Regular conditional probability: Difference between revisions

Revision as of 19:34, 6 December 2023

Revision as of 19:37, 6 December 2023

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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}]</math>

=== Regularity ===

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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==