Conditional independence

In probability theory, two events R and B are conditionally independent given a third event Y precisely if the occurrence of R and the occurrence of B are independent events in their conditional probability distribution given Y. In other words, R and B are conditionally independent given Y if and only if, given knowledge that Y occurs, knowledge of whether R occurs provides no information on the likelihood of B occurring, and knowledge of whether B occurs provides no information on the likelihood of R occurring.

In the standard notation of probability theory, R and B are conditionally independent given Y if and only if

or equivalently,

Two random variables X and Y are conditionally independent given a third random variable Z if and only if they are independent in their conditional probability distribution given Z. That is, X and Y are conditionally independent given Z if and only if, given any value of Z, the probability distribution of X is the same for all values of Y and the probability distribution of Y is the same for all values of X.

Two events R and B are conditionally independent given a σ-algebra Σ if

where ${\displaystyle \Pr(A\mid \Sigma )}$ denotes the conditional expectation of the indicator function of the event ${\displaystyle A}$, ${\displaystyle \chi _{A}}$, given the sigma algebra ${\displaystyle \Sigma }$. That is,

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