Bayesian cluster analysis: Point estimation and credible balls

We show that model-based Bayesian clustering, the probabilistically most systematic approach to the partitioning of data, can be mapped into a statistical physics problem for a gas of particles, and as a result becomes amenable to a detailed quantitative analysis. A central role in the resulting statistical physics framework is played by an entropy function. We demonstrate that there is a relevant parameter regime where mean-field analysis of this function is exact, and that,...

Broadly, the goal when clustering data is to separate observations into meaningful subgroups. The rich variety of methods for clustering reflects the fact that the relevant notion of meaningful clusters varies across applications. The classical Bayesian approach clusters observations by their association with components of a mixture model; the choice in class of components allows flexibility to capture a range of meaningful cluster notions. However, in practice the range is s...

Unsupervised learning is widely recognized as one of the most important challenges facing machine learning nowa- days. However, in spite of hundreds of papers on the topic being published every year, current theoretical understanding and practical implementations of such tasks, in particular of clustering, is very rudimentary. This note focuses on clustering. I claim that the most signif- icant challenge for clustering is model selection. In contrast with other common computa...

A key problem in statistical modeling is model selection, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number ofclusters in mixture models or the number of factors in factor analysis. In this tutorial we describe Bayesian nonparametric methods, a class of methods that side-steps this issue by allowing the data to determine the complexity of the model. This tutorial is a high-level introduc...

A fundamental problem in network analysis is clustering the nodes into groups which share a similar connectivity pattern. Existing algorithms for community detection assume the knowledge of the number of clusters or estimate it a priori using various selection criteria and subsequently estimate the community structure. Ignoring the uncertainty in the first stage may lead to erroneous clustering, particularly when the community structure is vague. We instead propose a coherent...

Variable clustering is important for explanatory analysis. However, only few dedicated methods for variable clustering with the Gaussian graphical model have been proposed. Even more severe, small insignificant partial correlations due to noise can dramatically change the clustering result when evaluating for example with the Bayesian Information Criteria (BIC). In this work, we try to address this issue by proposing a Bayesian model that accounts for negligible small, but no...

We develop a set of scalable Bayesian inference procedures for a general class of nonparametric regression models. Specifically, nonparametric Bayesian inferences are separately performed on each subset randomly split from a massive dataset, and then the obtained local results are aggregated into global counterparts. This aggregation step is explicit without involving any additional computation cost. By a careful partition, we show that our aggregated inference results obtain...

Clustering provides a common means of identifying structure in complex data, and there is renewed interest in clustering as a tool for the analysis of large data sets in many fields. A natural question is how many clusters are appropriate for the description of a given system. Traditional approaches to this problem are based either on a framework in which clusters of a particular shape are assumed as a model of the system or on a two-step procedure in which a clustering crite...

In recent years, large-scale Bayesian learning draws a great deal of attention. However, in big-data era, the amount of data we face is growing much faster than our ability to deal with it. Fortunately, it is observed that large-scale datasets usually own rich internal structure and is somewhat redundant. In this paper, we attempt to simplify the Bayesian posterior via exploiting this structure. Specifically, we restrict our interest to the so-called well-clustered datasets a...

This paper presents a clustering approach that allows for rigorous statistical error control similar to a statistical test. We develop estimators for both the unknown number of clusters and the clusters themselves. The estimators depend on a tuning parameter alpha which is similar to the significance level of a statistical hypothesis test. By choosing alpha, one can control the probability of overestimating the true number of clusters, while the probability of underestimation...