Conditional dependence

In probability theory, conditional dependence is a relationship between two or more events that are dependent when a third event occurs.^[1][2] For example, if ${\displaystyle A}$ and ${\displaystyle B}$ are two events that individually increase the probability of a third event ${\displaystyle C,}$ and do not directly affect each other, then initially (when it has not been observed whether or not the event ${\displaystyle C}$ occurs)^[3][4] $${\displaystyle \operatorname {P} (A\mid B)=\operatorname {P} (A)\quad {\text{ and }}\quad \operatorname {P} (B\mid A)=\operatorname {P} (B)}$$ (${\displaystyle A{\text{ and }}B}$ are independent).

But suppose that now ${\displaystyle C}$ is observed to occur. If event ${\displaystyle B}$ occurs then the probability of occurrence of the event ${\displaystyle A}$ will decrease because its positive relation to ${\displaystyle C}$ is less necessary as an explanation for the occurrence of ${\displaystyle C}$ (similarly, event ${\displaystyle A}$ occurring will decrease the probability of occurrence of ${\displaystyle B}$). Hence, now the two events ${\displaystyle A}$ and ${\displaystyle B}$ are conditionally negatively dependent on each other because the probability of occurrence of each is negatively dependent on whether the other occurs. We have^[5] $${\displaystyle \operatorname {P} (A\mid C{\text{ and }}B)<\operatorname {P} (A\mid C).}$$

Conditional dependence of A and B given C is the logical negation of conditional independence ${\displaystyle ((A\perp \!\!\!\perp B)\mid C)}$.^[6] In conditional independence two events (which may be dependent or not) become independent given the occurrence of a third event.^[7]