# The Kleisli Category of the Monad of Measures of Finite Support and its composition formula

In this post, I was introduced to the monad of finitely supported measures.

$$HX$$ is the set of finitely supported measures on $$X$$, with monad structure defined as for the Giry monad. Let's call this monad $$H$$.

The Giry Monad is a structure we find in the Category Theoretical analysis of probability theory. In particular, it consists of three things. First, it consists of a functor $$R : Meas \rightarrow Meas$$ on the category of measurable spaces that takes a space $$M$$ to the set of probability measures on $$M$$. The monad also has two natural transformations $$\mu, \eta$$ where $$\mu : R \cdot R \rightarrow R$$ and $$\eta : 1_{Meas} \rightarrow R$$, where $$1_{Meas}$$ is the identity functor on $$Meas$$. The product, $$\mu$$ is understood as follows:

A probability measure on an affine space has an average value (also called expectation or integral), which is a point in that affine space. Apply this to the affine space of probability measures on X. In other words, a measure on the space of measures determines a (weighted) average of those measures.

I am interested in the Kleisli category of the finite version of the Giry monad as described in the link above. In Panangaden 1999, we see an analysis of the Kleisli category of the Giry monad, $$G$$. My first guess is that the Kleisli category of $$H$$ and $$G$$ are almost identical. I do not have a guess as to how they differ. I am wondering what the differences are. In particular I am interested in how composition differs. In Panangaden 1999, he define the compostion this way:

$$k: (Y, \sigma_Y) \rightarrow (Z, \sigma_Z)$$ $$h: X \times \Sigma_Y \rightarrow [0,1]$$ Define: $$k \circ h : (X, \sigma_X) \rightarrow (Z, \sigma_Z)$$ by the formula $$k \circ h (x,C) = \int_Y k(y,C)h(x,dy)$$

Is it the same in both categories? What other differences are there in the categories?

• It would help the reader if you spared a couple of words to explain the Giry monad... – André Henriques Jan 5 at 21:53
• The Kleisli category of the distribution monad has sets as objects, and stochastic matrices with finitely-supported rows (or columns, depending on the convention used) as morphisms. Composition is matrix multiplication. This is direct from the definitions. – Robert Furber Jan 6 at 22:49