标题： Gibbs型先验是否是Dirichlet过程的最自然推广？ 显示英文标题

标题： Are Gibbs-type priors the most natural generalization of the Dirichlet process?

摘要： 离散随机概率测度及其诱导的可交换随机划分是贝叶斯推断中解决各种估计和预测问题的关键工具。事实上，许多流行的非参数先验，如狄利克雷过程先验和Pitman-Yor过程先验，几乎肯定会选择离散概率分布，并因此自动诱导可交换的随机划分。 在这里，我们专注于Gibbs型先验族，这是狄利克雷过程和Pitman-Yor过程先验的一个近期且优雅的推广。这些随机概率测度具有从理论和应用角度来看都很吸引人的性质：(i) 它们可以通过其预测结构以直观的方式刻画，从而基于学习机制的精确假设来证明其使用的合理性；(ii) 它们在数学上易于处理；(iii) 它们除了狄利克雷过程和Pitman-Yor过程外，还包括几个有趣的特殊情况。本文的目标是系统地统一处理Gibbs型先验，并突出它们对贝叶斯非参数推断的意义。我们将讨论它们的分布性质、由此产生的估计器、频率渐近验证以及时间依赖版本的构造。应用方面，主要涉及分层混合模型和物种采样，将用来传达主要思想。这种先验类别的直觉以及从中可以得出的清晰结果促使人们思考它是否实际上代表了狄利克雷过程最自然的推广。 显示英文摘要

摘要： Discrete random probability measures and the exchangeable random partitions they induce are key tools for addressing a variety of estimation and prediction problems in Bayesian inference. Indeed, many popular nonparametric priors, such as the Dirichlet and the Pitman-Yor process priors, select discrete probability distributions almost surely and, therefore, automatically induce exchangeable random partitions. Here we focus on the family of Gibbs-type priors, a recent and elegant generalization of the Dirichlet and the Pitman-Yor process priors. These random probability measures share properties that are appealing both from a theoretical and an applied point of view: (i) they admit an intuitive characterization in terms of their predictive structure justifying their use in terms of a precise assumption on the learning mechanism; (ii) they stand out in terms of mathematical tractability; (iii) they include several interesting special cases besides the Dirichlet and the Pitman-Yor processes. The goal of our paper is to provide a systematic and unified treatment of Gibbs-type priors and highlight their implications for Bayesian nonparametric inference. We will deal with their distributional properties, the resulting estimators, frequentist asymptotic validation and the construction of time-dependent versions. Applications, mainly concerning hierarchical mixture models and species sampling, will serve to convey the main ideas. The intuition inherent to this class of priors and the neat results that can be deduced for it lead one to wonder whether it actually represents the most natural generalization of the Dirichlet process.