this paper, we propose some solutions to these problems. Our goal is to come up with a simple, practical method for estimating the density. This is an interesting problem in its own right, as well as a first step towards solving other inference problems, such as providing more flexible distributions in hierarchical models. To see why the posterior is improper under the usual reference prior, we write the model in the following way. Let Z = (Z 1 ; : : : ; Z n ) and X = (X 1 ; : : : ; X n ). The Z

We give conditions that guarantee that the posterior probability of every Hellinger...

A Dirichlet mixture of normal densities is a useful choice for a prior distribution on densities in the problem of Bayesian density estimation. In the recent years, efficient Markov chain Monte Carlo method for the computation of the posterior distribution has been developed. The method has been applied to data arising from different fields of interest. The important issue of consistency was however left open. In this paper, we settle this issue in affirmative. 1. Introduction. Recent

Assume fP ` : ` 2 \Thetag is a set of probability distributions with a common dominating measure on a complete separable metric space Y . A state ` 2 \Theta is chosen by Nature. A statistician gets n independent observations Y 1 ; : : : ; Y n from Y distributed according to P ` . For each time t between 1 and n, based on the observations Y 1 ; : : : ; Y t\Gamma1 , the statistician produces an estimated distribution P t for P ` , and suffers a loss L(P ` ; P t ). The cumulative risk for the statistician is the average total loss up to time n. Of special interest in information theory, data compression, mathematical finance, computational learning theory and statistical mechanics is the special case when the loss L(P ` ; P t ) is the relative entropy between the true distribution P ` and the estimated distribution P t . Here the cumulative Bayes risk from time 1 to n is the mutual information between the random parameter \Theta and the observations Y 1 ; : : : ;...

We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include infinite dimensional cases. The rates are driven by two quantities: the size of the space, as measure by metric entropy or bracketing entropy, and the degree to which the prior concentrates in a small ball around the true parameter. We apply the results to several examples. In some cases, natural priors give sub-optimal rates of convergence and better rates can be obtained by using sievebased priors such as those introduced by Zhao (1993, 1998). AMS 1990 classification: Primary, 62A15, Secondary: 62E20, 62G15. KEYWORDS: Bayesian inference, asymptotic inference, non-parametric models, sieves. 1 Introduction. Nonparametric Bayesian methods have become quite popular lately, largely because of advances in computing; see Dey, Mueller and Sinha (1998) for a recent account. Because of their growing popularity, it is important to understand ...

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: Bose-Einstein distribution, Dirichlet process, identification, method of sieves, random probability measure, relative entropy, weak convergence.

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This paper proposes a new approach to sparsity called the horseshoe estimator. The horseshoe is a close cousin of other widely used Bayes rules arising from, for example, double-exponential and Cauchy priors, in that it is a member of the same family of multivariate scale mixtures of normals. But the horseshoe enjoys a number of advantages over existing approaches, including its robustness, its adaptivity to different sparsity patterns, and its analytical tractability. We prove two theorems that formally characterize both the horseshoe’s adeptness at large outlying signals, and its super-efficient rate of convergence to the correct estimate of the sampling density in sparse situations. Finally, using a combination of real and simulated data, we show that the horseshoe estimator corresponds quite closely to the answers one would get by pursuing a full Bayesian model-averaging approach using a discrete mixture prior to model signals and noise.

Ghosh and Ramamoorthi (1995) studied the posterior consistency for survival models and showed that the posterior was consistent, when the prior on the distribution of survival times was the Dirichlet process prior. In this paper, we study the posterior consistency of survival models with neutral to the right process priors which include Dirichlet process priors. A set of sufficient conditions for the posterior consistency with neutral to the right process priors is given. Interestingly, not all the neutral to the right process priors have consistent posteriors, but most of the popular priors such as Dirichlet processes, beta processes and gamma processes have consistent posteriors. With a class of priors which includes beta processes, a necessary and sufficient condition for the consistency is also established. An interesting counter intuitive phenomenon is found. Suppose there are two priors centered at the true parameter value with finite variances. Surprisingly, the posterior with s...

Density estimation, especially multivariate density estimation, is a fundamental problem in nonparametric inference. Dirichlet mixture priors are often used in practice for such problem. However, asymptotic properties of such priors have only been studied in the univariate case. We extend L1-consistency of Dirichlet mixutures in the multivariate density estimation setting. We obtain such a result by showing that the Kullback-Leibler property of the prior holds and that the size of the sieve in the parameter space in term of L1-metric entropy is not larger than the order of n. However, it seems that the usual technique of choosing a sieve by controlling prior probabilities is unable to lead to a useful bound on the metric entropy required for the application of a general posterior consistency theorem for the multivariate case. We overcome this difficulty by using a structural property of Dirichlet mixtures. Our result is general and can be applied to a multivariate normal kernel even when the kernel has a general variance-covariance matrix.