We argue for the use of separate exchangeability as a modeling principle in Bayesian nonparametric (BNP) inference. Separate exchangeability is \emph{de facto} widely applied in the Bayesian parametric case, e.g., it naturally arises in simple mixed models. However, while in some areas, such as random graphs, separate and (closely related) joint exchangeable models are widely used, they are curiously underused for several other applications in BNP. We briefly review the definition of separate exchangeability, focusing on the implications of such a definition in Bayesian modeling. We then discuss two tractable classes of models that implement separate exchangeability, which are the natural counterparts of familiar partially exchangeable BNP models. The first is nested random partitions for a data matrix, defining a partition of columns and nested partitions of rows, nested within column clusters. Many recent models for nested partitions implement partially exchangeable models related to variations of the well-known nested Dirichlet process. We argue that inference under such models in some cases ignores important features of the experimental setup. We obtain the separately exchangeable counterpart of such partially exchangeable partition structures. The second class is about setting up separately exchangeable priors for a nonparametric regression model when multiple sets of experimental units are involved. We highlight how a Dirichlet process mixture of linear models, known as ANOVA DDP, can naturally implement separate exchangeability in such regression problems. Finally, we illustrate how to perform inference under such models in two real data examples.

翻译：我们主张将可分离可交换性作为贝叶斯非参数（BNP）推理中的建模原则。可分离可交换性在贝叶斯参数情形下实际上已被广泛应用，例如在简单混合模型中自然出现。然而，尽管在随机图等某些领域中，可分离及（密切相关的）联合可交换模型被广泛使用，但在BNP的其他若干应用中却出人意料地未被充分利用。我们简要回顾了可分离可交换性的定义，重点关注此类定义在贝叶斯建模中的含义。随后，我们讨论了实现可分离可交换性的两类可处理模型，它们分别是常见部分可交换BNP模型的自然对应物。第一类是针对数据矩阵的嵌套随机划分，它定义了列的划分以及嵌套于列簇内的行的嵌套划分。许多近期关于嵌套划分的模型实现了与著名嵌套狄利克雷过程变体相关的部分可交换模型。我们认为，此类模型下的推理在某些情况下忽略了实验设置的重要特征。我们获得了此类部分可交换划分结构的可分离可交换对应物。第二类涉及在涉及多组实验单元时，为非参数回归模型建立可分离可交换先验。我们强调线性模型的狄利克雷过程混合（称为ANOVA DDP）如何能在此类回归问题中自然实现可分离可交换性。最后，我们通过两个真实数据示例说明如何在此类模型下进行推理。