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- Conditional_independence abstract "In probability theory, two events R and B are conditionally independent given a third event Y precisely if the occurrence or non-occurrence of R and the occurrence or non-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 ifor 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 ifwhere denotes the conditional expectation of the indicator function of the event , , given the sigma algebra . That is,Two random variables X and Y are conditionally independent given a σ-algebra Σ if the above equation holds for all R in σ(X) and B in σ(Y).Two random variables X and Y are conditionally independent given a random variable W if they are independent given σ(W): the σ-algebra generated by W. This is commonly written:orThis is read "X is independent of Y, given W"; the conditioning applies to the whole statement: "(X is independent of Y) given W".If W assumes a countable set of values, this is equivalent to the conditional independence of X and Y for the events of the form [W = w].Conditional independence of more than two events, or of more than two random variables, is defined analogously.The following two examples show that X ⊥ Yneither implies nor is implied by X ⊥ Y | W.First, suppose W is 0 with probability 0.5 and is the value 1 otherwise. WhenW = 0 take X and Y to be independent, each having the value 0 with probability 0.99 and the value 1 otherwise. When W = 1, X and Y are again independent, but this time they take the value 1with probability 0.99. Then X ⊥ Y | W. But X and Y are dependent, because Pr(X = 0) < Pr(X = 0|Y = 0). This is because Pr(X = 0) = 0.5, but if Y = 0 then it's very likely that W = 0 and thus that X = 0 as well, so Pr(X = 0|Y = 0) > 0.5. For the second example, suppose X ⊥ Y, each taking the values 0 and 1 with probability 0.5. Let W be the product X×Y. Then when W = 0, Pr(X = 0) = 2/3, but Pr(X = 0|Y = 0) = 1/2, so X ⊥ Y | W is false.This is also an example of Explaining Away. See Kevin Murphy's tutorialwhere X and Y take the values "brainy" and "sporty".".
- Conditional_independence thumbnail Conditional_independence.svg?width=300.
- Conditional_independence wikiPageID "801135".
- Conditional_independence wikiPageRevisionID "605819363".
- Conditional_independence hasPhotoCollection Conditional_independence.
- Conditional_independence subject Category:Probability_theory.
- Conditional_independence subject Category:Statistical_dependence.
- Conditional_independence comment "In probability theory, two events R and B are conditionally independent given a third event Y precisely if the occurrence or non-occurrence of R and the occurrence or non-occurrence of B are independent events in their conditional probability distribution given Y.".
- Conditional_independence label "Conditional independence".
- Conditional_independence sameAs m.03d1zz.
- Conditional_independence sameAs Q5159264.
- Conditional_independence sameAs Q5159264.
- Conditional_independence wasDerivedFrom Conditional_independence?oldid=605819363.
- Conditional_independence depiction Conditional_independence.svg.
- Conditional_independence isPrimaryTopicOf Conditional_independence.
