Conditional independence

Definition

Let $(\Omega, \mathcal{F}, P)$ be a probability space and let $A, B, C \in \mathcal{F}$ with $P(C) > 0$. The events $A$ and $B$ are conditionally independent given $C$ if $$P(A \cap B \mid C) = P(A \mid C) \, P(B \mid C),$$ where $P(\cdot \mid C)$ is the conditional probability given $C$.

Remarks

If also $P(B \cap C) > 0$, the condition is equivalent to $P(A \mid B \cap C) = P(A \mid C)$. Conditional independence does not imply independence, and independence does not imply conditional independence. Example: choose one of two coins at random, a fair coin or a coin with two heads, and toss it two times. Given the coin, the two tosses are independent. Without this knowledge, they are dependent. For random variables $X$ and $Y$ given $Z$, the definition uses the conditional distributions given $Z$.

Explanations

L1 human

Two results can be linked only through a hidden common cause. When you know the cause, the link disappears. This is conditional independence given the cause.