  Selective inference (post-selection inference) is a methodology that has
attracted much attention in recent years in the fields of statistics and
machine learning. Naive inference based on data that are also used for model
selection tends to show an overestimation, and so the selective inference
conditions the event that the model was selected. In this paper, we develop
selective inference in propensity score analysis with a semiparametric
approach, which has become a standard tool in causal inference. Specifically,
for the most basic causal inference model in which the causal effect can be
written as a linear sum of confounding variables, we conduct Lasso-type
variable selection by adding an $\ell_1$ penalty term to the loss function that
gives a semiparametric estimator. Confidence intervals are then given for the
coefficients of the selected confounding variables, conditional on the event of
variable selection, with asymptotic guarantees. An important property of this
method is that it does not require modeling of nonparametric regression
functions for the outcome variables, as is usually the case with semiparametric
propensity score analysis.
