Regular conditional probability: Difference between revisions

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Revision as of 22:20, 21 November 2023

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* For all <math>A\in\mathcal F</math>, <math>\nu(\cdot, A)</math> (a mapping <math>E \to [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(circ T^{-1})(\mathrm{d }x)\big).</math>

where <math>P\circ T^{-1}</math> is the [[pushforward measure]] <math>T_*P</math> of the distribution of the random element <math>T</math>,

<math>x\in\mathrm{supp}\,T,</math> i.e. the [[Support (measure theory)|support]] of the <math>T_* P</math>.

Specifically, if we take <math>B=E</math>, then <math>A \cap T^{-1}(E) = A</math>, and so

:<math>P(A) = \int_E \nu(x,A) \, (P\big(circ T^{-1})(\mathrm{d }x)\big)</math>,

where <math>\nu(x, A)</math> can be denoted, using more familiar terms <math>P(A\ |\ T=x)</math>.