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:

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

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

Definition

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

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

To express this in our more familiar notation:

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

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

Alternate definition

We may also define a regular conditional probability for an event A given a particular value t of the random variable T in the following manner:

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

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 $\epsilon >0,$ there exists an open neighborhood U of t, such that for every open V with $t\in V\subset U,$

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

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

Example

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

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

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

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

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

Regularity versus completeness

Standard probability space	Radon space
Lebesgue measure	Borel measure
Complete measure	Regular measure
Conditional probability	Regular conditional probability
Extremely complicated and weak.	Simple and powerful.
Pathological cases.	No pathological cases.
$\lambda (\mathbb {Q} \cap [0,1])=0.$	$\mu (\mathbb {Q} \cap [0,1])$ is undefined.
Probability is $\sigma$-additive	except for sets with isolated points.

Note: In this article we use the Fraktur ${\mathfrak {P}}$ (whose shape is somewhat reminiscent of ${\mathfrak {B}}$ for Borel) to indicate a probability based on a regular measure as opposed to one based on a complete measure. The notions of regularity and completeness are incompatible in a separable space.

See also

• Conditioning (probability)
• Disintegration theorem
• Adherent point
• Limit point

External links

• Regular Conditional Probability on PlanetMath