  We propose new inference tools for forward stepwise regression, least angle
regression, and the lasso. Assuming a Gaussian model for the observation vector
y, we first describe a general scheme to perform valid inference after any
selection event that can be characterized as y falling into a polyhedral set.
This framework allows us to derive conditional (post-selection) hypothesis
tests at any step of forward stepwise or least angle regression, or any step
along the lasso regularization path, because, as it turns out, selection events
for these procedures can be expressed as polyhedral constraints on y. The
p-values associated with these tests are exactly uniform under the null
distribution, in finite samples, yielding exact type I error control. The tests
can also be inverted to produce confidence intervals for appropriate underlying
regression parameters. The R package "selectiveInference", freely available on
the CRAN repository, implements the new inference tools described in this
paper.
