Conditional independence

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of <a href="/facts/Conditional_probability/QcN2UERV">conditional probability</a>, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability without. If 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is the hypothesis, and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and 
 
 
 
 C
 
 
 {\displaystyle C}
 
 are observations, conditional independence can be stated as an equality:

P
        (
        A
        ∣
        B
        ,
        C
        )
        =
        P
        (
        A
        ∣
        C
        )
      
    
    {\displaystyle P(A\mid B,C)=P(A\mid C)}

where 
 
 
 
 P
 (
 A
 ∣
 B
 ,
 C
 )
 
 
 {\displaystyle P(A\mid B,C)}
 
 is the probability of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 given both 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and 
 
 
 
 C
 
 
 {\displaystyle C}
 
. Since the probability of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 given 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is the same as the probability of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 given both 
 
 
 
 B
 
 
 {\displaystyle B}
 
 and 
 
 
 
 C
 
 
 {\displaystyle C}
 
, this equality expresses that 
 
 
 
 B
 
 
 {\displaystyle B}
 
 contributes nothing to the certainty of 
 
 
 
 A
 
 
 {\displaystyle A}
 
. In this case, 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 are said to be conditionally independent given 
 
 
 
 C
 
 
 {\displaystyle C}
 
, written symbolically as: 
 
 
 
 (
 A
 ⊥
 
 
 
 ⊥
 B
 ∣
 C
 )
 
 
 {\displaystyle (A\perp \!\!\!\perp B\mid C)}
 
.
The concept of conditional independence is essential to graph-based theories of statistical inference, as it establishes a mathematical relation between a collection of conditional statements and a <a href="/facts/Graphoid/fdqjpV04">graphoid</a>.

Conditional independence open-in-new

Conditional independence