- Regular conditional probability
**Regular conditional probability**is a concept that has developed to overcome certain difficulties in formally defining conditional probabilities forcontinuous probability distribution s. It is defined as an alternativeprobability measure conditioned on a particular value of arandom 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 arandom variable "X" with auniform 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 arandom variable , defined as ameasurable function from $Omega$ to itsstate 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 [*] :$mathfrak\; Pig(Acap\; T^\{-1\}(B)ig)\; =\; int\_B\; u(x,A)\; ,dmathfrak\; Pig(T^\{-1\}(x)ig).$*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 $T\; *\; mathfrak\; P\; =\; mathfrak\; Pig(T^\{-1\}(cdot)ig).$ 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 allprobability measure s $mathfrak\; P$ on $(Omega,\; mathcal\; F),$ allrandom variable s on $(Omega,\; mathcal\; F,\; mathfrak\; P)$ admit a regular conditional probability. ARadon space , in particular, has this property: the underlying measurable space of anyis Radon (if its topology is chosen appropriately).standard probability space **Example**To continue with our motivating example above, where "X" is a real-valued random variable, we may write:$mathfrak\; P(A|X=x\_0)\; =\; u(x\_0,A)\; =\; lim\_\{epsilon\; ightarrow\; 0+\}\; frac\; \{mathfrak\; P(Acap\{x\_0-epsilon\; <\; X\; <\; x\_0+epsilon\})\}\{mathfrak\; P(\{x\_0-epsilon\; <\; X\; <\; x\_0+epsilon\})\},$(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 $mathfrak\; P(\{x\_0-epsilon\; <\; X\; <\; x\_0+epsilon\})=0.$

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*]

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