A Semi-Bayesian Nonparametric Estimator of the Maximum Mean Discrepancy Measure: Applications in Goodness-of-Fit Testing and Generative Adversarial Networks

Abstract

A classic inferential problem in statistics is the goodness-of-fit (GOF) test. Performing such tests can be challenging when the hypothesized parametric model has an intractable likelihood and its distributional form is not available. Bayesian methods for GOF testing can be appealing due to their ability to incorporate expert knowledge through prior distributions. However, standard Bayesian methods for this test often require strong distributional assumptions on the data and their relevant parameters. To address this issue, we propose a semi-Bayesian nonparametric (semi-BNP) procedure based on the maximum mean discrepancy (MMD) measure that can be applied to the GOF test. We introduce a novel Bayesian estimator for the MMD, which enables the development of a measure-based hypothesis test for intractable models. Through extensive experiments, we demonstrate that our proposed test outperforms frequentist MMD-based methods by achieving a lower false rejection and acceptance rate of the null hypothesis. Furthermore, we showcase the versatility of our approach by embedding the proposed estimator within a generative adversarial network (GAN) framework. It facilitates a robust BNP learning approach as another significant application of our method. With our BNP procedure, this new GAN approach can enhance sample diversity and improve inferential accuracy compared to traditional techniques.