  Many statistical models are given in the form of non-normalized densities
with an intractable normalization constant. Since maximum likelihood estimation
is computationally intensive for these models, several estimation methods have
been developed which do not require explicit computation of the normalization
constant, such as noise contrastive estimation (NCE) and score matching.
However, model selection methods for general non-normalized models have not
been proposed so far. In this study, we develop information criteria for
non-normalized models estimated by NCE or score matching. They are
approximately unbiased estimators of discrepancy measures for non-normalized
models. Simulation results and applications to real data demonstrate that the
proposed criteria enable selection of the appropriate non-normalized model in a
data-driven manner.
