Information criteria for non-normalized models

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 derived as approximately unbiased estimators of discrepancy measures for non-normalized models. Simulation results and application to natural image and RNAseq data demonstrate that the proposed criteria enable selection of the appropriate non-normalized model in a data-driven manner. Extension to a finite mixture of non-normalized models is also discussed.