conditional independence

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

Conditional Independence Given an Event

Given an event $C\in ℱ$:

1. Two events $A$ and $B$ in $ℱ$ 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 ${ℱ}_1,{ℱ}_2$ of $ℱ$ are conditionally independent given $C$ if any two events $A\in {ℱ}_1$ and $B\in {ℱ}_2$ are conditionally independent given $C$.

3. Two real random variables $X,Y:\mathrm{\Omega }\to ℝ$ are conditionally independent given event $C$ if ${ℱ}_X$ and ${ℱ}_Y$, 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 $𝒢$ of $ℱ$:

1. Two events $A$ and $B$ in $ℱ$ 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):

   $$P(A\cap B|𝒢)=P(A|𝒢)P(B|𝒢).$$

2. Two sub sigma algebras ${ℱ}_1,{ℱ}_2$ of $ℱ$ are conditionally independent given $\mathrm{G}$ if any two events $A\in {ℱ}_1$ and $B\in {ℱ}_2$ are conditionally independent given $𝒢$.

3. Two real random variables $X,Y:\mathrm{\Omega }\to ℝ$ are conditionally independent given event $\mathrm{G}$ if ${ℱ}_X$ and ${ℱ}_Y$, the sub sigma algebras generated by $X$ and $Y$ are conditionally independent given $𝒢$.

4. Finally, we can define conditional idependence given a random variable, say $Z:\mathrm{\Omega }\to ℝ$ in each of the above three items by setting $𝒢={ℱ}_Z$.