Markov kernel: Difference between revisions

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== Formal definition ==

Let <math>(X,\mathcal A)</math> and <math>(Y,\mathcal B)</math> be [[measurable space]]s. A ''Markov kernel'' with source <math>(X,\mathcal A)</math> and target <math>(Y,\mathcal B)</math>, sometimes written as <math>\kappa:(X,\mathcal{A})\to(Y,\mathcal{B})</math>, is a mapfunction <math>\kappa : \mathcal B \times X \to [0,1]</math> with the following properties:

# For every (fixed) <math>B_0 \in \mathcal B</math>, the map <math> x \mapsto \kappa(B_0, x)</math> is <math>\mathcal A</math>-[[measurable function|measurable]]

# For every (fixed) <math> x_0 \in X</math>, the map <math> B \mapsto \kappa(B, x_0)</math> is a [[probability measure]] on <math>(Y, \mathcal B)</math>

In other words it associates to each point <math>x \in X</math> a [[probability measure]] <math>\kappa(dy|x): B \mapsto \kappa(B, x)</math> on <math>(Y,\mathcal B)</math> such that, for every measurable set <math>B\in\mathcal B</math>, the map <math>x\mapsto \kappa(B, x)</math> is measurable with respect to the [[Σ-algebra|<math>\sigma</math>-algebra <math>\mathcal A</math>]].<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability Theory: A Comprehensive Course|series=Universitext |year=2014 |publisher=Springer|page=180|edition=2|doi=10.1007/978-1-4471-5361-0|isbn=978-1-4471-5360-3 }}</ref>

== Examples ==

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where <math> x </math> is the number of element at the state <math> n </math>, <math>\xi_i</math> are [[Independent and identically distributed random variables|i.i.d.]] [[random variable]]s (usually with mean 0) and where <math>\mathbf{1}_B</math> is the indicator function. For the simple case of [[Bernoulli distribution|coin flips]] this models the different levels of a [[Galton board]].

== Composition of Markov Kernels and the Markov Category==

Given measurable spaces <math>(X, \mathcal A)</math>, <math>(Y, \mathcal B) </math> we consider a Markov kernel <math>\kappa: \mathcal B \times X \to [0,1]</math> as a morphism <math>\kappa: X \to Y</math>. Intuitively, rather than assigning to each <math>x \in X</math> a sharply defined point <math> y \in Y</math> the kernel assigns a "fuzzy" point in <math>Y</math> which is only known with some level of uncertainty, much like actual physical measurements. If we have a third measurable space <math>(Z, \mathcal C)</math>, and probability kernels <math>\kappa: X \to Y</math> and <math>\lambda: Y \to Z</math>, we can define a composition <math>\lambda \circ \kappa : X \to Z</math> by the [[Chapman-Kolmogorov equation]]

:<math>(\lambda \circ \kappa) (dz|x) = \int_Y \lambda(dz | y)\kappa(dy|x)</math>.

The composition is associative by the Monotone Convergence Theorem and the identity function considered as a Markov kernel (i.e. the delta measure <math> \kappa_{1}(dx'|x) = \delta_x(dx')</math>) is the unit for this composition.

This composition defines the structure of a [[category (mathematics)|category]] on the measurable spaces with Markov kernels as morphisms, first defined by Lawvere.,<ref>{{cite web|author = F. W. Lawvere|title = The Category of Probabilistic Mappings|date = 1962|url = https://ncatlab.org/nlab/files/lawvereprobability1962.pdf}}</ref> Thethe [[category has the empty set as initial object and the one point set <math>*</math> as the terminal object. From this point of view a probability space <math>(\Omega, \mathcal A, \mathbb P)</math> is the same thing as a pointed space <math>* \to \Omega</math> in the [[Markov categorykernels]].

== Probability Space defined by Probability Distribution and a Markov Kernel==