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Dirichlet Prior Sieves in Finite Normal Mixtures
 Statistica Sinica
, 2002
"... Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive ..."
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Abstract: The use of a finite dimensional Dirichlet prior in the finite normal mixture model has the effect of acting like a Bayesian method of sieves. Posterior consistency is directly related to the dimension of the sieve and the choice of the Dirichlet parameters in the prior. We find that naive use of the popular uniform Dirichlet prior leads to an inconsistent posterior. However, a simple adjustment to the parameters in the prior induces a random probability measure that approximates the Dirichlet process and yields a posterior that is strongly consistent for the density and weakly consistent for the unknown mixing distribution. The dimension of the resulting sieve can be selected easily in practice and a simple and efficient Gibbs sampler can be used to sample the posterior of the mixing distribution. Key words and phrases: BoseEinstein distribution, Dirichlet process, identification, method of sieves, random probability measure, relative entropy, weak convergence.
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
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Density estimation with stagewise optimization of the empirical risk
, 2006
"... We consider multivariate density estimation with identically distributed observations. We study a density estimator which is a convex combination of functions in a dictionary and the convex combination is chosen by minimizing the L2 empirical risk in a stagewise manner. We derive the convergence rat ..."
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We consider multivariate density estimation with identically distributed observations. We study a density estimator which is a convex combination of functions in a dictionary and the convex combination is chosen by minimizing the L2 empirical risk in a stagewise manner. We derive the convergence rates of the estimator when the estimated density belongs to the L2 closure of the convex hull of a class of functions which satisfies entropy conditions. The L2 closure of a convex hull is a large nonparametric class but under suitable entropy conditions the convergence rates of the estimator do not depend on the dimension, and density estimation is feasible also in high dimensional cases. The variance of the estimator does not increase when the number of components of the estimator increases. Instead, we control the biasvariance tradeoff by the choice of the dictionary from which the components are chosen.
CONVERGENCE OF LATENT MIXING MEASURES IN FINITE AND INFINITE MIXTURE MODELS
"... This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and fdivergence functionals such as Hellinge ..."
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This paper studies convergence behavior of latent mixing measures that arise in finite and infinite mixture models, using transportation distances (i.e., Wasserstein metrics). The relationship between Wasserstein distances on the space of mixing measures and fdivergence functionals such as Hellinger and Kullback–Leibler distances on the space of mixture distributions is investigated in detail using various identifiability conditions. Convergence in Wasserstein metrics for discrete measures implies convergence of individual atoms that provide support for the measures, thereby providing a natural interpretation of convergence of clusters in clustering applications where mixture models are typically employed. Convergence rates of posterior distributions for latent mixing measures are established, for both finite mixtures of multivariate distributions and infinite mixtures based on the Dirichlet process. 1. Introduction. A
CONSISTENT ORDER ESTIMATION AND THE LOCAL GEOMETRY OF MIXTURES BY ELISABETH GASSIAT AND RAMON VAN HANDEL
, 2010
"... Consider an i.i.d. sequence of random variables whose distribution f ⋆ lies in one of a nested family of models(Mq)q∈N,Mq ⊂ Mq+1. The smallest index q ⋆ such that Mq ⋆ contains f ⋆ is called the model order. We establish strong consistency of the penalized likelihood order estimator in a general set ..."
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Consider an i.i.d. sequence of random variables whose distribution f ⋆ lies in one of a nested family of models(Mq)q∈N,Mq ⊂ Mq+1. The smallest index q ⋆ such that Mq ⋆ contains f ⋆ is called the model order. We establish strong consistency of the penalized likelihood order estimator in a general setting with penalties of order η(q)loglogn, where η(q) is a dimensional quantity. Moreover, such penalties are shown to be minimal. In contrast to previous work, an a priori upper bound on the model order is not assumed. The local dimension η(q) of the model Mq is defined in terms of the bracketing entropy of a class of weighted densities, whose computation is a nonstandard problem which is of independent interest. We perform the requisite computations for the case of onedimensional location mixtures, thus demonstrating the consistency of the penalized likelihood mixture order estimator. The proof requires a delicate analysis of the local geometry of the mixture family Mq in a neighborhood of f ⋆ , for q> q ⋆. The extension to more general mixture models remains an open problem. 1. Introduction. Let (Xk)k∈N