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Random partition models are widely used in Bayesian methods for various clustering tasks, such as mixture models, topic models, and community detection problems. While the number of clusters induced by random partition models has been studied extensively, another important model property regarding the balancedness of partition has been largely neglected. We formulate a framework to define and theoretically study the balancedness of exchangeable random partition models, by analyzing how a model assigns probabilities to partitions with different levels of balancedness. We demonstrate that the "rich-get-richer" characteristic of many existing popular random partition models is an inevitable consequence of two common assumptions: product-form exchangeability and projectivity. We propose a principled way to compare the balancedness of random partition models, which gives a better understanding of what model works better and what doesn’t for different applications. We also introduce the "rich-get-poorer" random partition models and illustrate their application to entity resolution tasks. 

Graphs have been commonly used to represent complex data structures. In models dealing with graph-structured data, multivariate parameters may not only exhibit sparse patterns but have structured sparsity and smoothness in the sense that both zero and non-zero parameters tend to cluster together. We propose a new prior for high-dimensional parameters with graphical relations, referred to as the Tree-based Low-rank Horseshoe (T-LoHo) model, that generalizes the popular univariate Bayesian horseshoe shrinkage prior to the multivariate setting to detect structured sparsity and smoothness simultaneously. The T-LoHo prior can be embedded in many high-dimensional hierarchical models. To illustrate its utility, we apply it to regularize a Bayesian high-dimensional regression problem where the regression coefficients are linked by a graph, so that the resulting clusters have flexible shapes and satisfy the cluster contiguity constraint with respect to the graph. We design an efficient Markov chain Monte Carlo algorithm that delivers full Bayesian inference with uncertainty measures for model parameters such as the number of clusters. We offer theoretical investigations of the clustering effects and posterior concentration results. Finally, we illustrate the performance of the model with simulation studies and a real data application for anomaly detection on a road network. The results indicate substantial improvements over other competing methods such as the sparse fused lasso.