In mathematics, the category of Markov kernels, often denoted Stoch, is the category whose objects are measurable spaces and whose morphisms are Markov kernels.
[1][2][3][4] It is analogous to the category of sets and functions, but where the arrows can be interpreted as being stochastic.
Several variants of this category are used in the literature.
[6] Also, one can take as morphisms equivalence classes of Markov kernels under almost sure equality;[7] see below.
Recall that a Markov kernel between measurable spaces
which is measurable as a function on
, which suggests an interpretation as conditional probability.
The category Stoch has:[4] This composition formula is sometimes called the Chapman-Kolmogorov equation.
[4] This composition is unital, and associative by the monotone convergence theorem, so that one indeed has a category.
The terminal object of Stoch is the one-point space
[4] Morphisms in the form
can be equivalently seen as probability measures on
with values for every measurable subset
, a measure-preserving Markov kernel
such that for every measurable subset
,[7] Probability spaces and measure-preserving Markov kernels form a category, which can be seen as the slice category
defines canonically a Markov kernel
This construction preserves identities and compositions, and is therefore a functor from Meas to Stoch.
By functoriality, every isomorphism of measurable spaces (in the category Meas) induces an isomorphism in Stoch.
However, in Stoch there are more isomorphisms, and in particular, measurable spaces can be isomorphic in Stoch even when the underlying sets are not in bijection.
is left adjoint, it preserves colimits.
[8] Because of this, all colimits in the category of measurable spaces are also colimits in Stoch.
This in particular implies that the product of measurable spaces is not a product in Stoch in general.
Since the Giry monad is monoidal, however, the product of measurable spaces still makes Stoch a monoidal category.
[4] A limit of particular significance for probability theory is de Finetti's theorem, which can be interpreted as the fact that the space of probability measures (Giry monad) is the limit in Stoch of the diagram formed by finite permutations of sequences.
Sometimes it is useful to consider Markov kernels only up to almost sure equality, for example when talking about disintegrations or about regular conditional probability.
are almost surely equal if and only if for every measurable subset
[7] This defines an equivalence relation on the set of measure-preserving Markov kernels
Probability spaces and equivalence classes of Markov kernels under the relation defined above form a category.
When restricted to standard Borel probability spaces, the category is often denoted by Krn.