# Regular conditional probability

**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

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

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

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

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 "
".

## References

- 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