ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Deduction system
Zermelo-Fraenkel set theory
Binary cartesian set product
Binary relation
Simple map
Simple function
Measurable simple complex function
Simple integral
Unsigned basic integral
Unsigned basic expectation
Basic expectation
Random real number moment
Conditional expectation representative
Conditional expectation
Conditional probability
Conditionally independent event collection
Conditionally independent collection of event collections
Definition D4154
Conditionally independent collection of sigma-algebras
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $\mathcal{H} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$
(ii) $\mathcal{G}_j \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ for each $j \in J$
Then $ \{ \mathcal{G}_j \}_{j \in J}$ is a conditionally independent collection of sigma-algebras in $P$ given $\mathcal{H}$ if and only if \begin{equation} \forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \, j_1, \, \ldots, \, j_N \in J \left[ E_{j_1} \in \mathcal{G}_{j_1}, \, \ldots, \, E_{j_N} \in \mathcal{G}_{j_N} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \mid \mathcal{H} \right) \overset{a.s.}{=} \prod_{n = 1}^N \mathbb{P}(E_{j_n} \mid \mathcal{H}) \right] \end{equation}
Conditionally independent random collection