Markov kernel: Difference between revisions

Revision as of 18:16, 8 December 2022

Revision as of 18:16, 8 December 2022

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Let <math>(X, \mathcal A, P)</math> be a probability space and <math>\kappa</math> a Markov kernel from <math>(X, \mathcal A)</math> to some <math>(Y, \mathcal B)</math>. Then there exists a unique measure <math>Q</math> on <math>(X \times Y, \mathcal A \otimes \mathcal B)</math>, such that:

: <math>Q(A \times B) = \int_A \kappa(B|x)\,P(x)dx), \quad \forall A \in \mathcal A, \quad \forall B \in\mathcal B.</math>

=== Regular conditional distribution ===