A simple measure of conditional dependence
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We propose a coefficient of conditional dependence between two random variables Y and Z given a set of other variables X_1,...,X_p, based on an i.i.d. sample. The coefficient has a long list of desirable properties, the most important of which is that under absolutely no distributional assumptions, it converges to a limit in [0,1], where the limit is 0 if and only if Y and Z are conditionally independent given X_1,...,X_p, and is 1 if and only if Y is equal to a measurable function of Z given X_1,...,X_p. Using this statistic, we devise a new variable selection algorithm, called Feature Ordering by Conditional Independence (FOCI), which is model-free, has no tuning parameters, and is provably consistent under sparsity assumptions. A number of applications to synthetic and real datasets are worked out.
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