Random Partitions for Multi-view Data: How to Encode Repeated Measures Design into Nonparametric Bayesian Models.

Random partitions are fundamental probabilistic objects in Bayesian statistics, particularly in nonparametric models, as they enable flexible clustering and relax strong distributional assumptions about the data. Exchangeable partitions arise naturally in Dirichlet process mixtures, allowing the number of clusters to grow with the data, which are also assumed to be exchangeable. More general forms, such as partially exchangeable partitions and product partition models with covariates, extend their applicability to settings where, alongside the data, a set of covariates is also available. These models further highlight the connection between the symmetry assumptions imposed on the data-corresponding to the adopted experimental design-and those reflected in the law governing the partition.

A less explored experimental design for random partition models is that of repeated measurements, which has gained particular attention only in recent years. This setting arises when the inferential problem requires estimating a collection of partitions of the same items, commonly referred to as multi-view clustering models. These models should incorporate temporal dynamics or separate exchangeability assumptions within random partition frameworks.

This talk will provide an overview of recent advancements in this area, with a particular focus on conditional partial exchangeability (CPE), a unifying dependence condition for constructing dependent partitions of the same objects. CPE differs from traditional partial exchangeability due to its conditional nature and its requirement for marginal invariance. Together, these conditions ensure local dependence at the level of the items across partitions, aligning the symmetry assumptions in the partition law with the experimental design of observing multiple instances corresponding to the same items (i.e., repeated measures design). The theoretical and modeling advancements of CPE will be illustrated through two real data applications: clustering metabolomics data and analyzing diffusion tensor imaging data.