Markov kernel: Difference between revisions

Revision as of 13:01, 3 July 2019

Revision as of 14:04, 8 December 2019

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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==