Regular conditional probability: Difference between revisions

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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]].

{{short description|Alternative probability measure conditioned on a particular value of a random variable}}

'''Regular conditional probability''' is a concept that has developed to overcome certain difficulties in formally defining [[Conditional probability|conditional probabilities]] for [[continuous probability distribution]]s. It is defined as an alternative [[probability measure]] conditioned on a particular value of a [[random variable]].

==Motivation Definition ==

Normally we define the '''conditional probability''' of an event ''A'' given an event ''B'' as:

:<math>P(A|B)=\frac{P(A\cap B)}{P(B)}.</math>

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 (continuous)|uniform distribution]] on <math>[0,1],</math> and ''B'' is the event that <math>X=2/3.</math> Clearly, the probability of ''B,'' in this case, is <math>P(B)=0,</math> but nonetheless we would still like to assign meaning to a conditional probability such as <math>P(A|X=2/3).</math> To do so rigorously requires the definition of a regular conditional probability.

=== Conditional probability distribution ===

==Definition==

Let <math>(\Omega, \mathcal F, P)</math> be a [[probability space]], and let <math>T:\Omega\rightarrow E</math> be a [[random variable]], defined as a [[Borel measure|Borel-]][[measurable function]] from <math>\Omega</math> to its [[Probability space#Random variables|state space]] <math>(E, \mathcal E).</math> Then a '''regular conditional probability''' is defined as a function <math>\nu:E \times\mathcal F \rightarrow [0,1],</math> called a "transition probability", where <math>\nu(x,A)</math> is a valid probability measure (in its second argument) on <math>\mathcal F</math> for all <math>x\in E</math> and a measurable function in ''E'' (in its first argument) for all <math>A\in\mathcal F,</math> such that for all <math>A\in\mathcal F</math> and all <math>B\in\mathcal E</math><ref>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 [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>

:<math>P\big(A\cap T^{-1}(B)\big) = \int_B \nu(x,A) \,P\big(T^{-1}(d x)\big).</math>

Consider two random variables <math>X, Y : \Omega \to \mathbb{R}</math>. The ''conditional probability distribution'' of ''Y'' given ''X'' is a two variable function <math>\kappa_{Y\mid X}: \mathbb{R} \times \mathcal{B}(\mathbb{R}) \to [0,1]</math>

To express this in our more familiar notation:

:<math>P(A|T=x) = \nu(x,A),</math>

where <math>x\in\mathrm{supp}\,T,</math> i.e. the [[Support (measure theory)|topological support]] of the [[pushforward measure]] <math>T _* P = P\big(T^{-1}(\cdot)\big).</math> As can be seen from the integral above, the value of <math>\nu</math> 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''.

If the random variable ''X'' is discrete

The [[measurable space]] <math>(\Omega, \mathcal F)</math> is said to have the '''regular conditional probability property''' if for all [[probability measure]]s <math>P</math> on <math>(\Omega, \mathcal F),</math> all [[random variable]]s on <math>(\Omega, \mathcal F, P)</math> admit a regular conditional probability. A [[Radon space]], in particular, has this property.

:<math>\kappa_{Y\mid X}(x, A) = P(Y \in A \mid X = x) = \begin{cases}

\frac{P(Y \in A, X = x)}{P(X=x)} & \text{ if } P(X = x) > 0 \\[3pt]

\text{arbitrary value} & \text{ otherwise}.

\end{cases}</math>

If the random variables ''X'', ''Y'' are continuous with density <math>f_{X,Y}(x,y)</math>.

See also [[Conditional expectation#Definition of conditional probability|conditional probability]] and [[Conditional probability distribution#Measure-Theoretic Formulation|conditional probability distribution]].

:<math>\kappa_{Y\mid X}(x, A) = \begin{cases}

\frac{\int_A f_{X,Y}(x, y) \, \mathrm{d}y}{\int_\mathbb{R} f_{X,Y}(x, y) \mathrm{d}y} &

\text{ if } \int_\mathbb{R} f_{X,Y}(x, y) \, \mathrm{d}y > 0 \\[3pt]

\text{arbitrary value} & \text{ otherwise}.

