Regular conditional probability

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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[edit]

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 distribution on 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 definition of a regular conditional probability.

Definition[edit]

Let be a probability space, and let be a random variable, defined as a Borel-measurable function from to its state space Then 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 [1]

To express this in our more familiar notation:

where i.e. the topological support of the pushforward measure As 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.

The measurable space is 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.

See also conditional probability and conditional probability distribution.

Alternate definition[edit]

Consider a Radon space (that is a probability measure defined on a Radon space endowed with the Borel sigma-algebra) and a real-valued random variable T. As discussed above, in this case there exists a regular conditional probability with respect to T. Moreover, we can alternatively define the regular conditional probability for an event A given a particular value t of the random variable T in the following manner:

where the limit is taken over the net of open neighborhoods U of t as they become smaller with respect to set inclusion. This limit is defined if and only if the probability space is Radon, and only in the support of T, as described in the article. This is the restriction of the transition probability to the support of T. To describe this limiting process rigorously:

For every there exists an open neighborhood U of the event {T=t}, such that for every open V with

where is the limit.

Example[edit]

To continue with our motivating example above, we consider a real-valued random variable X and 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 "".

See also[edit]

References[edit]

  1. ^ 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 PDF

External links[edit]