A classic inferential statistical problem is the goodness-of-fit (GOF) test. Such a test can be challenging when the hypothesized parametric model has an intractable likelihood and its distributional form is not available. Bayesian methods for GOF 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 in the context of the maximum mean discrepancy (MMD) measure that can be applied to the GOF test. Our method introduces a novel Bayesian estimator for the MMD, enabling 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.

翻译：经典的统计推断问题是拟合优度检验。当假设的参数模型具有难以处理的似然函数且其分布形式不可用时，此类检验可能具有挑战性。基于贝叶斯方法的拟合优度检验因其能够通过先验分布整合专家知识而具有吸引力，然而，此类检验的标准贝叶斯方法通常需要对数据及其相关参数做出较强的分布假设。为解决这一问题，我们提出了一种基于最大均值差异度量的半贝叶斯非参数方法，该方法可应用于拟合优度检验。我们针对MMD引入了一种新型贝叶斯估计量，从而能够对难以处理的模型开展基于度量的假设检验。通过大量实验证明，我们提出的方法在降低原假设的错误拒绝率和错误接受率方面优于频率学派基于MMD的方法。此外，通过将所提出的估计量嵌入生成对抗网络框架中，我们展示了该方法的广泛适用性。作为我们方法的另一重要应用，该框架实现了一种鲁棒的贝叶斯非参数学习方法。与传统技术相比，采用我们的贝叶斯非参数方法，这种新型生成对抗网络方法能够增强样本多样性并提高推断精度。