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Concept
Regular conditional probability
Summary
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 .
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