A Bernoulli Mixture Model (BMM) is a finite mixture of random binary vectors with independent dimensions. The problem of clustering BMM data arises in a variety of real-world applications, ranging from population genetics to activity analysis in social networks. In this paper, we analyze the clusterability of BMMs from a theoretical perspective, when the number of clusters is unknown. In particular, we stipulate a set of conditions on the sample complexity and dimension of the model in order to guarantee the Probably Approximately Correct (PAC)-clusterability of a dataset. To the best of our knowledge, these findings are the first non-asymptotic bounds on the sample complexity of learning or clustering BMMs.
@cite_0: In this paper, I review the main results on the asymptotic properties of the posterior distribution in nonparametric or high-dimensional models. In particular, I explain how posterior concentration rates can be derived and what we learn from such analysis in terms of the impact of the prior distribution on high-dimensional models. These results concern fully Bayes and empirical Bayes procedures. I also describe some of the results that have been obtained recently in semiparametric models, focusing mainly on the Bernstein–von Mises property. Although these results are theoretical in nature, they shed light on some subtle behaviors of the prior models and sharpen our understanding of the family of functionals that can be well estimated for a given prior model.
@cite_36: We propose the hierarchical Dirichlet process (HDP), a nonparametric Bayesian model for clustering problems involving multiple groups of data. Each group of data is modeled with a mixture, with the number of components being open-ended and inferred automatically by the model. Further, components can be shared across groups, allowing dependencies across groups to be modeled effectively as well as conferring generalization to new groups. Such grouped clustering problems occur often in practice, e.g. in the problem of topic discovery in document corpora. We report experimental results on three text corpora showing the effective and superior performance of the HDP over previous models.
@cite_7: A key problem in statistical modeling is model selection, that is, how to choose a model at an appropriate level of complexity. This problem appears in many settings, most prominently in choosing the number of clustersinmixturemodelsorthenumberoffactorsinfactoranalysis.Inthistutorial,wedescribeBayesian 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 introduction to Bayesian nonparametric methods and contains several examples of their application.
@cite_39: 
Our work is also related to Bayesian non-parametric approaches in the sense that the number of clusters is open-ended, and will be inferred based on the observed data. Some good reviews on non-parametric approaches in statistics can be found, for example, in @cite_0 @cite_39 @cite_7 . In particular, @cite_36 has proposed a unified non-parametric framework for model-based clustering with the use of hierarchical Dirichlet mixtures. Almost all of the studies reviewed so far share a common property: at their best, authors have only proved convergence to a sub-optimal likelihood value, rather than providing guarantees on the accuracy of the final clustering learning.