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BET on Independence
We study the problem of model-free dependence detection. This problem can be difficult even when the marginal distributions are known. We explain this difficulty by showing the impossibility to uniformly consistently distinguish degeneracy from independence with any single test. To make model-free dependence detection a tractable problem, we introduce the concept of binary expansion statistics (BEStat) and propose the binary expansion testing (BET) framework. Through simple mathematics, we convert the dependence detection problem to a multiple testing problem. Besides being model-free, the BET also enjoys many other advantages which include (1) invariance to monotone marginal transformations, (2) clear interpretability of local relationships upon rejection, and (3) close connections to computing for efficient algorithms. We illustrate the BET by studying the distribution of the brightest stars in the night sky.