Regular conditional probability: Difference between revisions

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==Definition==

Let <math>(\Omega, \mathcal F, P)</math> be a [[probability space]], and let <math>T:\Omega\rightarrow E</math> be a [[random variable]], defined as a [[Borel measure|Borel-]][[measurable function]] from <math>\Omega</math> to its [[Probability_spaceProbability space#Random_variablesRandom variables|state space]] <math>(E, \mathcal E).</math> Then a '''regular conditional probability''' is defined as a function <math>\nu:E \times\mathcal F \rightarrow [0,1],</math> called a "transition probability", where <math>\nu(x,A)</math> is a valid probability measure (in its second argument) on <math>\mathcal F</math> for all <math>x\in E</math> and a measurable function in ''E'' (in its first argument) for all <math>A\in\mathcal F,</math> such that 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>

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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.

See also [[Conditional_expectationConditional expectation#Definition_of_conditional_probabilityDefinition of conditional probability|conditional probability]] and [[Conditional_probability_distributionConditional probability distribution#Measure-Theoretic_FormulationTheoretic Formulation|conditional probability distribution]].

==Alternate definition==

{{disputeabout|'''this way leads to irregular conditional probability'''|Non-regular conditional probability|date=September 2009}}

Consider a Radon space <math> \Omega </math> (that is a probability measure defined on a Radon space endowed with the Borel sigma-algebra) and a real-valued random variable ''T''. As discussed above, in this case there exists a regular conditional probability with respect to ''T''. Moreover, we can alternatively define the '''regular conditional probability''' for an event ''A'' given a particular value ''t'' of the random variable ''T'' in the following manner:

:<math> P (A|T=t) = \lim_{U\supset \{T= t\}} \frac {P(A\cap U)}{P(U)},</math>