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Definition D5508
Conditionally independent collection of event collections

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 D78: Subset of $\mathcal{F}$ for each $j \in J$
Then $\mathcal{G} = \{ \mathcal{G}_j \}_{j \in J}$ is a conditionally independent collection of event collections in $P$ given $\mathcal{H}$ if and only if $$\forall \, N \in 1, 2, 3, \ldots : \forall \, \text{distinct } j_1, \ldots, j_N \in J \left[ E_{j_1} \in \mathcal{G}_{j_1}, \ldots, E_{j_N} \in \mathcal{G}_{j_N} \quad \implies \quad \mathbb{P} \left( \bigcap_{n = 1}^N E_{j_n} \mid \mathcal{H} \right) \overset{a.s.}{=} \prod_{n = 1}^N \mathbb{P}(E_{j_n} \mid \mathcal{H}) \right]$$
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 ▶ Conditionally independent collection of sigma-algebras