Minimax optimal conditional independence testing

We consider the problem of conditional independence testing X and Y given Z where X,Y are three real random variables is continuous. focus on two main cases—when both discrete, when In view recent results [Ann. Statist. 48 (2020) 1514–1538], one cannot hope to design nontrivial tests, which control type I error for all absolutely continuous conditionally independent distributions, while still ensuring power against interesting alternatives. Consequently, we identify various, natural smoothness assumptions distributions X,Y|Z=z as z varies in support Z, study hardness under these assumptions. derive matching lower upper bounds critical radius separation between null alternative hypotheses total variation metric. The tests easily implementable rely binning variable Z. To complement results, provide a new proof result Shah Peters 1514–1538].