  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 exchangeability are widely used,
it is 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 that 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.
