  Parameter selection in high-dimensional models is typically finetuned in a
way that keeps the (relative) number of false positives under control. This is
because otherwise the few true positives may be dominated by the many possible
false positives. This happens, for instance, when the selection follows from a
naive optimisation of an information criterion, such as AIC or Mallows's Cp. It
can be argued that the overestimation of the selection comes from the
optimisation process itself changing the statistics of the selected variables,
in a way that the information criterion no longer reflects the true divergence
between the selection and the data generating process. In lasso, the
overestimation can also be linked to the shrinkage estimator, which makes the
selection too tolerant of false positive selections. For these reasons, this
paper works on refined information criteria, carefully balancing false
positives and false negatives, for use with estimators without shrinkage. In
particular, the paper develops corrected Mallows's Cp criteria for structured
selection in trees and graphical models.
