Conditional probability distribution: Difference between revisions

Revision as of 13:55, 25 August 2021

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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{B})(\omega) </math> is a [[probability measure]] on <math>(\Omega, \mathcal{F})</math> for all <math>\omega \in \Omega</math> a.e.

Special cases:

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* 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 below).

Let <math>X : \Omega \to \mathbb{R}E</math> be a real-valued random variable (measurable with respect to the Borel <math>\sigma</math>-field(E, <math>\mathcal{RE}^1)</math>-valued onrandom <math>\mathbb{R}</math>)variable. ItFor caneach then be shown that there exists<refmath>[[#billingsley95|BillingsleyB (1995)]], p. 439</ref> a function <math>\mu :in \mathcal{R}^1 \times \Omega \to \mathbb{RE}</math>, such thatdefine <math display="block">\mu(mu_{X \cdot, | \omega)</math>, is a probability measure on <math>\mathcal{RG}}^1</math> for(B each <math>\omega \in \Omega</math> (i.e., it is [[Regular conditional probability|'''regular''']]) and <math>\mu(H, \cdotmathcal{G}) = \mathrm{P} (X^{-1}( H B)\mid \mathcal{G})</math>, (almost surely) for every <math>H| \in, \mathcal{RG}^1).</math>. For any <math>\omega \in \Omega</math>, the function <math>\mumu_{X \, | \mathcal{G}}(\cdot \, | \mathcal{G}) (\omega) : \mathcal{RE}^1 \to \mathbb{R}</math> is called athe '''[[Conditional expectation#Definition of conditional probability|conditional probability]] distribution''' of <math>X</math> given <math>\mathcal{G}</math>. InIf thisit is a probability measure on case,<math>(E[X \mid, \mathcal{GE}] = \int_{-\infty}^\infty x \, \mu(d x, \cdot)</math>, almostthen surelyit 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.

=== Relation to conditional expectation ===

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:<math>\operatorname{E}(\mathbf{1}_A) = \operatorname{P}(A). \; </math>

Given a <math>\sigma</math>-field <math>\mathcal{G} \subseteq \mathcal{F}</math>, the conditional probability <math> \operatorname{P}(A\mid\mathcal{G})</math> is a version of the [[conditional expectation]] of the indicator function for <math>A</math>:

:<math>\operatorname{P}(A\mid\mathcal{B}) = \operatorname{E}(\mathbf{1}_A\mid\mathcal{B}) \; </math>