Markov kernel: Difference between revisions

Revision as of 14:12, 13 September 2020

Revision as of 11:22, 11 December 2020

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

# For every (fixed) <math> x \in X</math>, the map <math> B \mapsto \kappa(B, x)</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 <math>\sigma</math>-algebra <math>\mathcal A.</math>.<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability Theory: A Comprehensive Course|publisher=Springer|page=180|edition=2|doi=10.1007/978-1-4471-5361-0}}</ref>.

== Examples ==

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then <math> \kappa(dy |x) = k(y, x)\nu(dy) </math> i.e. the mapping

:<math>\begin{cases} \kappa:\mathcal B \times X \to [0,1] \\ \kappa(B|x)=\int_{B}k(y, x)\nu(\mathrm{d} y) \end{cases}</math>

defines a Markov kernel.<ref>{{cite book|last1=Erhan|first1=Cinlar|title=Probability and Stochastics|date=2011|publisher=Springer|___location=New York|isbn=978-0-387-87858-4|pages=37–38}}</ref>. This example generalises the countable Markov process example where <math>\nu</math> 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 <math>X = Y = \mathbb R</math> with <math>\nu(dx) = dx </math> standard Lebesgue measure and

:<math>k_t(y, x) = \frac{1}{\sqrt{2\pi}t}e^{-(y - x)^2/(2t^2)}</math>.

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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 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>. The category has the empty set as initial object and the one point set <math>*</math> as the terminal object.

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