BET on Independence

We study the problem of nonparametric dependence detection when no assumption is made about the joint distribution. We approach this problem by introducing the new concept of binary expansion statistics (BEStat), which studies dependence through a filtration induced by the binary expansion of a uniformly distributed variable. In particular, for the nonparametric dependence detection problem, we propose the binary expansion testing (BET) framework to test independence up to certain resolution. Through an orthogonal decomposition of the $\chi^2$ test into interactions of the Bernoulli variables in the marginal binary expansions, we convert the dependence detection problem to a multiple testing problem. In addition to being distribution-free, the BET also improves upon a wide class of commonly used testing methods with (1) substantial power gains against a large class of alternatives compared to existing methods which rely on the clustering intuition, (2) freedom from the problem of non-uniform consistency, (3) clear interpretability of local relationships upon rejection of independence, and (4) close connections to computing to allow efficient implementations. We illustrate the BET by studying the distribution of the brightest stars in the night sky.