  For testing conditional independence (CI) of a response Y and a predictor X
given covariates Z, the recently introduced model-X (MX) framework has been the
subject of active methodological research, especially in the context of MX
knockoffs and their successful application to genome-wide association studies.
In this paper, we study the power of MX CI tests, yielding quantitative
insights into the role of machine learning and providing evidence in favor of
using likelihood-based statistics in practice. Focusing on the conditional
randomization test (CRT), we find that its conditional mode of inference allows
us to reformulate it as testing a point null hypothesis involving the
conditional distribution of X. The Neyman-Pearson lemma then implies that a
likelihood-based statistic yields the most powerful CRT against a point
alternative. We also obtain a related optimality result for MX knockoffs.
Switching to an asymptotic framework with arbitrarily growing covariate
dimension, we derive an expression for the limiting power of the CRT against
local semiparametric alternatives in terms of the prediction error of the
machine learning algorithm on which its test statistic is based. Finally, we
exhibit a resampling-free test with uniform asymptotic Type-I error control
under the assumption that only the first two moments of X given Z are known, a
significant relaxation of the MX assumption.
