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Convergence rates of posterior distributions
 Ann. Statist
, 2000
"... We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, D ..."
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Cited by 43 (11 self)
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We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, Dirichlet processes and interval censoring. 1. Introduction. Suppose
Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities
 Ann. Statist
, 2001
"... We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or locationscale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is ..."
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Cited by 34 (10 self)
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We study the rates of convergence of the maximum likelihood estimator (MLE) and posterior distribution in density estimation problems, where the densities are location or locationscale mixtures of normal distributions with the scale parameter lying between two positive numbers. The true density is also assumed to lie in this class with the true mixing distribution either compactly supported or having subGaussian tails. We obtain bounds for Hellinger bracketing entropies for this class, and from these bounds, we deduce the convergence rates of (sieve) MLEs in Hellinger distance. The rate turns out to be �log n � κ / √ n, where κ ≥ 1 is a constant that depends on the type of mixtures and the choice of the sieve. Next, we consider a Dirichlet mixture of normals as a prior on the unknown density. We estimate the prior probability of a certain KullbackLeibler type neighborhood and then invoke a general theorem that computes the posterior convergence rate in terms the growth rate of the Hellinger entropy and the concentration rate of the prior. The posterior distribution is also seen to converge at the rate �log n � κ / √ n in, where κ now depends on the tail behavior of the base measure of the Dirichlet process. 1. Introduction. A
Convergence rates for density estimation with Bernstein polynomials
 Ann. Statist
, 2001
"... Mixture models for density estimation provide a very useful set up for the Bayesian or the maximum likelihood approach. For a density on the unit interval, mixtures of beta densities form a flexible model. The class of Bernstein densities is a muchsmaller subclass of the beta mixtures defined by Ber ..."
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Cited by 23 (5 self)
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Mixture models for density estimation provide a very useful set up for the Bayesian or the maximum likelihood approach. For a density on the unit interval, mixtures of beta densities form a flexible model. The class of Bernstein densities is a muchsmaller subclass of the beta mixtures defined by Bernstein polynomials, which can approximate any continuous density. A Bernstein polynomial prior is obtained by putting a prior distribution on the class of Bernstein densities. The posterior distribution of a Bernstein polynomial prior is consistent under very general conditions. In this article, we present some results on the rate of convergence of the posterior distribution. If the underlying distribution generating the data is itself a Bernstein density, then we show that the posterior distribution converges at “nearly parametric rate ” �log n� / √ n for the Hellinger distance. If the true density is not of the Bernstein type, we show that the posterior converges at a rate n −1/3 �log n � 5/6 provided that the true density is twice differentiable and bounded away from 0. Similar rates are also obtained for sieve maximum likelihood estimates. These rates are inferior to the pointwise convergence rate of a kernel type estimator. We show that the Bayesian bootstrap method gives a proxy for the posterior distribution and has a convergence rate at par with that of the kernel estimator. 1. Introduction. Mixture models
Posterior Consistency in Nonparametric Regression Problems under Gaussian Process Priors
, 2004
"... Posterior consistency can be thought of as a theoretical justification of the Bayesian method. One of the most popular approaches to nonparametric Bayesian regression is to put a nonparametric prior distribution on the unknown regression function using Gaussian processes. In this paper, we study pos ..."
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Cited by 17 (1 self)
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Posterior consistency can be thought of as a theoretical justification of the Bayesian method. One of the most popular approaches to nonparametric Bayesian regression is to put a nonparametric prior distribution on the unknown regression function using Gaussian processes. In this paper, we study posterior consistency in nonparametric regression problems using Gaussian process priors. We use an extension of the theorem of Schwartz (1965) for nonidentically distributed observations, verifying its conditions when using Gaussian process priors for the regression function with normal or double exponential (Laplace) error distributions. We define a metric topology on the space of regression functions and then establish almost sure consistency of the posterior distribution. Our metric topology is weaker than the popular L 1 topology. With additional assumptions, we prove almost sure consistency when the regression functions have L 1 topologies. When the covariate (predictor) is assumed to be a random variable, we prove almost sure consistency for the joint density function of the response and predictor using the Hellinger metric.
Consistency of Bayes estimates for nonparametric regression: normal theory
 Bernoulli
, 1998
"... Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at ..."
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Cited by 9 (1 self)
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Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at
DYNAMICS OF BAYESIAN UPDATING WITH DEPENDENT DATA AND MISSPECIFIED MODELS
, 2009
"... Recent work on the convergence of posterior distributions under Bayesian updating has established conditions under which the posterior will concentrate on the truth, if the latter has a perfect representation within the support of the prior, and under various dynamical assumptions, such as the data ..."
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Cited by 9 (2 self)
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Recent work on the convergence of posterior distributions under Bayesian updating has established conditions under which the posterior will concentrate on the truth, if the latter has a perfect representation within the support of the prior, and under various dynamical assumptions, such as the data being independent and identically distributed or Markovian. Here I establish sufficient conditions for the convergence of the posterior distribution in nonparametric problems even when all of the hypotheses are wrong, and the datagenerating process has a complicated dependence structure. The main dynamical assumption is the generalized asymptotic equipartition (or “ShannonMcMillanBreiman”) property of information theory. I derive a kind of large deviations principle for the posterior measure, and discuss the advantages of predicting using a combination of models known to be wrong. An appendix sketches connections between the present results and the “replicator dynamics” of evolutionary theory.
A note on the consistency of Bayes factors for testing point null versus nonparametric alternatives
 Journal of Statistical Planning and Inference
, 2004
"... When testing a point null hypothesis versus an alternative that is vaguely specified, a Bayesian test usually proceeds by putting a nonparametric prior on the alternative and then computing a Bayes factor based on the observations. This paper addresses the question of consistency, that is, whether ..."
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Cited by 5 (0 self)
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When testing a point null hypothesis versus an alternative that is vaguely specified, a Bayesian test usually proceeds by putting a nonparametric prior on the alternative and then computing a Bayes factor based on the observations. This paper addresses the question of consistency, that is, whether the Bayes factor is correctly indicative of the null or the alternative as the sample size increases. We establish several consistency results in the affirmative under fairly general conditions. Consistency of Bayes factors for testing a point null versus a parametric alternative has long been known. The results here can also be viewed as the nonparametric extension of the parametric counterpart. MSC: 62G20; 62C10
Bayesian Density Estimation
, 1998
"... . This is a brief exposition of posterior consistency issues in Bayesian nonparametrics especially in the context of Bayesian Density estimation, 1991 Mathematics Subject Classification: 62A15,62G99. Keywords and Phrases: Dirichlet mixtures,density estimation 1 Introduction We describe popular meth ..."
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Cited by 1 (0 self)
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. This is a brief exposition of posterior consistency issues in Bayesian nonparametrics especially in the context of Bayesian Density estimation, 1991 Mathematics Subject Classification: 62A15,62G99. Keywords and Phrases: Dirichlet mixtures,density estimation 1 Introduction We describe popular methods of Bayesian density estimation and explore sufficient conditions for the posterior given data to converge to a true underlying distribution P 0 as the data size increases. One of the advantages of Bayesian density estimates is that,unlike classical frequentist methods,choice of the right amount of smoothing is not such a serious problem. Section 2 provides a general background to infinite dimensional problems of inference such as Bayesian nonparametrics, semiparametrics and density estimation. Bayesian nonparametrics has been around for about twenty five years but the other two areas,specially the last, is of more recent vintage. Section 3 indicates in broad terms why different tools ar...