Minimax optimal conditional independence testing

By Matey Neykov, Sivaraman Balakrishnan, Larry A. Wasserman. Annals of Statistics, 2020

54 citations recorded by the source corpus.

We consider the problem of conditional independence testing of $X$ and $Y$ given $Z$ where $X,Y$ and $Z$ are three real random variables and $Z$ is continuous. We focus on two main cases -- when $X$ and $Y$ are both discrete, and when $X$ and $Y$ are both continuous. In view of recent results on conditional independence testing (Shah and Peters 2018), one cannot hope to design non-trivial tests, which control the type I error for all absolutely continuous conditionally independent distributions, while still ensuring power against interesting alternatives. Consequently, we identify various, natural smoothness assumptions on the conditional distributions of $X,Y|Z=z$ as $z$ varies in the support of $Z$, and study the hardness of conditional independence testing under these smoothness assumptions. We derive matching lower and upper bounds on the critical radius of separation between the null and alternative hypotheses in the total variation metric. The tests we consider are easily implementable and rely on binning the support of the continuous variable $Z$. To complement these results, we provide a new proof of the hardness result of Shah and Peters and show that in the absence of smoothness assumptions conditional independence testing remains difficult even when $X,Y$ are discrete variables of finite (and not scaling with the sample-size) support.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/5d080b3b2fb7983132add2460337d69781611074

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Citations in the corpus

On Azadkia–Chatterjee’s conditional dependence coefficient (2021, methodology): “…on the other hand, is based on a brute-force calculation of the mean and variance of the Azadkia–Chatterjee conditional dependence coefficient along a special non-standard local alternative sequence that serves as the “worst case” in the minimax lower bound construction of Neykov et al. (2021).”
Minimax optimality of permutation tests (2020, methodology): “When the conditional variable is continuous, however, this strategy does not work and this has led several authors to use “local permutation” heuristics (see for instance Doran et al., 2014; Fukumizu et al., 2008; Neykov et al., 2019).”