ThmDex – An index of mathematical definitions, results, and conjectures.
 ▼ Set of symbols ▼ Alphabet ▼ Deduction system ▼ Theory ▼ Zermelo-Fraenkel set theory ▼ Set ▼ Binary cartesian set product ▼ Binary relation ▼ Map ▼ Simple map ▼ Simple function ▼ Measurable simple complex function ▼ Simple integral ▼ Unsigned basic integral ▼ Unsigned basic expectation ▼ Basic expectation ▼ Random real number moment ▼ Expectation ▼ Conditional expectation representative ▼ Conditional expectation ▼ Conditional probability
Definition D2795
Conditionally independent event collection

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
 (i) $\mathcal{G} \subseteq \mathcal{F}$ is a D470: Subsigma-algebra of $\mathcal{F}$ on $\Omega$ (ii) $E_j \in \mathcal{F}$ is an D1716: Event in $P$ for each $j \in J$
Let $\mathcal{P}_{\mathsf{finite}}(J)$ be the D2337: Set of finite subsets of $J$.
Then $E = \{ E_j \}_{j \in J}$ is a conditionally independent event collection in $P$ given $\mathcal{G}$ if and only if $$\forall \, I \in \mathcal{P}_{\mathsf{finite}}(J) : \mathbb{P} \left( \bigcap_{i \in I} E_i \mid \mathcal{G} \right) \overset{a.s.}{=} \prod_{i \in I} \mathbb{P}(E_i \mid \mathcal{G})$$
Children
 ▶ Conditionally independent collection of event collections