Conditional Independence

Independence and Conditional Independence

The concepts of independence and conditional independence concern whether information about one variable also contains information about another variable. Two variables are said to be independent if information about one gives no information about the other. For example, one might expect that whether an individual is left-handed or right-handed gives no information about the likelihood of developing pneumonia; it would then be said that being left-handed or right-handed is independent of the development of pneumonia. More formally, if P(Y = y|Z = z) is the probability that Y = y given Z = z and if P(Y = y) is the overall probability that Y = y, then the variables Y and Z are said to be independent if P(Y = y|Z = z) = P(Y = y); in other words, Y and Z are independent if the information that Z = z gives no information about the distribution of Y; equivalently, Y and Z are independent if P(Z = z|Y = y) = P(Z = z). When two variables X and Y are not independent, they are said to be correlated or to be statistically associated. Independence is also often referred to as “marginal independence” or “unconditional independence” to distinguish it from conditional independence.

The concept of conditional independence is a natural extension of the concept of independence. Conditional independence is similar to independence, except that it involves conditioning on a third variable (or set of variables). Thus suppose that one is interested in the relationship between X and Y within the strata of some third variable C. The two variables, X and Y, are said to be conditionally independent given C if information about X gives no information about Y once one knows the value of C. For example, a positive clinical breast exam is predictive of the presence of breast cancer; that is to say, a positive clinical breast exam and the presence of breast cancer are not independent; they are statistically associated. Suppose, however, that in addition to the results of a clinical breast exam, information is also available on further evaluation procedures such as mammogram and biopsy results. In this case, once [Page 160]one has information on these further evaluation procedures, the results from the clinical breast exam give no additional information about the likelihood of breast cancer beyond the mammogram and biopsy results; that is to say the presence of breast cancer is conditionally independent of the clinical breast exam results given the results from the mammogram and biopsy. More formally, if P(Y = y|Z = z, C = c) is the probability that Y = y given that Z = z and C = c and if P(Y = y|C = c) is the probability that Y = y given that C = c, then the variables Y and Z are said to be conditionally independent given C if P(Y = y|Z = z, C = c) = P(Y = y|C = c). When two variables Y and Z are not conditionally independent given C, then they are said to be associated conditionally on C or to be conditionally associated given C. The notation YIIZ|C is sometimes used to denote that Y and Z are conditionally independent given C; the notation YIIZ is used to denote that Y and Z are unconditionally independent. A. P. Dawid's article “Conditional Independence in Statistical Theory” gives an overview of some of the technical statistical properties concerning conditional independence. The focus here will be the relevance of the idea of conditional independence in medical decision making.

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