Regular conditional probability: Difference between revisions

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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

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

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

Then the conditional probability distribution is given by

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

=== Regularity ===

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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 ===

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:<math>

\begin{aligned}

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

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

\end{aligned}

</math>

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

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

== Formal definition==

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

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==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>