Score-Based Hypothesis Testing for Unnormalized Models

Unnormalized statistical models play an important role in machine learning, statistics, and signal processing. In this paper, we derive a new hypothesis testing procedure for unnormalized models. Our approach is motivated by the success of score matching techniques that avoid the intensive computational costs of normalization constants in many high-dimensional settings. Our proposed test statistic is the difference between Hyvärinen scores corresponding to the null and alternative hypotheses. Under some reasonable conditions, we prove that the asymptotic distribution of this statistic is Chi-squared. We outline a bootstrap approach to learn the test critical values, particularly when the distribution under the null hypothesis cannot be expressed in a closed form, and provide consistency guarantees. Finally, we conduct extensive numerical experiments and demonstrate that our proposed approach outperforms goodness-of-fit benchmarks in various settings.