Traditional Bayesian random partition models assume that the size of each cluster grows linearly with the number of data points. While this is appealing for some applications, this assumption is not appropriate for other tasks such as entity resolution, modeling of sparse networks, and DNA sequencing tasks. Such applications require models that yield clusters whose sizes grow sublinearly with the total number of data points -- the microclustering property. Motivated by these issues, we propose a general class of random partition models that satisfy the microclustering property with well-characterized theoretical properties. Our proposed models overcome major limitations in the existing literature on microclustering models, namely a lack of interpretability, identifiability, and full characterization of model asymptotic properties. Crucially, we drop the classical assumption of having an exchangeable sequence of data points, and instead assume an exchangeable sequence of clusters. In addition, our framework provides flexibility in terms of the prior distribution of cluster sizes, computational tractability, and applicability to a large number of microclustering tasks. We establish theoretical properties of the resulting class of priors, where we characterize the asymptotic behavior of the number of clusters and of the proportion of clusters of a given size. Our framework allows a simple and efficient Markov chain Monte Carlo algorithm to perform statistical inference. We illustrate our proposed methodology on the microclustering task of entity resolution, where we provide a simulation study and real experiments on survey panel data.
Many popular random partition models, such as the Chinese restaurant process and its two-parameter extension, fall in the class of exchangeable random partitions, and have found wide applicability in model-based clustering, population genetics, ecology or network analysis. While the exchangeability assumption is sensible in many cases, it has some strong implications. In particular, Kingman's representation theorem implies that the size of the clusters necessarily grows linearly with the sample size; this feature may be undesirable for some applications, as recently pointed out by Miller et al…
Recent advances in Bayesian models for random partitions have led to the formulation and exploration of Exchangeable Sequences of Clusters (ESC) models. Under ESC models, it is the cluster sizes that are exchangeable, rather than the observations themselves. This property is particularly useful for obtaining microclustering behavior, whereby cluster sizes grow sublinearly in the number of observations, as is common in applications such as record linkage, sparse networks and genomics. Unfortunately, the exchangeable clusters property comes at the cost of projectivity. As a consequence, in contr…
In database management, record linkage aims to identify multiple records that correspond to the same individual. This task can be treated as a clustering problem, in which a latent entity is associated with one or more noisy database records. However, in contrast to traditional clustering applications, a large number of clusters with a few observations per cluster is expected in this context. In this paper, we introduce a new class of prior distributions based on allelic partitions that is specially suited for the small cluster setting of record linkage. Our approach makes it straightforward t…
Gibbs-type random probability measures and the exchangeable random partitions they induce represent the subject of a rich and active literature. They provide a probabilistic framework for a wide range of theoretical and applied problems that are typically referred to as species sampling problems. In this paper, we consider the class of looking-backward species sampling problems introduced in Lijoi et al. (Ann. Appl. Probab. 18 (2008) 1519-1547) in Bayesian nonparametrics. Specifically, given some information on the random partition induced by an initial sample from a Gibbs-type random probabil…
Gibbs-type exchangeable random partitions, which is a class of multiplicative measures on the set of positive integer partitions, appear in various contexts, including Bayesian statistics, random combinatorial structures, and stochastic models of diversity in various phenomena. Some distributional results on ordered sizes in the Gibbs partition are established by introducing associated partial Bell polynomials and analysis of the generating functions. The combinatorial approach is applied to derive explicit results on asymptotic behavior of the extreme sizes in the Gibbs partition. Especially,…
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 "r…
The beta-negative binomial process (BNBP), an integer-valued stochastic process, is employed to partition a count vector into a latent random count matrix. As the marginal probability distribution of the BNBP that governs the exchangeable random partitions of grouped data has not yet been developed, current inference for the BNBP has to truncate the number of atoms of the beta process. This paper introduces an exchangeable partition probability function to explicitly describe how the BNBP clusters the data points of each group into a random number of exchangeable partitions, which are shared a…
Scalar-on-image regression aims to investigate changes in a scalar response of interest based on high-dimensional imaging data. We propose a novel Bayesian nonparametric scalar-on-image regression model that utilises the spatial coordinates of the voxels to group voxels with similar effects on the response to have a common coefficient. We employ the Potts-Gibbs random partition model as the prior for the random partition in which the partition process is spatially dependent, thereby encouraging groups representing spatially contiguous regions. In addition, Bayesian shrinkage priors are utilise…
Motivated by the fundamental problem of modeling the frequency of frequencies (FoF) distribution, this paper introduces the concept of a cluster structure to define a probability function that governs the joint distribution of a random count and its exchangeable random partitions. A cluster structure, naturally arising from a completely random measure mixed Poisson process, allows the probability distribution of the random partitions of a subset of a population to be dependent on the population size, a distinct and motivated feature that makes it more flexible than a partition structure. This …
We consider the task of modeling a dependent sequence of random partitions. It is well-known that a random measure in Bayesian nonparametrics induces a distribution over random partitions. The community has therefore assumed that the best approach to obtain a dependent sequence of random partitions is through modeling dependent random measures. We argue that this approach is problematic and show that the random partition model induced by dependent Bayesian nonparametric priors exhibit counter-intuitive dependence among partitions even though the dependence for the sequence of random probabilit…
Hierarchical and k-medoids clustering are deterministic clustering algorithms based on pairwise distances. Using these same pairwise distances, we propose a novel stochastic clustering method based on random partition distributions. We call our method CaviarPD, for cluster analysis via random partition distributions. CaviarPD first samples clusterings from a random partition distribution and then finds the best cluster estimate based on these samples using algorithms to minimize an expected loss. We compare CaviarPD with hierarchical and k-medoids clustering through eight case studies. Cluster…
In big data analysis, a simple task such as linear regression can become very challenging as the variable dimension $p$ grows. As a result, variable screening is inevitable in many scientific studies. In recent years, randomized algorithms have become a new trend and are playing an increasingly important role for large scale data analysis. In this article, we combine the ideas of variable screening and random partitioning to propose a new iterative variable screening method. For moderate sized $p$ of order $O(n^{2-δ})$, we propose a basic algorithm that adopts a distribution-based thresholding…
A popular approach for testing if two univariate random variables are statistically independent consists of partitioning the sample space into bins, and evaluating a test statistic on the binned data. The partition size matters, and the optimal partition size is data dependent. While for detecting simple relationships coarse partitions may be best, for detecting complex relationships a great gain in power can be achieved by considering finer partitions. We suggest novel consistent distribution-free tests that are based on summation or maximization aggregation of scores over all partitions of a…
Similarity-based Random Partition Distribution for Clustering Functional Data
Graduate School of Economics, The University of Tokyo（东京大学经济学研究科）；Faculty of Economics, Keio University（庆应义塾大学经济学部）；School of Commerce, Meiji University（明治大学商学部）
Change point analyses are concerned with identifying positions of an ordered stochastic process that undergo abrupt local changes of some underlying distribution. When multiple processes are observed, it is often the case that information regarding the change point positions is shared across the different processes. This work describes a method that takes advantage of this type of information. Since the number and position of change points can be described through a partition with contiguous clusters, our approach develops a joint model for these types of partitions. We describe computational …
Sufficient dimension reduction (SDR) is continuing an active research field nowadays for high dimensional data. It aims to estimate the central subspace (CS) without making distributional assumption. To overcome the large-$p$-small-$n$ problem we propose a new approach for SDR. Our method combines the following ideas for high dimensional data analysis: (1) Randomly partition the covariates into subsets and use distance correlation (DC) to construct a sketch of envelope subspace with low dimension. (2) Obtain a sketch of the CS by applying conventional SDR method within the constructed envelope…
