Separate Exchangeability as Modeling Principle in Bayesian Nonparametrics

We argue for the use of separate exchangeability as a modeling principle in Bayesian inference, especially for nonparametric Bayesian models. While in some areas, such as random graphs, separate and (closely related) joint exchangeability are widely used, and it naturally arises for example in simple mixed models, it is curiously underused for other applications. We briefly review the definition of separate exchangeability. We then discuss two specific models that implement separate exchangeability. One example is about nested random partitions for a data matrix, defining a partition of columns and nested partitions of rows, nested within column clusters. Many recently proposed models for nested partitions implement partially exchangeable models. We argue that inference under such models in some cases ignores important features of the experimental setup. The second example is about setting up separately exchangeable priors for a nonparametric regression model when multiple sets of experimental units are involved.