  The semiparametric estimation approach, which includes
inverse-probability-weighted and doubly robust estimation using propensity
scores, is a standard tool in causal inference, and it is rapidly being
extended in various directions. On the other hand, although model selection is
indispensable in statistical analysis, an information criterion for selecting
an appropriate regression structure has just started to be developed. In this
paper, based on the original definition of Akaike information criterion (AIC;
\citealt{Aka73}), we derive an AIC-type criterion for propensity score
analysis. Here, we define a risk function based on the Kullback-Leibler
divergence as the cornerstone of the information criterion and treat a general
causal inference model that is not necessarily a linear one. The causal effects
to be estimated are those in the general population, such as the average
treatment effect on the treated or the average treatment effect on the
untreated. In light of the fact that this field attaches importance to doubly
robust estimation, which allows either the model of the assignment variable or
the model of the outcome variable to be wrong, we make the information
criterion itself doubly robust so that either one can be wrong and it will
still be an asymptotically unbiased estimator of the risk function. In
simulation studies, we compare the derived criterion with an existing criterion
obtained from a formal argument and confirm that the former outperforms the
latter. Specifically, we check that the divergence between the estimated
structure from the derived criterion and the true structure is clearly small in
all simulation settings and that the probability of selecting the true or
nearly true model is clearly higher. Real data analyses confirm that the
results of variable selection using the two criteria differ significantly.
