Conditional dependence

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, conditional dependence is a relationship between two or more <a href="/facts/Event_(probability_theory)/aJ0l4oQb">events</a> that are <a href="/facts/Dependence_(probability_theory)/NUzQtnUL">dependent</a> when a third event occurs. For example, if 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 are two events that individually increase the probability of a third event 
 
 
 
 C
 ,
 
 
 {\displaystyle C,}
 
 and do not directly affect each other, then initially (when it has not been observed whether or not the event 
 
 
 
 C
 
 
 {\displaystyle C}
 
 occurs)

P
 ⁡
 (
 A
 ∣
 B
 )
 =
 P
 ⁡
 (
 A
 )
 
 
  and 
 
 
 P
 ⁡
 (
 B
 ∣
 A
 )
 =
 P
 ⁡
 (
 B
 )
 
 
 {\displaystyle \operatorname {P} (A\mid B)=\operatorname {P} (A)\quad {\text{ and }}\quad \operatorname {P} (B\mid A)=\operatorname {P} (B)}
 
 (
 
 
 
 A
 
  and 
 
 B
 
 
 {\displaystyle A{\text{ and }}B}
 
 are independent).
But suppose that now 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is observed to occur. If event 
 
 
 
 B
 
 
 {\displaystyle B}
 
 occurs then the probability of occurrence of the event 
 
 
 
 A
 
 
 {\displaystyle A}
 
 will decrease because its positive relation to 
 
 
 
 C
 
 
 {\displaystyle C}
 
 is less necessary as an explanation for the occurrence of 
 
 
 
 C
 
 
 {\displaystyle C}
 
 (similarly, event 
 
 
 
 A
 
 
 {\displaystyle A}
 
 occurring will decrease the probability of occurrence of 
 
 
 
 B
 
 
 {\displaystyle B}
 
). Hence, now the two events 
 
 
 
 A
 
 
 {\displaystyle A}
 
 and 
 
 
 
 B
 
 
 {\displaystyle B}
 
 are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have

P
 ⁡
 (
 A
 ∣
 C
 
  and 
 
 B
 )
 <
 P
 ⁡
 (
 A
 ∣
 C
 )
 .
 
 
 {\displaystyle \operatorname {P} (A\mid C{\text{ and }}B)<\operatorname {P} (A\mid C).}

Conditional dependence of A and B given C is the logical negation of <a href="/facts/Conditional_independence/udUXFaMw">conditional independence</a> 
 
 
 
 (
 (
 A
 ⊥
 
 
 
 ⊥
 B
 )
 ∣
 C
 )
 
 
 {\displaystyle ((A\perp \!\!\!\perp B)\mid C)}
 
. In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.

Conditional dependence open-in-new

Conditional dependence