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(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] ] :mathfrak Pig(Acap T^{-1}(B)ig) = int_B u(x,A) ,dmathfrak Pig(T^{-1}(x)ig).

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 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: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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