\end{cases}</math>

A more general definition can be given in terms of [[conditional expectation]]. Consider a function <math> e_{Y \in A} : \mathbb{R} \to [0,1]</math> satisfying

==Alternate definition==

:<math>e_{Y \in A}(X(\omega)) = \operatorname E[1_{Y \in A} \mid X](\omega)</math>

{{disputeabout|'''this way leads to irregular conditional probability'''|Non-regular conditional probability|date=September 2009}}

for almost all <math>\omega</math>.

Consider a Radon space <math> \Omega </math> (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:

Then the conditional probability distribution is given by

:<math>\kappa_{Y\mid X}(x, A) = e_{Y \in A}(x).</math>

As with conditional expectation, this can be further generalized to conditioning on a sigma algebra <math>\mathcal{F}</math>. In that case the conditional distribution is a function <math>\Omega \times \mathcal{B}(\mathbb{R}) \to [0, 1]</math>:

:<math> P (A|T=t) = \lim_{U\supset \{T= t\}} \frac {P(A\cap U)}{P(U)},</math>

:<math> \kappa_{Y\mid\mathcal{F}}(\omega, A) = \operatorname E[1_{Y \in A} \mid \mathcal{F}](\omega)</math>

=== Regularity ===

where the [[Limit (mathematics)|limit]] is taken over the [[Net (mathematics)|net]] of [[Open set|open]] [[Neighbourhood (mathematics)|neighborhoods]] ''U'' of ''t'' as they become [[Subset|smaller with respect to set inclusion]]. This limit is defined if and only if the probability space is [[Radon space|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 working with <math>\kappa_{Y\mid X}</math>, it is important that it be ''regular'', that is:

For every <math>\epsilon > 0,</math> there exists an open neighborhood ''U'' of the event {''T=t''}, such that for every open ''V'' with <math>\{T=t\} \subset V \subset U,</math>

# For almost all ''x'', <math>A \mapsto \kappa_{Y\mid X}(x, A)</math> is a probability measure

:<math>\left|\frac {P(A\cap V)}{P(V)}-L\right| < \epsilon,</math>

where# For all ''A'', <math>Lx =\mapsto P\kappa_{Y\mid X}(x, A|T=t)</math> is thea measurable limit.function

In other words <math>\kappa_{Y\mid X}</math> 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 <math>\Omega \to S</math> in a [[Radon space]] ''S'', there exists a <math>\kappa_{Y\mid X}</math> that satisfies the first condition.<ref>{{cite book |last1=Klenke |first1=Achim |title=Probability theory : a comprehensive course |date=30 August 2013 |___location=London |isbn=978-1-4471-5361-0 |edition=Second}}</ref> It is possible to construct more general spaces where a regular conditional probability distribution does not exist.<ref>Faden, A.M., 1985. The existence of regular conditional probabilities: necessary and sufficient conditions. ''The Annals of Probability'', 13(1), pp.&nbsp;288–298.</ref>

=== Relation to conditional expectation ===

For discrete and continuous random variables, the [[conditional expectation]] can be expressed as

:<math>

\begin{aligned}

\operatorname E[Y\mid X=x] &= \sum_y y \, P(Y=y\mid X=x) \\

\operatorname E[Y\mid X=x] &= \int y \, f_{Y\mid X}(x, y) \, \mathrm{d}y

\end{aligned}

</math>

where <math>f_{Y\mid X}(x, y)</math> is the [[conditional density]] of {{mvar|Y}} given {{mvar|X}}.

This result can be extended to measure theoretical conditional expectation using the regular conditional probability distribution:

:<math>\operatorname E[Y\mid X](\omega) = \int y \, \kappa_{Y\mid\sigma(X)}(\omega, \mathrm{d}y).</math>

== Formal definition==

Let <math>(\Omega, \mathcal F, P)</math> be a [[probability space]], and let <math>T:\Omega\rightarrow E</math> be a [[random variable]], defined as a [[Borel measure|Borel-]][[measurable function]] from <math>\Omega</math> to its [[Probability space#Random variables|state space]] <math>(E, \mathcal E)</math>.

