In probability theory, a Markov kernel (also known as a stochastic kernel or probability kernel) is a map that in the general theory of Markov processes plays the role that the transition matrix does in the theory of Markov processes with a finite state space.

Formal definition

Let and be measurable spaces. A Markov kernel with source and target, sometimes written as , is a function with the following properties: In other words it associates to each point x \in X a probability measure on such that, for every measurable set, the map is measurable with respect to the \sigma-algebra \mathcal A.

Examples

Simple random walk on the integers

Take X=Y=\Z, and (the power set of \Z). Then a Markov kernel is fully determined by the probability it assigns to singletons for each : Now the random walk \kappa that goes to the right with probability p and to the left with probability 1 - p is defined by where \delta is the Kronecker delta. The transition probabilities for the random walk are equivalent to the Markov kernel.

General Markov processes with countable state space

More generally take X and Y both countable and. Again a Markov kernel is defined by the probability it assigns to singleton sets for each i \in X We define a Markov process by defining a transition probability where the numbers K_{ji} define a (countable) stochastic matrix (K_{ji}) i.e. We then define Again the transition probability, the stochastic matrix and the Markov kernel are equivalent reformulations.

Markov kernel defined by a kernel function and a measure

Let \nu be a measure on, and a measurable function with respect to the product \sigma-algebra such that then i.e. the mapping defines a Markov kernel. This example generalises the countable Markov process example where \nu was the counting measure. Moreover it encompasses other important examples such as the convolution kernels, in particular the Markov kernels defined by the heat equation. The latter example includes the Gaussian kernel on with standard Lebesgue measure and

Measurable functions

Take and arbitrary measurable spaces, and let f:X \to Y be a measurable function. Now define i.e. Note that the indicator function is \mathcal{A}-measurable for all iff f is measurable. This example allows us to think of a Markov kernel as a generalised function with a (in general) random rather than certain value. That is, it is a multivalued function where the values are not equally weighted.

Galton–Watson process

As a less obvious example, take, and the real numbers \R with the standard sigma algebra of Borel sets. Then where x is the number of element at the state n, \xi_i are i.i.d. random variables (usually with mean 0) and where is the indicator function. For the simple case of coin flips this models the different levels of a Galton board.

Composition of Markov Kernels

Given measurable spaces, we consider a Markov kernel as a morphism. Intuitively, rather than assigning to each x \in X a sharply defined point y \in Y the kernel assigns a "fuzzy" point in Y which is only known with some level of uncertainty, much like actual physical measurements. If we have a third measurable space, and probability kernels and , we can define a composition by the Chapman-Kolmogorov equation The composition is associative by the Monotone Convergence Theorem and the identity function considered as a Markov kernel (i.e. the delta measure ) is the unit for this composition. This composition defines the structure of a category on the measurable spaces with Markov kernels as morphisms, first defined by Lawvere, the category of Markov kernels.

Probability Space defined by Probability Distribution and a Markov Kernel

A composition of a probability space and a probability kernel defines a probability space, where the probability measure is given by

Properties

Semidirect product

Let be a probability space and \kappa a Markov kernel from to some. Then there exists a unique measure Q on , such that:

Regular conditional distribution

Let (S,Y) be a Borel space, X a (S,Y)-valued random variable on the measure space and a sub-\sigma-algebra. Then there exists a Markov kernel \kappa from to (S,Y), such that is a version of the conditional expectation for every B \in Y, i.e. It is called regular conditional distribution of X given \mathcal G and is not uniquely defined.

Generalizations

Transition kernels generalize Markov kernels in the sense that for all x \in X, the map can be any type of (non negative) measure, not necessarily a probability measure.