  The knockoff filter introduced by Barber and Cand\`es 2016 is an elegant
framework for controlling the false discovery rate in variable selection. While
empirical results indicate that this methodology is not too conservative, there
is no conclusive theoretical result on its power. When the predictors are
i.i.d. Gaussian, it is known that as the signal to noise ratio tend to
infinity, the knockoff filter is consistent in the sense that one can make FDR
go to 0 and power go to 1 simultaneously. In this work we study the case where
the predictors have a general covariance matrix $\Sigma$. We introduce a simple
functional called effective signal deficiency (ESD) of the covariance matrix
$\Sigma$ that predicts consistency of various variable selection methods. In
particular, ESD reveals that the structure of the precision matrix
$\Sigma^{-1}$ plays a central role in consistency and therefore, so does the
conditional independence structure of the predictors. To leverage this
connection, we introduce Conditional Independence knockoff, a simple procedure
that is able to compete with the more sophisticated knockoff filters and that
is defined when the predictors obey a Gaussian tree graphical models (or when
the graph is sufficiently sparse). Our theoretical results are supported by
numerical evidence on synthetic data.
