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The double category of mixed probabilities [lcc-000T]

March 30, 2024
Eigil Fjeldgren Rischel

In probability theory, a random variable is a (measurable) map $x: \Omega \to S$ from some implicit "background space" $\Omega$ , which is taken to be equipped with a probability measure $\mathbb {P}$ --- the image measure $x(\mathbb {P})$ describes the distribution of $x$ .

Since the background space itself is supposed not to matter too much, one might thing we could just as well replace $x$ with its image distribution, but in fact the explicit presentation plays a very important role, most notably because, given two variables $x,y: \Omega \to S$ , we get to talk about their joint distribution on $S^2$

(Of course, we could simply study this distribution in itself, but first, this is not always possible for infinite-dimensional cases such as the trajectories of stochastic dynamical systems, and second, it is often more convenient to work with variables than joint distributions)

From a categorical point of view, we can think of "parametrized stochastic maps" as parametrized deterministic maps equipped with a distribution on the parameterizing object. Much of this drops out of well-known constructions: Given any monoidal category $\mathcal {C}$ , the slice category $\mathcal {C}_{/I}$ is again monoidal, and has a forgetful functor to $\mathcal {C}$ , and therefore acts on it. We can therefore form the double category $\mathsf {\mathbb Para}_{\mathcal {C}_{/I}}(\mathcal {C})$ of parametrized morphisms (see para construction under "Generalizations, and this talk by David Jaz Myers). We will be restrict.ting to the sub-double category of this containing only those horizontal morphisms where the map $\Omega \otimes X \to Y$ is deterministic.

Since we are generally only interested in morphisms up to almost-certain equality, we may want to expand the set of 2-cells, requiring only that the square commutes up to $\mathbb {P}$ -almost certain equality. This has the result of rendering two horizontal maps isomorphic if they are equal almost certainly, but note that for this to have well-defined composition requires some assumptions on $\mathcal {C}$ (see Fritz 2019, Prop. 13.9)

Explicitly, we have the following definition:

Definition [lcc-000S]

March 26, 2024
Eigil Fjeldgren Rischel

Fix any Markov category $\mathcal {C}$

There is a (pseudo) double category where

The objects are objects of $\mathcal {C}$
The vertical category is precisely $\mathcal {C}$
The horizontal morphisms $X \to Y$ are tuples $(\Omega \in \mathcal {C}, \mathbb {P}: I \to \Omega , f: \Omega \otimes X \to Y)$ , where $f$ is deterministic, with composition given in the obvious way by tensoring and composing
A 2-cell consists of a map $\Omega _1 \to \Omega _2$ in $\mathsf {Prob}(\mathcal {C})$ making the obvious square commute

If $\mathcal {C}$ is causal, there is an extension of this double category with the same 0- and 1-cells, but where the 2-cells $(\Omega _1, \mathbb {P}_1, f_1) \to (\Omega _2, \mathbb {P}_2, f_2)$ are required to make the diagram commute only $\mathbb {P}_1$ -almost certainly

Given a parametrized map $f: \Omega \otimes X \to Y, \mathbb {P}: I \to \Omega$ , there is an obvious stochastic map $\hat {f}: X \to Y$ given by composing. This is a sort of half-companion to $f$ : we have a canonical 2-cell filling this square:

where the map

I \to \Omega

giving the reparametrization is simply

\mathbb {P}

. Note that this requires us to allow stochastic reparametrizations, but we're not considering stochastic parametrized maps (nor do we want to), and so this is taking us further outside the realm of the

\mathsf {Para}

construction.

