Markov kernel: Difference between revisions

Revision as of 12:35, 20 August 2020

Revision as of 01:22, 11 September 2020

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A probability measure on a measurable space <math>(X, \mathcal A)</math> is the same thing as a morphism <math>* \to X</math>

in the Markov category also denoted by <math>P</math>. By composition, 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>. It is concretely defined by

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

== Properties ==