A SIMPLE MEASURE OF CONDITIONAL DEPENDENCE

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. Moreover, it has a natural interpretation as a nonlinear generalization of the familiar partial R-2 statistic for measuring conditional dependence by regression. 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 data sets are worked out.

Automated summary

This study introduces a coefficient for measuring the conditional dependence between random variables without distributional assumptions, and develops a new variable selection algorithm based on this coefficient. The method is model-free, parameter-free, and provably consistent under sparsity assumptions, with applications to both synthetic and real data sets.