One should think of <math>T</math> as a way to "disintegrate" the sample space <math>\Omega</math> into <math>\{ T^{-1}(x) \}_{x \in E}</math>.

Using the [[disintegration theorem]] from the measure theory, it allows us to "disintegrate" the measure <math>P</math> into a collection of measures,

one for each <math>x \in E</math>. Formally, a '''regular conditional probability''' is defined as a function <math>\nu:E \times\mathcal F \rightarrow [0,1],</math> called a "transition probability", where:

* For every <math>x \in E</math>, <math>\nu(x, \cdot)</math> is a probability measure on <math>\mathcal F</math>. Thus we provide one measure for each <math>x \in E</math>.

* For all <math>A\in\mathcal F</math>, <math>\nu(\cdot, A)</math> (a mapping <math>E \to [0,1]</math>) is <math>\mathcal E</math>-measurable, and

* For all <math>A\in\mathcal F</math> and all <math>B\in\mathcal E</math><ref>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 [http://www.scielo.cl/pdf/proy/v23n1/art02.pdf PDF]</ref>

:: <math>P\big(A\cap T^{-1}(B)\big) = \int_B \nu(x,A) \,(P\big(circ T^{-1})(\mathrm{d }x)\big).</math>

where <math>P\circ T^{-1}</math> is the [[pushforward measure]] <math>T_*P</math> of the distribution of the random element <math>T</math>,

<math>x\in\operatorname{supp}T,</math> i.e. the [[Support (measure theory)|support]] of the <math>T_* P</math>.

Specifically, if we take <math>B=E</math>, then <math>A \cap T^{-1}(E) = A</math>, and so

:<math>P(A|T=x) = \int_E \nu(x,A) \, (P\circ T^{-1})(\mathrm{d}x),</math>

where <math>\nu(x, A)</math> can be denoted, using more familiar terms <math>P(A\ |\ T=x)</math>.

==Alternate definition==

{{disputeabout|'''this way leads to irregular conditional probability'''|Non-regular conditional probability|date=September 2009}}

Consider a [[Radon space]] <math> \Omega </math> (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:

:<math> P (A|\mid T=t) = \lim_{U\supset \{T= t\}} \frac {P(A\cap U)}{P(U)},</math>

==Example==

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

:<math>P(A|X=x_0) = \nu(x_0,A) = \lim_{\epsilon\rightarrow 0+} \frac {P(A\cap\{x_0-\epsilon < X < x_0+\epsilon\})}{P(\{x_0-\epsilon < X < x_0+\epsilon\})},</math>

(where <math>x_0=2/3</math> for the example given.) This limit, if it exists, is a regular conditional probability for ''X'', restricted to <math>\mathrm{supp}\,X.</math>

where the [[Limit (mathematics)|limit]] is taken over the [[Net (mathematics)|net]] of [[Open set|open]] [[Neighbourhood (mathematics)|neighborhoods]] ''U'' of ''t'' as they become [[Subset|smaller with respect to set inclusion]]. This limit is defined if and only if the probability space is [[Radon space|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 &nbsp;''T''. To describe this limiting process rigorously:

In any case, it is easy to see that this limit fails to exist for <math>x_0</math> 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 [[Neighbourhood (mathematics)|neighborhood]] has positive probability, for every point <math>x_0</math> outside the support of ''X'' (by definition) there will be an <math>\epsilon > 0</math> such that <math>P(\{x_0-\epsilon < X < x_0+\epsilon\})=0.</math>

For every <math>\epsilonvarepsilon > 0,</math> there exists an open neighborhood ''U'' of the event {''T''&nbsp;=&nbsp;''t''}, such that for every open ''V'' with <math>\{T=t\} \subset V \subset U,</math>

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

:<math>P(A\left|B)=\frac {P(A\cap BV)}{P(BV)}.-L\right| < \varepsilon,</math>

where <math>L = P (A\mid T=t)</math> is the limit.

==See also==