- 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 measureconditioned on a particular value of a random variable.
Normally we define the conditional probability of an event "A" given an event "B" as::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 distributionon and "B" is the event that Clearly the probability of "B" in this case is but nonetheless we would still like to assign meaning to a conditional probability such as To do so rigorously requires the definiton of a regular conditional probability.
Let be a
probability space, and let be a random variable, defined as a measurable functionfrom to its state spaceThen a regular conditional probability is defined as a function called a "transition probability", where is a valid probability measure (in its second argument) on for all and a measurable function in "E" (in its first argument) for all such that for all and all [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::where i.e. the topological support of the
pushforward measureAs can be seen from the integral above, the value of 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".
measurable spaceis said to have the regular conditional probability property if for all probability measures on all random variables on admit a regular conditional probability. A Radon space, in particular, has this property: the underlying measurable space of any standard probability spaceis Radon (if its topology is chosen appropriately).
To continue with our motivating example above, where "X" is a real-valued random variable, we may write:(where for the example given.) This limit, if it exists, is a regular conditional probability for "X", restricted to
In any case, it is easy to see that this limit fails to exist for 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 outside the support of "X" (by definition) there will be an such that
Thus if "X" is distributed uniformly on it is truly meaningless to condition a probability on "".
* [http://planetmath.org/encyclopedia/ConditionalProbabilityMeasure.html Regular Conditional Probability] on [http://planetmath.org/ PlanetMath]
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