Markov kernel: Difference between revisions

Revision as of 23:31, 17 March 2022

Revision as of 18:23, 10 May 2022

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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. From this point of view a probability space is <math>(\Omega, \mathcal A, \mathbb P)</math> is the same thing as a pointed space <math>* \to \Omega</math> in the Markov category.

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