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.
The conditional probability distribution of Y given X is a two variable function
If the random variable X is discrete If the random variables X, Y are continuous with density
A more general definition can be given in terms of conditional expectation.
Then the conditional probability distribution is given by As with conditional expectation, this can be further generalized to conditioning on a sigma algebra
In that case the conditional distribution is a function
, it is important that it be regular, that is: In other words
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 is a random element
in a Radon space S, there exists a
[1] It is possible to construct more general spaces where a regular conditional probability distribution does not exist.
[2] For discrete and continuous random variables, the conditional expectation can be expressed as where
This result can be extended to measure theoretical conditional expectation using the regular conditional probability distribution: Let
be a probability space, and let
be a random variable, defined as a Borel-measurable function from
to its state space
as a way to "disintegrate" the sample space
Using the disintegration theorem from the measure theory, it allows us to "disintegrate" the measure
Formally, a regular conditional probability is defined as a function
called a "transition probability", where: where
is the pushforward measure
of the distribution of the random element
can be denoted, using more familiar terms
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