Contents

### Context

#### Measure and probability theory

measure theory

probability theory

# Contents

## Idea

The Giry monad (Giry 80) is the monad on a category of suitable spaces which sends each suitable space $X$ to the space of suitable probability measures on $X$.

## History

The Giry monad was originally developed by Lawvere in 1962, prior to the recognition of the explicit relationship between monads and adjunctions. It wasn’t until 1965 that the constructions of Eilenberg-Moore, and Kleisi, showed that every adjoint pair gives rise to a monad.

Lawvere’s construction was written up as an appendix to a proposal to the International Atomic Energy Commission. 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, Giry was collaborating with another French mathematician at that time who was also working with the French intelligence agency, and was able to obtain a copy of the appendix. Giry then developed and extended some of the ideas in the appendix.

Gian-Carlo Rota had (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. (Lawvere 62)

## Definition

The Giry monad is defined on the category of measurable spaces, assigning to each measurable space $X$ the space of all probability measures on $X$, $G(X)$, 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 sends a point $x \in X$ to the Dirac measure at $x$, $\delta_x$, while the counit of the monad is defined by the natural transformation

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

given by

$\mu_X (Q)(U) := \int_{q \in P(X)} ev_U(q) dQ.$

This makes the endofunctor $G$ into a monad, and this is the Giry monad on Measurable spaces, as originally given by Lawvere.

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 (topological) spaces, $Pol$, which are separable metric spaces for which a complete metric exists.

Write

$P \colon Pol \to Pol$

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

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

This makes the endofunctor $P$ into a monad, and this is the Giry monad on Polish spaces. The shortcoming of restricting the Giry monad to Polish spaces is that discrete spaces have no $P$-algebras (even though they do have $G$-algebras). For example, take the discrete (measurable) space $2$, which we can also view as a topological discrete space. The measurable map $\epsilon_{2}: G(2) \to 2$ defined by $\epsilon_{2}( (1-\alpha) \delta_0 + \alpha \delta_1) = 0$ for all $\alpha \in [0,1)$, and by $\epsilon_{2}(\delta_1)=1$ shows that $(2,\epsilon_2)$ is a $G$-algebra, yet it is not a $P$-algebra.
(As Doberkat has noted, a $P$-algebra must be connected, but that is impossible for discrete spaces.) The map $\epsilon_{2}$ plays a critical role in the analysis of the $G$-algebras, because it is the unique affine map from the geometric (convex) space $G(2)$, which is isomorphic to the unit interval $[0,1]$ with its natural convex structure, to the discrete (combinatorial) convex space $2$.

## Properties

### Algebras over the Giry monad

(Doberkat 03) works out the algebras for the Giry monad using the Giry monad defined on $Pol$. We want measurable maps between $P(X)$ and $X$, 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 $X$.

Now, as ever, $P(X)$ will support an algebra, $\mu_{X}: P(P(X)) \to P(X)$. This is the analogue of a free group being an algebra of the group monad. But just as there are many interesting groups which are not free, we should want to find algebras of Giry’s monad which are not of the $\mu_{X}$ form. Doberkat shows that for such an algebra $X$ must be connected, and suggests this example

$h: P([0, 1]) \ni \tau \mapsto \int_{0}^{1} t \tau (dt) \in [0, 1].$

(One author believes that this might be $\mu_{\{0, 1\}}$. After all, probability measures on $\{0, 1\}$ are just binomial, parameterised by $p \in [0, 1]$.)

The other example he gives has $X$ a bounded, closed and convex subset of $\mathbb{R}^n$, and probability measures being sent to their barycentre.

Doberkat has a longer article on Eilenberg-Moore algebras of the Giry monad as item 5 here. (Unfortunately, the monograph ‘Stochastic Relations: Foundations for Markov Transition Systems’ doesn’t appear to be available.) There are two monads being treated here, one which sends a Polish space to the space of all probability measures, the other to the space of all subprobability measures. The extra structure relating to these monads, is that of a (positive) convex structure. In the case of a convex structure, this intuitively captures the idea that a weighted sum of points in the space has barycentre within the space.

