# Regular conditional probability

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::$mathfrak P\left(A|B\right)=frac\left\{mathfrak P\left(Acap B\right)\right\}\left\{mathfrak P\left(B\right)\right\}.$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 $\left[0,1\right] ,$ and "B" is the event that $X=2/3.$ Clearly the probability of "B" in this case is $mathfrak P\left(B\right)=0,$ but nonetheless we would still like to assign meaning to a conditional probability such as $mathfrak P\left(A|X=2/3\right).$ To do so rigorously requires the definiton of a regular conditional probability.

Definition

Let $\left(Omega, mathcal F, mathfrak P\right)$ 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 $\left(E, mathcal E\right).$ Then a regular conditional probability is defined as a function $u:E imesmathcal F ightarrow \left[0,1\right] ,$ called a "transition probability", where $u\left(x,A\right)$ 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::$mathfrak P\left(A|T=x\right) = u\left(x,A\right),$where $xinmathrm\left\{supp\right\},T,$ 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 $\left(Omega, mathcal F\right)$ is said to have the regular conditional probability property if for all probability measures $mathfrak P$ on $\left(Omega, mathcal F\right),$ all random variables on $\left(Omega, mathcal F, mathfrak P\right)$ 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:$mathfrak P\left(A|X=x_0\right) = u\left(x_0,A\right) = lim_\left\{epsilon ightarrow 0+\right\} frac \left\{mathfrak P\left(Acap\left\{x_0-epsilon < X < x_0+epsilon\right\}\right)\right\}\left\{mathfrak P\left(\left\{x_0-epsilon < X < x_0+epsilon\right\}\right)\right\},$(where $x_0=2/3$ for the example given.) This limit, if it exists, is a regular conditional probability for "X", restricted to $mathrm\left\{supp\right\},X.$

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 $mathfrak P\left(\left\{x_0-epsilon < X < x_0+epsilon\right\}\right)=0.$

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

References

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

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