🤖 AI Summary
This paper addresses the underutilization of separate exchangeability in Bayesian nonparametric (BNP) modeling, noting that existing partially exchangeable models—such as nested Dirichlet process variants—often neglect multidimensional experimental structures (e.g., matrix-valued data, multiple treatment groups), leading to misalignment between prior specification and experimental design. To resolve this, the paper introduces the first systematic BNP framework grounded in separate exchangeability, proposing two novel models: nested random partitioning and ANOVA-type dependent Dirichlet processes (ANOVA-DDP). Both explicitly encode hierarchical experimental structure, ensuring theoretical consistency between prior construction and statistical inference. Empirical evaluation on real-world datasets demonstrates substantial improvements in regression prediction accuracy and clustering interpretability. The proposed framework establishes the first BNP paradigm that simultaneously satisfies rigorous theoretical foundations and practical applicability for complex experimental designs.
📝 Abstract
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.