conditionally independent events

条件独立

Probability · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Other names / variants: conditional independence of events

Translation evidence: Literature usage

Meaning in context

For an event CC with P(C)>0P(C)>0, events A,BA,B are independent given CC when P(A∩B∣C)=P(A∣C)P(B∣C)P(A\cap B\mid C)=P(A\mid C)P(B\mid C).

当条件事件C的概率为正时，在C所诱导的概率律下交事件概率等于各条件概率之积。

Example

For independent fair bits, conditioning on the bits being equal makes the two 'bit is 1' events dependent: 1/2≠(1/2)21/2\ne(1/2)^2.

Usage and distinctions

Unconditional independence and independence after conditioning do not imply one another; this entry uses event conditioning.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. 李东风 — 应用随机过程（北京大学）§1.5.3 定义1.18，条件独立的数学公式 · Checked 2026-10-14

Translation sources

  1. 李东风 — 应用随机过程（北京大学）§1.5.3 关于一个随机变量的条件期望，定义1.18 · Checked 2026-10-14
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