Regular conditional probability: Difference between revisions

Revision as of 19:12, 9 January 2021

Revision as of 01:11, 10 January 2021

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one for each <math>x \in E</math>. Formally, a '''regular conditional probability''' is defined as a function <math>\nu:E \times\mathcal F \rightarrow [0,1],</math> called a "transition probability", where:

* For every <math>x \in E</math>, <math>\nu(x, \cdot)</math> is a probability measure on <math>\mathcal F</math>. Thus we provide one measure for each <math>x \in E</math>.

* For all <math>A\in\mathcal F</math>, <math>\nu(\cdot, A)</math> (a mapping <math>E \mapstoto [0,1]</math>) is <math>\mathcal E</math>-measurable, and

* For all <math>A\in\mathcal F</math> and all <math>B\in\mathcal E</math><ref>D. Leao Jr. et al. ''Regular conditional probability, disintegration of probability and Radon spaces.'' Proyecciones. Vol. 23, No. 1, pp. 15–29, May 2004, Universidad Católica del Norte, Antofagasta, Chile [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>

:<math>P\big(A\cap T^{-1}(B)\big) = \int_B \nu(x,A) \,P\big(T^{-1}(d x)\big).</math>