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-NikodymRadon–Nikodym theorem]] implies that there is{{sfnp|Billingsley|1995|p=430}} 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:

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An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation.

=== Interpretation of conditioning on a Sigma Field ===

Consider the probability space <math>(\Omega, \mathcal{F}, \mathbb{P})</math>

and a sub-sigma field <math>\mathcal{A} \subset \mathcal{F}</math>.

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Consider a probability space on the unit interval, <math>\Omega = [0, 1]</math>. Let <math>\mathcal{G}</math> be the sigma-field of all countable sets and sets whose complement is countable. So each set in <math>\mathcal{G}</math> has measure <math>0</math> or <math>1</math> and so is independent of each event in <math>\mathcal{F}</math>. However, notice that <math>\mathcal{G}</math> also contains all the singleton events in <math>\mathcal{F}</math> (those sets which contain only a single <math>\omega \in \Omega</math>). So knowing which of the events in <math>\mathcal{G}</math> occurred is equivalent to knowing exactly which <math>\omega \in \Omega</math> occurred! So in one sense, <math>\mathcal{G}</math> contains no information about <math>\mathcal{F}</math> (it is independent of it), and in another sense it contains all the information in <math>\mathcal{F}</math>.{{sfnp|Billingsley|2012}}{{Page needed|date=May 2025}}

== See also ==

* [[Conditioning (probability)]]

* [[Conditional probability]]

* [[Regular conditional probability]]

* [[Bayes' theorem]]

== References ==

=== Citations ===

{{Reflist}}

=== Sources ===

{{refbegin}}

* {{cite book |last= Billingsley |first= Patrick |date= 1995 |title= Probability and Measure |edition= 3rd |publisher= John Wiley and Sons |___location= New York |isbn= 0-471-00710-2 |author-link= Patrick Billingsley |url= https://books.google.com/books?id=a3gavZbxyJcC }}