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This paper is concerned with test of the conditional independence. We firstestablish an equivalence between the conditional independence and the mutualindependence. Based on the equivalence, we propose an index to measure theconditional dependence by quantifying the mutual dependence among thetransformed variables. The proposed index has several appealing properties. (a)It is distribution free since the limiting null distribution of the proposedindex does not depend on the population distributions of the data. Hence thecritical values can be tabulated by simulations. (b) The proposed index rangesfrom zero to one, and equals zero if and only if the conditional independenceholds. Thus, it has nontrivial power under the alternative hypothesis. (c) Itis robust to outliers and heavy-tailed data since it is invariant toconditional strictly monotone transformations. (d) It has low computationalcost since it incorporates a simple closed-form expression and can beimplemented in quadratic time. (e) It is insensitive to tuning parametersinvolved in the calculation of the proposed index. (f) The new index isapplicable for multivariate random vectors as well as for discrete data. Allthese properties enable us to use the new index as statistical inference toolsfor various data. The effectiveness of the method is illustrated throughextensive simulations and a real application on causal discovery.

A Distribution Free Conditional Independence Test with Applications to Causal Discovery