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. Consider two random variables . 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. Consider a function satisfying for almost all . 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 : For working with , it is important that it be regular, that is: For almost all x, is a probability measure For all A, is a measurable function 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 that satisfies the first condition. It is possible to construct more general spaces where a regular conditional probability distribution does not exist. For discrete and continuous random variables, the conditional expectation can be expressed as where is the conditional density of Y given X. 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 . One should think of as a way to "disintegrate" the sample space into . Using the disintegration theorem from the measure theory, it allows us to "disintegrate" the measure into a collection of measures, one for each . Formally, a regular conditional probability is defined as a function called a "transition probability", where: For every , is a probability measure on .