Conditional probability distribution: Difference between revisions

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==Measure-theoretic formulation==

Let <math>(\Omega, \mathcal{F}, P)</math> be a probability space, <math>\mathcal{G} \subseteq \mathcal{F}</math> a <math>\sigma</math>-field in <math>\mathcal{F}</math>. Given <math>A\in \mathcal{F}</math>, the [[Radon-Nikodym theorem]] implies that there is<ref>[[#billingsley95|Billingsley (1995)]], p. 430</ref> a <math>\mathcal{G}</math>-measurable random variable <math>P(A\mid\mathcal{G}):\Omega\to \mathbb{R}</math>, called the '''[[conditional probability''']], such that<math display="block">\int_G P(A\mid\mathcal{G})(\omega) dP(\omega)=P(A\cap G)</math>for every <math>G\in \mathcal{G}</math>, and such a random variable is uniquely defined up to sets of probability zero. A conditional probability is called [[Regular conditional probability|'''regular''']] if <math> \operatorname{P}(\cdot\mid\mathcal{G})(\omega) </math> is a [[probability measure]] on <math>(\Omega, \mathcal{F})</math> for all <math>\omega \in \Omega</math> a.e.

Special cases:

* For the trivial sigma algebra <math>\mathcal G= \{\emptyset,\Omega\}</math>, the conditional probability is the constant function <math>\operatorname{P}\!\left( A\mid \{\emptyset,\Omega\} \right) = \operatorname{P}(A).</math>

* If <math>A\in \mathcal{G}</math>, then <math>\operatorname{P}(A\mid\mathcal{G})=1_A</math>, the indicator function (defined [[#Relation to conditional expectation|below]]).

Let <math>X : \Omega \to E</math> be a <math>(E, \mathcal{E})</math>-valued random variable. For each <math>B \in \mathcal{E}</math>, define <math display="block">\mu_{X \, | \, \mathcal{G}} (B \, |\, \mathcal{G}) = \mathrm{P} (X^{-1}(B) \, | \, \mathcal{G}).</math>For any <math>\omega \in \Omega</math>, the function <math>\mu_{X \, | \mathcal{G}}(\cdot \, | \mathcal{G}) (\omega) : \mathcal{E} \to \mathbb{R}</math> is called the '''[[Conditional expectation#Definition of conditional probability|conditional probability]] distribution''' of <math>X</math> given <math>\mathcal{G}</math>. If it is a probability measure on <math>(E, \mathcal{E})</math>, then it is called [[Regular conditional probability|'''regular''']].

For a real-valued random variable (with respect to the Borel <math>\sigma</math>-field <math>\mathcal{R}^1</math> on <math>\mathbb{R}</math>), every conditional probability distribution is regular.<ref>[[#billingsley95|Billingsley (1995)]], p. 439</ref> In this case,<math>E[X \mid \mathcal{G}] = \int_{-\infty}^\infty x \, \mu(d x, \cdot)</math> almost surely.