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 (http://planetmath.org/MathcalFMeasurableFunction) $X$ and $Y$ are conditionally independent given $C$ .

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) (http://planetmath.org/ProbabilityConditioningOnASigmaAlgebra):

$P(A\cap B|\mathcal{G})=P(A|\mathcal{G})P(B|\mathcal{G}).$
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 $\mathcal{G}=\mathcal{F}_{Z}$ .