Regular conditional probability: Difference between revisions

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* 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>

where <math>T_* P = 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)|topological support]] of the <math>T_* P</math>.

In more familiar terms:

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

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

where <math>x\in\mathrm{supp}\,T,</math> i.e. the [[Support (measure theory)|topological support]] of the <math>T_* P</math>. As can be seen from the integral above, the value of <math>\nu</math> for points ''x'' outside the support of the random variable is meaningless; its significance as a conditional probability is strictly limited to the support of ''T''.

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

(which is "defined" to be conditional probability of <math>A</math> given <math>x</math>, where this

conditional probability can be undefined in elementary constructions of conditional probability).

where <math>x\in\mathrm{supp}\,T,</math> i.e. the [[Support (measure theory)|topological support]] of the <math>T_* P</math>. As can be seen from the integral above, the value of <math>\nu</math> for points ''x'' outside the support of the random variable is meaningless; its significance as a conditional probability is strictly limited to the support of ''T''.

The [[measurable space]] <math>(\Omega, \mathcal F)</math> is said to have the '''regular conditional probability property''' if for all [[probability measure]]s <math>P</math> on <math>(\Omega, \mathcal F),</math> all [[random variable]]s on <math>(\Omega, \mathcal F, P)</math> admit a regular conditional probability. A [[Radon space]], in particular, has this property.