  For propensity score analysis and sparse estimation, we develop an
information criterion for determining the regularization parameters needed in
variable selection. First, for Gaussian distribution-based causal inference
models, we extend Stein's unbiased risk estimation theory, which leads to a
generalized Cp criterion that has almost no weakness in conventional sparse
estimation, and derive an inverse-probability-weighted sparse estimation
version of the criterion without resorting to asymptotics. Next, for general
causal inference models that are not necessarily Gaussian distribution-based,
we extend the asymptotic theory on LASSO for propensity score analysis, with
the intention of implementing doubly robust sparse estimation. From the
asymptotic theory, an AIC-type information criterion for
inverse-probability-weighted sparse estimation is given, and then a criterion
with double robustness in itself is derived for doubly robust sparse
estimation. Numerical experiments compare the proposed criterion with the
existing criterion derived from a formal argument and verify that the proposed
criterion is superior in almost all cases, that the difference is not
negligible in many cases, and that the results of variable selection differ
significantly. Real data analysis confirms that the difference between variable
selection and estimation by these criteria is actually large. Finally,
generalizations to general sparse estimation using group LASSO, elastic net,
and non-convex regularization are made in order to indicate that the proposed
criterion is highly extensible.
