Gibbs type priors have been shown to be natural generalizations of Dirichlet process (DP) priors used for intricate applications of Bayesian nonparametric methods. This includes applications to mixture models and to species sampling models arising in populations genetics. Notably these latter applications, and also applications where power law behavior such as that arising in natural language models are exhibited, provide instances where the DP model is wholly inadequate. Gibbs type priors include the DP, the also popular Pitman-Yor process and closely related normalized generalized gamma process as special cases. However, there is in fact a richer infinite class of such priors, where, despite knowledge about the exchangeable marginal structures produced by sampling $n$ observations, descriptions of the corresponding posterior distribution, a crucial component in Bayesian analysis, remain unknown. This paper presents descriptions of the posterior distributions for the general class, utilizing a novel proof that leverages the exclusive Gibbs properties of these models. The results are applied to several specific cases for further illustration.

翻译：Gibbs型先验已被证明是狄利克雷过程（DP）先验的自然推广，广泛应用于贝叶斯非参数方法的复杂应用场景。这包括混合模型以及群体遗传学中的物种抽样模型。值得注意的是，后一类应用以及呈现幂律行为（如自然语言模型中的表现）的应用，均表明DP模型完全无法胜任。Gibbs型先验以DP、同样流行的Pitman-Yor过程及其密切相关的归一化广义伽马过程为特例。然而，实际上存在一个更丰富的无穷先验类；尽管已知通过抽样$n$个观测值产生的可交换边际结构，但贝叶斯分析中关键组成部分——对应后验分布的描述仍属未知。本文利用一种新颖的证明方法，借助这些模型独有的Gibbs性质，描述了该类先验的后验分布，并将其应用于多个具体案例以作进一步阐释。