This 2-cell has a sort of universal property---if given another 2-cell from the identity horizontal cell $1_A: A = A$ to $f$ , it must factor over the above one (this amounts to the claim that, given such as 2-cell, the kernel $A \to Y$ is the composition of the kernel $A \to X$ and $\hat {f},$ which is not hard to see)

In the other direction, we have no hope of building a 2-cell, since that would require two maps $X \times \Omega \to Y$ to be (almost certainly) equal, one of which is given by first applying the deletion $\Omega \to I$ . We shouldn't't expect these two maps to be proper companions, since this would entail that if $\hat {f} = \hat {g}$ then $f = g$ , defeating the whole point of the exercise

On Random Trajectories in Categorical Systems Theory [lcc-000M]

March 13, 2024
Eigil Fjeldgren Rischel

The Categorical Systems Theory of David Jaz Myers is a very general language for talking about "systems"---as a special case, dynamical systems. In that framework, the set of trajectories of a system is a representable functor, given by mapping out of some other system. For example, if $s: X \to TX$ is a vector field giving a smooth dynamical system, a trajectory in the usual sense is a map from the dynamical system $\mathbb {R} \to T(\mathbb {R})$ given by $\dfrac {\partial }{\partial x}$ , ie the system which moves right at constant speed $1$ .

This framework handles this type of trajectories very well, but it is less clear how to talk about stochastic dynamical systems, which generally have random trajectories.

Let us look for a moment at discrete-time stochastic dynamical systems which are closed (that is, not open, with trivial interface). This consists of a set equipped with a map $X \to \Delta (X)$ , thought of as a section of the projection $X \times \Delta (X) \to X$ . A trajectory between such systems is then a map $X \to Y$ so that the obvious diagram commutes.

The first thing we notice is that, if the composite $X \to Y \to \Delta (Y)$ is stochastic, the diagram has no hope of commuting unless either $X \to \Delta (X)$ is also stochastic, or the map $\Delta (X) \to \Delta (Y)$ is stochastic. Thus if we restrict ourselves to deterministic maps between systems, we can't have trajectories typed by deterministic systems in a stochastic system. But this doesn't seem to be desirable---we do want to have some system representing the (random) trajectories in the ordinary sense of a stochastic system, but if we look just at deterministic maps, this won't be possible

But on the other hand, a stochastic map $f: X \to Y$ is also not what we want, because it doesn't allow us to talk about the correlation between $f(x)$ for different $x$ , which is exactly what the system is supposed to talk about.

Indeed, in the ordinary theory of stochastic differential equations and stochastic processes, a solution is not a time-parametrized family of distributions, but a random function. And that is precisely what is necessary in this case as well. When mapping from $X \times \Delta (X) \to Y \times \Delta (Y),$ we need to express the idea that the two values we end up with are not supposed to be independent, but result from the same (random) trajectory applied to the current position $x_t$ and the distribution of possible next states $x_{t+1}$ .

On the other hand, the notion of commutativity we have to ask for is just that two particular joint distributions are the same. Let $s: \mathbb {N} \to \mathbb {N}$ be the successor function considered as a (deterministic) dynamical system, and let $s_X: X \to \Delta (X)$ for some set $X$ be a discrete-time stochastic process. Let $x : \mathbb {N} \to X,$ written $x_n$ be a random function between them. Then we have the random diagram in $\mathsf {FinStoch}$ :

Each path around this diagram describes a random element of $\operatorname {\mathrm {Hom}}(\mathbb {N}, \Delta (X \times X))$ . The first takes $n$ to the distribution on $X \times X$ where the first coordinate is constantly $x_n$ , and the second is distributed according to $s(x_n)$ . The other path takes $n$ to the distribution which is deterministically $x_n, x_{n+1}$ . The condition for the random path $x_n$ to be a trajectory of the dynamical system $s_X: X \to \Delta (X)$ is that, for all $n \in \mathbb {N}$ , the two distributions obtained by taking the expectation over the distribution of $x$ agree---which amounts to saying that the conditional of $x_{n+1}$ given $x_n$ is exactly given by $s$ .

Note that the only place where we needed $x_n$ to be a random map, as opposed to a stochastic map, was to define the parallel composite $\mathbb {N} \times \mathbb {N} \to X \times X$ ---here we need to express the idea that both of inputs are to be processed according to the same random trajectory, not to be sampled independently from some distributions $x_n, x_m \in \Delta (X)$ .

It is not too hard to extend this discussion to trajectories of open systems with a fixed interface $\binom {I}{O}$ , but how to extend this idea to work with the rest of David Jaz Myers' framework seems to be an extremely complicated problem