### Category Theoretic Probability Theory

#### Posted by David Corfield

Having noticed (e.g., here and here) that what I do in my day job (statistical learning theory) has much to do with my hobby (things discussed here), I ought to be thinking about probability theory in category theoretic terms. What would seem to be the most promising approach is described by Prakash Panangaden in Probabilistic Relations.

The category SRel (stochastic relations) has 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

- $\forall B \in \Sigma_Y . \lambda x \in X . h(x, B)$ is a bounded measurable function,
- $\forall x \in X . \lambda B \in \Sigma_Y . h(x, B)$ 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 $(k \cdot h)(x, C) = \int_Y k(y, C)h(x, d y)$.

Apparently this is based on work by Michele Giry, which in turn was based on earlier work by Lawvere. This definition differs from Giry’s in the second clause where subprobability measures are allowed, rather than ordinary probability measures.

Panangaden notes that something very similar to the way that the category of relations can be constructed from the powerset functor is at stake. Just as the category of relations is the Kleisli category of the powerset functor over the category of sets, 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.

Now I’d like to know

- What is gained by the move from probability measures to subprobability measures?
- How to put this work into relation with the fact that there is a natural choice of metric with which to give a riemannian geometry to a space of probability distributions, and also to a space of conditional distributions (see section 6, p. 39, of this).

## Re: Category Theoretic Probability Theory

Can you tell us what a subprobability measure is? It doesn’t seem to be mentioned in Panangaden’s paper.

Also, you might like to take a look at Alex Simpson’s very interesting talk notes, Probabilistic Observations and Valuations.