---
tag: 0ZJ87ZZ8PK
kind: definition
name: Conditional independence
msc: [60A05]
defines: 0THJ7416DX
uses: [0FJB06AD2P, 0P0MZ4X682]
sources:
  - url: https://en.wikipedia.org/wiki/Conditional_independence
    locator: Conditional independence of events
---

## 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|0FJB06AD2P]] 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.