Normalized random measures with independent increments represent a large class of Bayesian nonaprametric priors and are widely used in the Bayesian nonparametric framework. In this paper, we provide the posterior consistency analysis for normalized random measures with independent increments (NRMIs) through the corresponding Levy intensities used to characterize the completely random measures in the construction of NRMIs. Assumptions are introduced on the Levy intensities to analyze the posterior consistency of NRMIs and are verified with multiple interesting examples. A focus of the paper is the Bernstein-von Mises theorem for the normalized generalized gamma process (NGGP) when the true distribution of the sample is discrete or continuous. When the Bernstein-von Mises theorem is applied to construct credible sets, in addition to the usual form there will be an additional bias term on the left endpoint closely related to the number of atoms of the true distribution when it is discrete. We also discuss the affect of the estimators for the model parameters of the NGGP under the Bernstein-von Mises convergences. Finally, to further explain the necessity of adding the bias correction in constructing credible sets, we illustrate numerically how the bias correction affects the coverage of the true value by the credible sets when the true distribution is discrete.

翻译：具有独立增量的归一化随机测度代表了一类广泛的贝叶斯非参数先验，并在贝叶斯非参数框架中得到广泛应用。本文通过用于构造归一化随机测度（NRMIs）的完全随机测度的莱维强度，对NRMIs进行了后验一致性分析。我们引入莱维强度上的假设以分析NRMIs的后验一致性，并通过多个有趣的示例加以验证。本文的一个重点是当样本的真实分布为离散或连续时，归一化广义伽马过程（NGGP）的伯恩斯坦-冯·米塞斯定理。当运用伯恩斯坦-冯·米塞斯定理构造可信集时，除了通常的形式外，左端点处将出现一个额外的偏差项，该偏差项与真实分布（当其离散时）的原子数量密切相关。我们还讨论了在伯恩斯坦-冯·米塞斯收敛下，NGGP模型参数估计量所受的影响。最后，为进一步解释在构造可信集时加入偏差校正的必要性，我们通过数值模拟说明了当真实分布离散时，偏差校正如何影响可信集对真实值的覆盖。