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Wikipedia / Regular conditional probability

In probability theory, regular conditional probability is a concept that formalizes the notion of conditioning on the outcome of a random variable. The resulting conditional probability distribution is a parametrized family of probability measures called a Markov kernel.

Definition

Conditional probability distribution

. The conditional probability distribution of Y given X

If the random variable X is discrete

If the random variables X, Y.

A more general definition can be given in terms of conditional expectation satisfying

. Then the conditional probability distribution is given by

:

Regularity

, it is important that it be regular, that is:

  1. For almost all x is a probability measure

  2. For all A is a measurable function

is a Markov kernel.

The second condition holds trivially, but the proof of the first is more involved. It can be shown that if Y in a Radon space S that satisfies the first condition. It is possible to construct more general spaces where a regular conditional probability distribution does not exist.

Relation to conditional expectation

For discrete and continuous random variables, the conditional expectation can be expressed as

is the conditional density of given .

This result can be extended to measure theoretical conditional expectation using the regular conditional probability distribution:

Formal definition

be a probability space be a random variable, defined as a Borel-measurable function to its state space. Using the disintegration theorem. Formally, a regular conditional probability called a "transition probability", where:

  • .

  • -measurable, and

is the pushforward measure i.e. the support, and so

.

Alternate definition

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 ofT. To describe this limiting process rigorously:

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

is the limit.

See also

References