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# conditional independence

Let $(\Omega,\mathcal{F},P)$ be a probability space.

# Conditional Independence Given an Event

Given an event $C\in\mathcal{F}$:

1. Two events $A$ and $B$ in $\mathcal{F}$ are said to be

*conditionally independent given $C$*if we have the following equality of conditional probabilities:$P(A\cap B|C)=P(A|C)P(B|C).$ 2. Two sub sigma algebras $\mathcal{F}_{1},\mathcal{F}_{2}$ of $\mathcal{F}$ are

*conditionally independent given $C$*if any two events $A\in\mathcal{F}_{1}$ and $B\in\mathcal{F}_{2}$ are conditionally independent given $C$.3. Two real random variables $X,Y:\Omega\to\mathbb{R}$ are

*conditionally independent given event $C$*if $\mathcal{F}_{X}$ and $\mathcal{F}_{Y}$, the sub sigma algebras generated by $X$ and $Y$ are conditionally independent given $C$.

# Conditional Independence Given a Sigma Algebra

Given a sub sigma algebra $\mathcal{G}$ of $\mathcal{F}$:

1. Two events $A$ and $B$ in $\mathcal{F}$ are said to be

*conditionally independent given $\mathcal{G}$*if we have the following equality of conditional probabilities (as random variables):$P(A\cap B|\mathcal{G})=P(A|\mathcal{G})P(B|\mathcal{G}).$ 2. Two sub sigma algebras $\mathcal{F}_{1},\mathcal{F}_{2}$ of $\mathcal{F}$ are

*conditionally independent given $\mathcal{G}$*if any two events $A\in\mathcal{F}_{1}$ and $B\in\mathcal{F}_{2}$ are conditionally independent given $\mathcal{G}$.3. Two real random variables $X,Y:\Omega\to\mathbb{R}$ are

*conditionally independent given event $\mathcal{G}$*if $\mathcal{F}_{X}$ and $\mathcal{F}_{Y}$, the sub sigma algebras generated by $X$ and $Y$ are conditionally independent given $\mathcal{G}$.4. Finally, we can define

*conditional idependence given a random variable*, say $Z:\Omega\to\mathbb{R}$ in each of the above three items by setting $\mathcal{G}=\mathcal{F}_{Z}$.

## Mathematics Subject Classification

60A05*no label found*

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