  Applying standard statistical methods after model selection may yield
inefficient estimators and hypothesis tests that fail to achieve nominal type-I
error rates. The main issue is the fact that the post-selection distribution of
the data differs from the original distribution. In particular, the observed
data is constrained to lie in a subset of the original sample space that is
determined by the selected model. This often makes the post-selection
likelihood of the observed data intractable and maximum likelihood inference
difficult. In this work, we get around the intractable likelihood by generating
noisy unbiased estimates of the post-selection score function and using them in
a stochastic ascent algorithm that yields correct post-selection maximum
likelihood estimates. We apply the proposed technique to the problem of
estimating linear models selected by the lasso. In an asymptotic analysis the
resulting estimates are shown to be consistent for the selected parameters and
to have a limiting truncated normal distribution. Confidence intervals
constructed based on the asymptotic distribution obtain close to nominal
coverage rates in all simulation settings considered, and the point estimates
are shown to be superior to the lasso estimates when the true model is sparse.
