program. The authors wish to acknowledge helpful discussions with Lancelot James and Jim Pitman and the referees for useful comments. 1 We consider problems involving groups of data, where each observation within a group is a draw from a mixture model, and where it is desirable to share mixture components between groups. We assume that the number of mixture components is unknown a priori and is to be inferred from the data. In this setting it is natural to consider sets of Dirichlet processes, one for each group, where the well-known clustering property of the Dirichlet process provides a nonparametric prior for the number of mixture components within each group. Given our desire to tie the mixture models in the various groups, we consider a hierarchical model, specifically one in which the base measure for the child Dirichlet processes is itself distributed according to a Dirichlet process. Such a base measure being discrete, the child Dirichlet processes necessar-ily share atoms. Thus, as desired, the mixture models in the different groups necessarily share mixture components. We discuss representations of hierarchical Dirichlet processes in terms of

AbstractÐThis paper proposes an unsupervised algorithm for learning a finite mixture model from multivariate data. The adjective ªunsupervisedº is justified by two properties of the algorithm: 1) it is capable of selecting the number of components and 2) unlike the standard expectation-maximization (EM) algorithm, it does not require careful initialization. The proposed method also avoids another drawback of EM for mixture fitting: the possibility of convergence toward a singular estimate at the boundary of the parameter space. The novelty of our approach is that we do not use a model selection criterion to choose one among a set of preestimated candidate models; instead, we seamlessly integrate estimation and model selection in a single algorithm. Our technique can be applied to any type of parametric mixture model for which it is possible to write an EM algorithm; in this paper, we illustrate it with experiments involving Gaussian mixtures. These experiments testify for the good performance of our approach. Index TermsÐFinite mixtures, unsupervised learning, model selection, minimum message length criterion, Bayesian methods, expectation-maximization algorithm, clustering. 1

We define a probability distribution over equivalence classes of binary matrices with a finite number of rows and an unbounded number of columns. This distribution

We discuss a hierarchical probabilistic model whose predictions are similar to those of the popular language modelling procedure known as `smoothing'. A number of interesting differences from smoothing emerge. The insights gained from a probabilistic view of this problem point towards new directions for language modelling. The ideas of this paper are also applicable to other problems such as the modelling of triphomes in speech, and DNA and protein sequences in molecular biology. The new algorithm is compared with smoothing on a two million word corpus. The methods prove to be about equally accurate, with the hierarchical model using fewer computational resources. Contents 1 Introduction 2 1.1 The bigram language model with smoothing 2 1.2 Any rational predictive procedure can be made Bayesian 3 2 An explicit model using Dirichlet priors 4 2.1 The inferences we will make 4 2.2 The likelihood function 5 2.3 What prior? 5 2.4 A convenient family of priors: Dirichlet distributions 5 2.5 ...

. We propose a split-merge Markov chain algorithm to address the problem of inefficient sampling for conjugate Dirichlet process mixture models. Traditional Markov chain Monte Carlo methods for Bayesian mixture models, such as Gibbs sampling, can become trapped in isolated modes corresponding to an inappropriate clustering of data points. This article describes a Metropolis-Hastings procedure that can escape such local modes by splitting or merging mixture components. Our Metropolis-Hastings algorithm employs a new technique in which an appropriate proposal for splitting or merging components is obtained by using a restricted Gibbs sampling scan. We demonstrate empirically that our method outperforms the Gibbs sampler in situations where two or more components are similar in structure. Key words: Dirichlet process mixture model, Markov chain Monte Carlo, Metropolis-Hastings algorithm, Gibbs sampler, split-merge updates 1 Introduction Mixture models are often applied to density estim...

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Consider the problem of fitting a finite Gaussian mixture, with an unknown number of components, to observed data. This paper proposes a new minimum description length (MDL) type criterion, termed MMDL (for mixture MDL), to select the number of components of the model. MMDL is based on the identification of an "equivalent sample size", for each component, which does not coincide with the full sample size. We also introduce an algorithm based on the standard expectationmaximization (EM) approach together with a new agglomerative step, called agglomerative EM (AEM). The experiments here reported have shown that MMDL outperforms existing criteria of comparable computational cost. The good behavior of AEM, namely its good robustness with respect to initialization, is also illustrated experimentally.

The increased collection of high-dimensional data in various fields has raised a strong interest in clustering algorithms and variable selection procedures. In this paper, we propose a model-based method that addresses the two problems simultaneously. We introduce a latent binary vector to identify discriminating variables and use Dirichlet process mixture models to define the cluster structure. We update the variable selection index using a Metropolis algorithm and obtain inference on the cluster structure via a split-merge Markov chain Monte Carlo technique. We explore the performance of the methodology on simulated data and illustrate an application with a dna microarray study.

Maximum a posterJori optimization of parameters and the Laplace approximation for the marginal likelihood are both basis-dependent methods. This note compares two choices of basis for models parameterized by probabilities, showing that it is possible to improve on the traditional choice, the probability simplex, by transforming to the softmax' basis.

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