In other words it associates to each point a probability measure on such that, for every measurable set , the map is measurable with respect to the -algebra .[2]
Examples[]
Simple random walk on the integers[]
Take , and (the power set of ). Then a Markov kernel is fully determined by the probability it assigns to a singleton set with for each :
.
Now the random walk that goes to the right with probability and to the left with probability is defined by
where is the Kronecker delta. The transition probabilities for the random walk are equivalent to the Markov kernel.
defines a Markov kernel.[3] This example generalises the countable Markov process example where 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 be a measurable function. Now define i.e.
for all .
Note that the indicator function is -measurable for all iff 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.
Composition of Markov Kernels and the Markov Category[]
Given measurable spaces , we consider a Markov kernel as a morphism . Intuitively, rather than assigning to each a sharply defined point the kernel assigns a "fuzzy" point in 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 composition is associative by Tonelli's 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.[4] The category has the empty set as initial object and the one point set as the terminal object. From this point of view a probability space is is the same thing as a pointed space in the Markov category.
Probability Space defined by Probability Distribution and a Markov Kernel[]
A probability measure on a measurable space is the same thing as a morphism
in the Markov category also denoted by . By composition, a probability space and a probability kernel defines a probability space . It is concretely defined by
Properties[]
Semidirect product[]
Let be a probability space and a Markov kernel from to some . Then there exists a unique measure on , such that:
Regular conditional distribution[]
Let be a Borel space, a -valued random variable on the measure space and a sub--algebra. Then there exists a Markov kernel from to , such that is a version of the conditional expectation for every , i.e.
It is called regular conditional distribution of given and is not uniquely defined.
Generalizations[]
Transition kernels generalize Markov kernels in the sense that for all , the map
can be any type of (non negative) measure, not necessarily a probability measure.
References[]
^Reiss, R. D. (1993). "A Course on Point Processes". Springer Series in Statistics. doi:10.1007/978-1-4613-9308-5. ISBN978-1-4613-9310-8. Cite journal requires |journal= (help)
^Klenke, Achim. Probability Theory: A Comprehensive Course (2 ed.). Springer. p. 180. doi:10.1007/978-1-4471-5361-0.
^Erhan, Cinlar (2011). Probability and Stochastics. New York: Springer. pp. 37–38. ISBN978-0-387-87858-4.