conditional independence
Let $(\mathrm{\Omega},\mathcal{F},P)$ 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^{}:
$$P(A\cap BC)=P(AC)P(BC).$$ 
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^{} $X,Y:\mathrm{\Omega}\to \mathbb{R}$ 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 $C$.
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 $\mathrm{G}$ 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 $\mathrm{G}$ 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 $X,Y:\mathrm{\Omega}\to \mathbb{R}$ are conditionally independent given event $\mathrm{G}$ 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 $Z:\mathrm{\Omega}\to \mathbb{R}$ in each of the above three items by setting $\mathcal{G}={\mathcal{F}}_{Z}$.
Title  conditional independence 

Canonical name  ConditionalIndependence 
Date of creation  20130322 16:25:09 
Last modified on  20130322 16:25:09 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  4 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 60A05 
Defines  conditionally independent 