  The recent paper Cand\`es et al. (2018) introduced model-X knockoffs, a
method for variable selection that provably and non-asymptotically controls the
false discovery rate with no restrictions or assumptions on the dimensionality
of the data or the conditional distribution of the response given the
covariates. The one requirement for the procedure is that the covariate samples
are drawn independently and identically from a precisely-known (but arbitrary)
distribution. The present paper shows that the exact same guarantees can be
made without knowing the covariate distribution fully, but instead knowing it
only up to a parametric model with as many as $\Omega(n^{*}p)$ parameters,
where $p$ is the dimension and $n^{*}$ is the number of covariate samples
(which may exceed the usual sample size $n$ of labeled samples when unlabeled
samples are also available). The key is to treat the covariates as if they are
drawn conditionally on their observed value for a sufficient statistic of the
model. Although this idea is simple, even in Gaussian models conditioning on a
sufficient statistic leads to a distribution supported on a set of zero
Lebesgue measure, requiring techniques from topological measure theory to
establish valid algorithms. We demonstrate how to do this for three models of
interest, with simulations showing the new approach remains powerful under the
weaker assumptions.
