Markov kernel: Difference between revisions

Revision as of 10:43, 27 July 2024

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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'' with source <math>(X,\mathcal A)</math> and target <math>(Y,\mathcal B)</math>, sometimes written as <math>\kappa:(X,\mathcal{A})\to(Y,\mathcal{B})</math>, is a mapfunction <math>\kappa : \mathcal B \times X \to [0,1]</math> with the following properties:

# For every (fixed) <math>B_0 \in \mathcal B</math>, the map <math> x \mapsto \kappa(B_0, x)</math> is <math>\mathcal A</math>-[[measurable function|measurable]]

# For every (fixed) <math> x_0 \in X</math>, the map <math> B \mapsto \kappa(B, x_0)</math> is a [[probability measure]] on <math>(Y, \mathcal B)</math>