Markov kernel: Difference between revisions

Revision as of 18:16, 23 March 2020

Revision as of 12:35, 20 August 2020

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== Formal definition ==

Let <math>(X,\mathcal A)</math> and <math>(Y,\mathcal B)</math> be [[measurable space]]s. A ''Markov kernel'' <math>\kappa: X \to Y</math> with source <math>(X,\mathcal A)</math> and target <math>(Y,\mathcal B)</math> is a map <math>\kappa : \mathcal B \times X \to [0,1]</math> with the following properties:

# For every (fixed) <math>B \in \mathcal B</math>, the map <math> x \mapsto \kappa(B, x)</math> is <math>\mathcal A</math>-measurable

# For every (fixed) <math> x \in X</math>, the map <math> B \mapsto \kappa(B, x)</math> is a [[probability measure]] on <math>(Y, \mathcal B)</math>