Markov kernel: Difference between revisions

Revision as of 15:06, 30 November 2022

Revision as of 15:09, 30 November 2022

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

ByA composition, of a probability space <math>(X, \mathcal A, P_X)</math> and a probability kernel <math>\kappa: (X, \mathcal A) \to (Y, \mathcal B) </math> defines a probability space <math>(Y, \mathcal B, P_Y = \kappa \circ P_X)</math>., Itwhere isthe concretelyprobability measure is definedgiven by

:<math> P_Y(B) = \int_X \int_B \kappa(dy|x) P_X(dx) = \int_X \kappa(B|x)P_X(dx) = \mathbb{E}_{P_X}\kappa(B|\cdot) .</math>

== Properties ==