Regular conditional probability

Regular conditional probability is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities for continuous probability distributions. It is defined as an alternative probability measure conditioned on a particular value of a random variable.

Motivation

Normally we define the conditional probability of an event "A" given an event "B" as::$mathfrak\; P(A|B)=frac\{mathfrak\; P(Acap\; B)\}\{mathfrak\; P(B)\}.$The difficulty with this arises when the event "B" is too small to have a non-zero probability. For example, suppose we have a random variable "X" with a uniform distribution on $[0,1]\; ,$ and "B" is the event that $X=2/3.$ Clearly the probability of "B" in this case is $mathfrak\; P(B)=0,$ but nonetheless we would still like to assign meaning to a conditional probability such as $mathfrak\; P(A|X=2/3).$ To do so rigorously requires the definiton of a regular conditional probability.

Definition

Let $(Omega,\; mathcal\; F,\; mathfrak\; P)$ be a probability space, and let $T:Omega\; ightarrow\; E$ be a random variable, defined as a measurable function from $Omega$ to its state space $(E,\; mathcal\; E).$ Then a regular conditional probability is defined as a function $u:E\; imesmathcal\; F\; ightarrow\; [0,1]\; ,$ called a "transition probability", where $u(x,A)$ is a valid probability measure (in its second argument) on $mathcal\; F$ for all $xin\; E$ and a measurable function in "E" (in its first argument) for all $Ainmathcal\; F,$ such that for all $Ainmathcal\; F$ and all $Binmathcal\; E$ [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] ] :

To express this in our more familiar notation::$mathfrak\; P(A|T=x)\; =\; u(x,A),$where $xinmathrm\{supp\},T,$ i.e. the topological support of the pushforward measure As can be seen from the integral above, the value of $u$ 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 $(Omega,\; mathcal\; F)$ is said to have the regular conditional probability property if for all probability measures $mathfrak\; P$ on $(Omega,\; mathcal\; F),$ all random variables on $(Omega,\; mathcal\; F,\; mathfrak\; P)$ admit a regular conditional probability. A Radon space, in particular, has this property: the underlying measurable space of any standard probability space is Radon (if its topology is chosen appropriately).

Example

To continue with our motivating example above, where "X" is a real-valued random variable, we may write:(where $x\_0=2/3$ for the example given.) This limit, if it exists, is a regular conditional probability for "X", restricted to $mathrm\{supp\},X.$

In any case, it is easy to see that this limit fails to exist for $x\_0$ outside the support of "X": since the support of a random variable is defined as the set of all points in its state space whose every neighborhood has positive probability, for every point $x\_0$ outside the support of "X" (by definition) there will be an $epsilon\; >\; 0$ such that

Thus if "X" is distributed uniformly on $[0,1]\; ,$ it is truly meaningless to condition a probability on "$X=3/2$".

References

External links

* [http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]