Category of Markov kernels explained

In mathematics, the category of Markov kernels, often denoted Stoch, is the category whose objects are measurable spaces and whose morphisms are Markov kernels.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. For example, one can use subprobability kernels instead of probability kernels, or more general s-finite kernels.Also, one can take as morphisms equivalence classes of Markov kernels under almost sure equality; see below.

Definition

Recall that a Markov kernel between measurable spaces

(X,l{F})

and

(Y,l{G})

is an assignment

k:X x l{G}\toR

which is measurable as a function on

X

and which is a probability measure on

l{G}

. We denote its values by

k(B|x)

for

x\inX

and

B\inl{G}

, which suggests an interpretation as conditional probability.

The category Stoch has:

(X,l{F})

, the identity morphism is given by the kernel

\delta(A|x)=1A(x)=\begin{cases} 1&x\inA;\\ 0&x\notinA\end{cases}

for all

x\inX

and

A\inl{F}

;

k:(X,l{F})\to(Y,l{G})

and

h:(Y,l{G})\to(Z,l{H})

, the composite morphism

h\circk:(X,l{F})\to(Z,l{H})

is given by

(h\circk)(C|x)=\intYh(C|y)k(dy|x)

for all

x\inX

and

C\inl{H}

.This composition formula is sometimes called the Chapman-Kolmogorov equation.

This composition is unital, and associative by the monotone convergence theorem, so that one indeed has a category.

Basic properties

Probability measures

1

. Morphisms in the form

1\toX

can be equivalently seen as probability measures on

X

, since they correspond to functions

1\toPX

, i.e. elements of

PX

.

Given kernels

p:1\toX

and

k:X\toY

, the composite kernel

k\circp:1\toY

gives the probability measure on

Y

with values

(k\circp)(B)=\intXk(B|x)p(dx),

for every measurable subset

B

of

Y

.

(X,l{F},p)

and

(Y,l{G},q)

, a measure-preserving Markov kernel

(X,l{F},p)\to(Y,l{G},q)

is a Markov kernel

k:(X,l{F})\to(Y,l{G})

such that for every measurable subset

B\inl{G}

,

q(B)=\intXk(B|x)p(dx).

(HomStoch(1,-),Stoch)

.

Measurable functions

Every measurable function

f:(X,l{F})\to(Y,l{G})

defines canonically a Markov kernel

\deltaf:(X,l{F})\to(Y,l{G})

as follows,

\deltaf(B|x)=1B(f(x))=\begin{cases} 1&f(x)\inB;\\ 0&f(x)\notinB \end{cases}

for every

x\inX

and every

B\inl{G}

. This construction preserves identities and compositions, and is therefore a functor from Meas to Stoch.

Isomorphisms

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.

Relationship with other categories

HomStoch(X,Y)\congHomMeas(X,PY)

between Stoch and the category of measurable spaces.

L:Meas\toStoch

of the adjunction above is the identity on objects, and on morphisms it gives the canonical Markov kernel induced by a measurable function described above.

(HomStoch(1,-),Stoch)

.

(HomStoch(1,-),L)

.

Particular limits and colimits

Since the functor

L:Meas\toStoch

is left adjoint, it preserves colimits. Because of this, all colimits in the category of measurable spaces are also colimits in Stoch. For example,

In general, the functor

L

does not preserve limits. 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.

Almost sure version

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.

(X,l{F},p)

and

(Y,l{G},q)

, we say that two measure-preserving kernels

k,h:(X,l{F},p)\to(Y,l{G},q)

are almost surely equal if and only if for every measurable subset

B\inl{G}

,

k(B|x)=h(B|x)

for

p

-almost all

x\inX

.This defines an equivalence relation on the set of measure-preserving Markov kernels

k,h:(X,l{F},p)\to(Y,l{G},q)

.

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.

See also

References

Further reading

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Category of Markov kernels".

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