{E}_{\bullet}(id_X)$ are the components of a natural transformation

$$\mathcal{F} \circ \mathcal{U} \Rightarrow id_{\mathbf{Cvx}_{Meas}}.$$

Proof

Suppose $X$ and $Y$ are objects in $\mathbf{Cvx}_{Meas}$ and that $X \xrightarrow{f} Y$ is an affine measurable function. We need to verify that $f \circ \mathbb{E}_{\bullet}(id_X) = \mathbb{E}_{\bullet}(id_Y) \circ \mathcal{F}\mathcal{U}(f)$. Applying both sides of this equation to any $P \in \mathcal{F}\mathcal{U}(X)$ yields the desired equality

$$f\big(\mathbb{E}_{P}(id_X)\big) = \mathbb{E}_{P}(f) = \mathbb{E}_P(id_Y \circ f)=\mathbb{E}_{\mathcal{F}\mathcal{U}(f)P}(id_Y)=\mathbb{E}_{\bullet}(id_Y)\big(\mathcal{F}\mathcal{U} (f)\big)P.$$

Hence the required commutativity condition holds showing $\mathbb{E}$ is a natural transformation.

It is now easy to verify that $\langle \mathcal{F}, \mathcal{U}, \eta, \mathbb{E}_{\cdot}(id_{\bullet}) \rangle$ forms an adjunction and that the induced monad is the Giry monad $(G, \eta, \mu)$.

Now apply Beck’s monadicity theorem (Theorem 2.2) to prove that the comparison functor

$$\Phi : \mathbf{Cvx}_{Meas} \rightarrow \mathbf{Meas}^G$$

is an equivalence. (The monadicity theorem is Theorem 1, page 147 of CWM. Although see the note on strict monadicity at monadicity theorem.) In Theorem 1, Chapter VI, page 152 CWM, MacLane provides the proof that the comparison functor $K: \langle \Omega, E \rangle-\mathbf{Alg} \rightarrow \mathbf{Set}^{\mathcal{U}'\circ \mathcal{F}'}$ between any algebraic variety and the corresponding category of algebras induced by the free functor $\mathcal{F}'$ and forgetful functor $\mathcal{U}'$ of that algebraic variety is an equivalence. Indeed, the category $\mathbf{Cvx}$ is one such algebra, and it is convenient to view the proof as a proof that the comparison functor $K: \mathbf{Cvx} \rightarrow \mathbf{Set}^{\mathcal{U}'\circ \mathcal{F}'}$ is an equivalence. That free functor $\mathcal{F}'$ sends a set $X$ to the set of all formal finite affine sums, $\{ \sum_{i=1}^n p_i x_i \, | \, p_i \in [0,1], \sum_{i=1}^n p_i =1, x_i \in X, n \in \mathbb{N} \}$ with the natural convex space structure. The monad associated with the functor $\mathcal{F}':\mathbf{Set} \rightarrow \mathbf{Cvx}$ and the forgetful functor $\mathcal{U}': \mathbf{Cvx} \rightarrow \mathbf{Set}$ is sometimes referred to as the distribution monad. By changing the base space from $\mathbf{Set}$ to $\mathbf{Meas}$, and using the fact that all the affine sum operations are measurable in $\mathbf{Cvx}_{Meas}$ rather than just set functions, MacLane’s proof carries through verbatim to show that the comparison functor $\Phi:\mathbf{Cvx}_{Meas} \rightarrow \mathbf{Meas}^{\mathcal{U} \circ \mathcal{F}}$ is an equivalence. That completes the proof that $\mathbf{Cvx}_{Meas}$ is equivalent to $\mathbf{Alg}_G$.

The construction used to derive the algebras of the Giry monad on $\mathbf{Meas}$ can be applied to the Giry monad defined on $\mathbf{Pol}$ to obtain the algebras $\mathbf{Cvx}_{Pol}$. Simply replace the category $\mathbf{Meas}$ with $\mathbf{Pol}$ and in place of the evaluation maps $ev_U:G(X)\rightarrow [0,1]$ use $\int f \, d\bullet: P(X) \rightarrow \mathbb{R}$ in the above construction. One thus obtains the equivalence $\Phi:\mathbf{Cvx}_{Pol} \rightarrow \mathbf{Pol}^{P}$.

Related constructions

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 $SRel$ (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

1. $\forall B \in \Sigma_Y . \lambda x \in X . h(x, B)$ is a bounded measurable function,
2. $\forall x \in X . \lambda B \in \Sigma_Y . h(x, B)$ is a subprobability measure on $\Sigma_Y$.

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 $1 + -$.

Related concepts

• stochastic process

• monads of probability, measures, and valuations

• measure, probability measure, pushforward measure, convex mixture

• measure theory, probability theory, random variable

• Radon monad, distribution monad, extended probabilistic powerdomain

• categorical probability, Markov category

• Markov kernel, Stoch

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

• W. Lawvere,_The category of probabilistic mappings with author commentary_, July 14, 2025. [pdf]

The original copy of the appendix is

• 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.)

This idea 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

• Bill Lawvere, video discussion, 6 June 2020 (in the talk “The algebra of statistical theories and models” by Evan Patterson, at Categorical Probability and Statistics, 5–8 June 2020,

  (YouTube video, timestamp 1:00:20)

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

• Ruben Van Belle, Probability monads as codensity monads (2021) $[$arXiv:2111.01250$]$

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 $Pol$ - every continuous function $P{X} \rightarrow X$ is completely determined by its values on the countable dense subset of $P{X}$.

The article

• Kirk Sturtz, Categorical Probability Theory, $[$arXiv:1406.6030$]$

views probability measures via double dualization, restricted to weakly averaging affine maps. An extended version of the derivation of the algebras given above is in

• Kirk Sturtz, The Algebras of the Giry monad, $[$arXiv:2409.14861$]$

Some corrections from an earlier version of the Categorical Probability Theory article, were pointed out in

• Tom Avery, Codensity and the Giry monad, Journal of Pure and Applied Algebra 220 3 (2016) 1229-1251 [arXiv:1410.4432, doi:10.1016/j.jpaa.2015.08.017]

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)

Elisabeth Burroni derives stochastic automata as algebras for a suitable distributive law on the monoid and Giry monads in

• Elisabeth Burroni, Lois distributives. Applications aux automates stochastiques, TAC 22, 2009 pp.199-221 (journal page)

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:

• Progic posts

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:

• Cihan Okay, Simplicial Distributions and Contextuality, talk at CQTS (11 Oct 2023) [video:YT]

category: probability