Conditional Independence¶

Definition¶

Let $A$, $B$, and $C$ be events with $\mathbb{P}(C)>0$$0$. The events $A$ and $B$ are conditionally independent given $C$ if

Interpretation¶

Conditional independence means two events stop influencing each other once certain information is known. It is a statement about independence inside a restricted situation.

The diagram chooses a fair coin or a coin with head probability $0.9$, each with probability $1/2$, and tosses the chosen coin twice. Given its identity $C$, the two heads events are independent. Without this conditioning, $\mathbb{P}(H_1\cap H_2)=\frac{1}{2}(0.25)+\frac{1}{2}(0.81)=0.53$$0.53$, whereas $\mathbb{P}(H_1)\mathbb{P}(H_2)=0.7^2=0.49$$0.49$.

Conditional probability first restricts the model to $C$. The factorization for independent events is then applied inside that restricted model; ordinary independence before conditioning does not automatically imply it.

Key Results¶

• Conditional independence is not the same as ordinary independence.
• Two variables can be dependent marginally but conditionally independent after a common cause is known.
• Conditional independence is central in graphical and hierarchical probability models.

Examples¶

• Two symptoms may be dependent overall but conditionally independent given the disease status.
• Two students' answers may become conditionally independent once their preparation levels are known.
• Machine failures in different locations may be conditionally independent given the weather condition.

Prerequisites¶

• Conditional Probability
• Independent Events

Related Concepts¶

• Bayes Rule

Sources¶

• Probability Theory for EOR · Week 2

Keywords: conditionally independent events Symbols: intersection

Last updated: 16 October 2026 22:28