The results of Doberkat can be generalized to the Giry monad $G$ on all measurable spaces by using the factorization of the Giry monad through the category of super convex spaces $\mathbf{SCvx}$, by viewing the Giry monad itself as a functor into that category. (A super convex space is similar to a convex space except the structure requires that if $\{\alpha_i\}_{i=1}^{\infty}$ is any countable partition of unity (so the limit of the sum is one), then for any sequence of points in a super convex space $A$, the countable sum $\sum_{i=1}^{\infty} \alpha_i a_i$ is also an element of the space. The morphisms in the category preserve the countable affine sums. The right adjoint of that functor assigns to each convex space the measurable space, defined on the underlying set, with the initial $\sigma$-algebra generated by all the countably affine maps into the one point extension of the real line $\mathbb{R}_{\infty}$. The construction of the counit amounts to using the fact that the full subcategory consisting of the single object $\mathbb{R}_{\infty}$ is condense in $\mathbf{SCvx}$. This implies that every $\mathbb{R}_{\infty}$-generalized point of a super convex space $A$ is an evaluation map, at a unique point $a$ of $A$. Using this fact, given any arbitrary probability measure $P$ defined on $\Sigma A$, one takes the restriction of $P$, viewed as an operator $\mathbf{Meas}(\Sigma A, \mathbb{R}_{\infty}) \rightarrow \mathbb{R}_{\infty}$, mapping $f \mapsto \int_A f \, dP$, to the subset of countably affine maps, $\mathbf{SCvx}(A, \mathbb{R}_{\infty}) \rightarrow \mathbb{R}_{\infty}$. This restriction process yields an $\mathbb{R}_{\infty}$-generalized point in $\mathbf{SCvx}$ which is necessarily a unique point of $A$ since $\mathbb{R}_{\infty}$ is condense in $\mathbf{SCvx}$.

As an illustration, the half open interval $[0,\infty)$ is not a super convex space because one can take the countable partition of one given by $\{\frac{1}{2^i}\}_{i=1}^{\infty}$, and a set of points $\{i 2^i\}_{i=1}^{\infty}$ in $[0,\infty)$ so that the countably infinite sum $\{\sum_{i=1}^{\infty}\frac{i 2^i}{2^i}\}_{i=1}^{\infty}$ does not exist. This shows that $[0,\infty)$ is not a super convex space and explains why there is no barycenter map for this space. (Consider the half-Cauchy distribution.) On the other hand, the open unit interval, $(0,1)$ is a super convex space and does have a barycenter. This illustrates that compactness is not a requirement for a barycenter map to exist, only the property of being a super convex space is necessary. (But it does tie in a sequential completeness condition - thereby making a connection with the topological viewpoint.)

The category of super convex spaces is equivalent to the category of Giry algebras. (Sturtz 19)

### 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:

…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

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 $1 + -$.

W. Lawvere, The category of probabilistic mappings, ms. 12 pages, 1962 (Lawvere Probability 1962)

‘ The key idea … is that random maps between spaces are just maps in a category of convex spaces between “simplices” ‘ (W. Lawvere, catlist remark 25 oct 1998).

• 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.

In the paper, there are allegedly a few minor analytically incorrect points and gaps in proofs, observed by later authors.

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 factorization of the Giry monad, defined on the category of measurable spaces, through the category of super convex spaces, is described in

• Kirk Sturtz?, Erratum and Addendum: The factorization of the Giry monad, arXiv:1907.00372

while the result that the category of Giry-algebras is equivalent to the category of super convex spaces is given in

• K. Sturtz, The equivalence of the categories of Giry-algebras and super convex spaces, arXiv:1907.03209

Those two short articles generalize the earlier work

• K. Sturtz, Categorical Probability Theory, arXiv:1406.6030

which views probability measures via double dualization, restricted to weakly averaging affine maps which preserves limits. The preservation of limits (of countable affine sums) is automatic in the category of super convex spaces.

That article includes some corrections from an earlier version of the article, The Giry monad as a codensity monad, pointed out in

• T. Avery, Codensity and the Giry monad, arXiv:1410.4432 (2014). (pdf)

Apart from these papers, there are similar developments in

• Ernst-Erich Doberkat, Characterizing the Eilenberg-Moore algebras for a monad of stochastic relations (pdf)

• Ernst-Erich Doberkat, Kleisli morphisms and randomized congruences

• 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.

There is also relation with work of Jacobs et al.

• Robert Furber, Bart Jacobs, Towards a categorical account of conditional probability, arxiv/1306.0831

• B. Jacobs, Probabilities, distribution monads and convex categories, Theoretical Computer Science 412(28) (2011) pp.3323–3336. (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? discusses the Giry monad in:

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

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, arXiv1301.6201 (2013) pdf

To do: