A parametric distribution for exact post-selection inference with data
  carving
Erik Drysdale
Post-selection inference (PoSI) is a statistical technique for obtaining
valid confidence intervals and p-values when hypothesis generation and testing
use the same source of data. PoSI can be used on a range of popular algorithms
including the Lasso. Data carving is a variant of PoSI in which a portion of
held out data is combined with the hypothesis generating data at inference
time. While data carving has attractive theoretical and empirical properties,
existing approaches rely on computationally expensive MCMC methods to carry out
inference. This paper's key contribution is to show that pivotal quantities can
be constructed for the data carving procedure based on a known parametric
distribution. Specifically, when the selection event is characterized by a set
of polyhedral constraints on a Gaussian response, data carving will follow the
sum of a normal and a truncated normal (SNTN), which is a variant of the
truncated bivariate normal distribution. The main impact of this insight is
that obtaining exact inference for data carving can be made computationally
trivial, since the CDF of the SNTN distribution can be found using the CDF of a
standard bivariate normal. A python package sntn has been released to further
facilitate the adoption of data carving with PoSI.