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A classic inferential statistical problem is the goodness-of-fit (GOF) test.Such a test can be challenging when the hypothesized parametric model has anintractable likelihood and its distributional form is not available. Bayesianmethods for GOF can be appealing due to their ability to incorporate expertknowledge through prior distributions.  However, standard Bayesian methods for this test often require strongdistributional assumptions on the data and their relevant parameters. Toaddress this issue, we propose a semi-Bayesian nonparametric (semi-BNP)procedure in the context of the maximum mean discrepancy (MMD) measure that canbe applied to the GOF test. Our method introduces a novel Bayesian estimatorfor the MMD, enabling the development of a measure-based hypothesis test forintractable models. Through extensive experiments, we demonstrate that ourproposed test outperforms frequentist MMD-based methods by achieving a lowerfalse rejection and acceptance rate of the null hypothesis. Furthermore, weshowcase the versatility of our approach by embedding the proposed estimatorwithin a generative adversarial network (GAN) framework. It facilitates arobust BNP learning approach as another significant application of our method.With our BNP procedure, this new GAN approach can enhance sample diversity andimprove inferential accuracy compared to traditional techniques.

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