Markov kernel: Difference between revisions

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== Composition of Markov Kernels and the Markov Categorie==

Given measurable spaces <math>(X, \mathcal A)</math>, <math>(Y, \mathcal B) </math> and <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

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

The composition is associative by [[Fubini's theorem#Tonelli's_theorem_for_non-negative_measurable_function|Tonelli's theorem]] and the identity function considered as 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]] 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>. The category has the empty set as initial object and the one point set <math>*</math> as the terminal object. A probability measure on a measurable space <math>(X, \mathcal A)</math> is the same thing as morphisma frommorphism <math>* \to X</math> in this category also denoted by <math>P</math>. By composition, a probability space <math>(X, \mathcal A, P)</math> and a probability kernel <math>\kappa: (X, \mathcal A) \to (Y, \mathcal B) </math> defines a probability space <math>(Y, \mathcal B, \kappa \circ P)</math>.

== Properties ==