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conditional independence∗
CWoo†
2013-03-21 21:30:30
Let (Ω, F, P ) be a probability space.
Conditional Independence Given an Event
Given an event C ∈ 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 ∩ B|C) = P (A|C)P (B|C).
2. Two sub sigma algebras F1 , F2 of F are conditionally independent given
C if any two events A ∈ F1 and B ∈ F2 are conditionally independent
given C.
3. Two real random variables X, Y : Ω → R are conditionally independent
given event C if FX and FY , 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 G of F:
1. Two events A and B in F are said to be conditionally independent given
G if we have the following equality of conditional probabilities (as random
variables):
P (A ∩ B|G) = P (A|G)P (B|G).
2. Two sub sigma algebras F1 , F2 of F are conditionally independent given
G if any two events A ∈ F1 and B ∈ F2 are conditionally independent
given G.
∗ hConditionalIndependencei created: h2013-03-21i by: hCWooi version: h38569i Privacy
setting: h1i hDefinitioni h60A05i
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3. Two real random variables X, Y : Ω → R are conditionally independent
given event G if FX and FY , 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 : Ω → R in each of the above three items by setting G = FZ .
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