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:

${\displaystyle P(A|B)={\frac {P(A\cap B)}{P(B)}}.}$

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 ${\displaystyle [0,1],}$ and B is the event that ${\displaystyle X=2/3.}$ Clearly, the probability of B, in this case, is ${\displaystyle P(B)=0,}$ but nonetheless we would still like to assign meaning to a conditional probability such as ${\displaystyle P(A|X=2/3).}$ To do so rigorously requires the definition of a regular conditional probability.

Definition

Let ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ be a probability space, and let ${\displaystyle T:\Omega \rightarrow E}$ be a random variable, defined as a Borel-measurable function from ${\displaystyle \Omega }$ to its state space ${\displaystyle (E,{\mathcal {E}}).}$ Then a regular conditional probability is defined as a function ${\displaystyle \nu :E\times {\mathcal {F}}\rightarrow [0,1],}$ called a "transition probability", where ${\displaystyle \nu (x,A)}$ is a valid probability measure (in its second argument) on ${\displaystyle {\mathcal {F}}}$ for all ${\displaystyle x\in E}$ and a measurable function in E (in its first argument) for all ${\displaystyle A\in {\mathcal {F}},}$ such that for all ${\displaystyle A\in {\mathcal {F}}}$ and all ${\displaystyle B\in {\mathcal {E}}}$ [1]

${\displaystyle P{\big (}A\cap T^{-1}(B){\big )}=\int _{B}\nu (x,A)\,P{\big (}T^{-1}(dx){\big )}.}$

To express this in our more familiar notation:

${\displaystyle P(A|T=x)=\nu (x,A),}$

where ${\displaystyle x\in \mathrm {supp} \,T,}$ i.e. the topological support of the pushforward measure ${\displaystyle T_{*}P=P{\big (}T^{-1}(\cdot ){\big )}.}$ As can be seen from the integral above, the value of ${\displaystyle \nu }$ 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 ${\displaystyle (\Omega ,{\mathcal {F}})}$ is said to have the regular conditional probability property if for all probability measures ${\displaystyle P}$ on ${\displaystyle (\Omega ,{\mathcal {F}}),}$ all random variables on ${\displaystyle (\Omega ,{\mathcal {F}},P)}$ admit a regular conditional probability. A Radon space, in particular, has this property.

Alternate definition

Consider a Radon space ${\displaystyle \Omega }$ (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:

${\displaystyle P(A|T=t)=\lim _{U\supset \{T=t\}}{\frac {P(A\cap U)}{P(U)}},}$

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 ${\displaystyle \epsilon >0,}$ there exists an open neighborhood U of the event {T=t}, such that for every open V with ${\displaystyle \{T=t\}\subset V\subset U,}$

${\displaystyle \left|{\frac {P(A\cap V)}{P(V)}}-L\right|<\epsilon ,}$

where ${\displaystyle L=P(A|T=t)}$ is the limit.

Example

To continue with our motivating example above, we consider a real-valued random variable X and write

${\displaystyle P(A|X=x_{0})=\nu (x_{0},A)=\lim _{\epsilon \rightarrow 0+}{\frac {P(A\cap \{x_{0}-\epsilon

(where ${\displaystyle x_{0}=2/3}$ for the example given.) This limit, if it exists, is a regular conditional probability for X, restricted to ${\displaystyle \mathrm {supp} \,X.}$

In any case, it is easy to see that this limit fails to exist for ${\displaystyle x_{0}}$ 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 ${\displaystyle x_{0}}$ outside the support of X (by definition) there will be an ${\displaystyle \epsilon >0}$ such that ${\displaystyle P(\{x_{0}-\epsilon

Thus if X is distributed uniformly on ${\displaystyle [0,1],}$ it is truly meaningless to condition a probability on "${\displaystyle X=3/2}$ ".