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20
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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Cited by 40 (1 self)
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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.
Poisson process partition calculus with an application to Bayesian . . .
, 2005
"... This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The P ..."
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Cited by 32 (10 self)
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This article develops, and describes how to use, results concerning disintegrations of Poisson random measures. These results are fashioned as simple tools that can be tailormade to address inferential questions arising in a wide range of Bayesian nonparametric and spatial statistical models. The Poisson disintegration method is based on the formal statement of two results concerning a Laplace functional change of measure and a Poisson Palm/Fubini calculus in terms of random partitions of the integers {1,...,n}. The techniques are analogous to, but much more general than, techniques for the Dirichlet process and weighted gamma process developed in [Ann. Statist. 12
Approximate Dirichlet Process Computing in Finite Normal Mixtures: Smoothing and Prior Information
 JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
, 2000
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Some Further Developments for StickBreaking Priors: Finite and Infinite Clustering and Classification
 Sankhya Series A
, 2003
"... this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures ..."
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Cited by 15 (0 self)
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this paper will be to develop new surrounding theory for the hierarchical model (7) and show how these may be used to develop computational algorithms for computing posterior quantities. Our theoretical contributions include developing key properties for the class of extended stickbreaking measures, which includes establishing a conjugacy property of their random weights to i.i.d sampling, and a characterization of the posterior for the extended stickbreaking prior under i.i.d sampling. See Section 3. These properties then lead us in Section 4 to a general characterization for the posterior of (7). In Section 5 we outline a collapsed Gibbs sampling algorithm and an i.i.d SIS (sequential importance sampling) algorithm that can be used for inference in (7). One important implication is our ability to t the posterior of (6) subject to in nite dimensional stickbreaking measures. The paper begins with a brief discussion of stickbreaking priors in Section 2
Independent and Identically Distributed Monte Carlo Algorithms for Semiparametric Linear Mixed Models
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2002
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Bayesian finite mixtures with an unknown number of components: the allocation sampler
 University of Glasgow
, 2005
"... A new Markov chain Monte Carlo method for the Bayesian analysis of finite mixture distributions with an unknown number of components is presented. The sampler is characterized by a state space consisting only of the number of components and the latent allocation variables. Its main advantage is that ..."
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Cited by 10 (1 self)
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A new Markov chain Monte Carlo method for the Bayesian analysis of finite mixture distributions with an unknown number of components is presented. The sampler is characterized by a state space consisting only of the number of components and the latent allocation variables. Its main advantage is that it can be used, with minimal changes, for mixtures of components from any parametric family, under the assumption that the component parameters can be integrated out of the model analytically. Artificial and real data sets are used to illustrate the method and mixtures of univariate and of multivariate normals are explicitly considered. The problem of label switching, when parameter inference is of interest, is addressed in a postprocessing stage.
Poisson calculus for spatial neutral to the right processes
, 2003
"... In this paper we consider classes of nonparametric priors on spaces of distribution functions and cumulative hazard measures that are based on extensions of the neutral to the right (NTR) concept. In particular, spatial neutral to the right processes that extend the NTR concept from priors on the cl ..."
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Cited by 10 (1 self)
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In this paper we consider classes of nonparametric priors on spaces of distribution functions and cumulative hazard measures that are based on extensions of the neutral to the right (NTR) concept. In particular, spatial neutral to the right processes that extend the NTR concept from priors on the class of distributions on the real line to classes of distributions on general spaces are discussed. Representations of the posterior distribution of the spatial NTR processes are given. A different type of calculus than traditionally employed in the Bayesian literature, based on Poisson process partition calculus methods described in James (2002), is provided which offers a streamlined proof of posterior results for NTR models and its spatial extension. The techniques are applied to progressively more complex models ranging from the complete data case to semiparametric multiplicative intensity models. Refinements are then given which describes the underlying properties of spatial NTR processes analogous to those developed for the Dirichlet process. The analysis yields accessible moment formulae and characterizations of the posterior distribution and relevant marginal distributions. An EPPF formula and additionally a distribution related to the risk and death sets is computed. In the homogeneous case, these distributions turn out to be connected and overlap with recent work on regenerative compositions defined by suitable discretisation of subordinators. The formulae we develop for the marginal distribution of spatial NTR models provide clues on how to sample posterior distributions in complex settings. In addition the spatial NTR is further extended to the mixture model setting which allows for applicability of such processes to much more complex data structures. A description of a species sampling model derived from a spatial NTR model is also given.
Continuous Contour Monte Carlo for Marginal Density Estimation with an Application to Spatial Statistical Model
, 2006
"... The problem of marginal density estimation for a multivariate density function f(x) can be generally stated as a problem of density function estimation for a random vector λ(x) of dimension lower than that of x. In this paper, we propose a technique, the socalled continuous Contour Monte Carlo (CCM ..."
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Cited by 7 (3 self)
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The problem of marginal density estimation for a multivariate density function f(x) can be generally stated as a problem of density function estimation for a random vector λ(x) of dimension lower than that of x. In this paper, we propose a technique, the socalled continuous Contour Monte Carlo (CCMC) algorithm, for solving this problem. CCMC can be viewed as a continuous version of the contour Monte Carlo (CMC) algorithm recently proposed in the literature. CCMC abandons the use of sample space partitioning and incorporates the techniques of kernel density estimation into its simulations. CCMC is more general than other marginal density estimation algorithms. First, it works for any density functions, even for those having a rugged or unbalanced energy landscape. Second, it works for any transformation λ(x) regardless of the availability of the analytical form of the inverse transformation. In this paper, CCMC is applied to estimate the unknown normalizing constant function for a spatial autologistic model, and the estimate is then used in a Bayesian analysis for the spatial autologistic model in place of the true normalizing constant function. Numerical results on the US cancer mortality data indicate that the Bayesian method can produce much more accurate estimates than the MPLE and MCMLE methods for the parameters of the spatial autologistic model.
Bayesian finite mixtures: a note on prior specification and posterior computation
, 2005
"... A new method for the computation of the posterior distribution of the number k of components in a finite mixture is presented. Two aspects of prior specification are also studied: an argument is made for the use of a P oi(1) distribution as the prior for k; and methods are given for the selection of ..."
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Cited by 3 (1 self)
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A new method for the computation of the posterior distribution of the number k of components in a finite mixture is presented. Two aspects of prior specification are also studied: an argument is made for the use of a P oi(1) distribution as the prior for k; and methods are given for the selection of hyperparameter values in the mixture of normals model, with natural conjugate priors on the components parameters.
Nonasymptotic bounds for Bayesian order identification with application to mixtures
, 2005
"... The efficiency of two Bayesian order estimators is studied. By using nonparametric techniques, we prove new underestimation and overestimation bounds. The results apply to various models, including mixture models. In this case, the errors are shown to be O(e −an) and O((log n) b / √ n) (a,b> 0), re ..."
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Cited by 2 (1 self)
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The efficiency of two Bayesian order estimators is studied. By using nonparametric techniques, we prove new underestimation and overestimation bounds. The results apply to various models, including mixture models. In this case, the errors are shown to be O(e −an) and O((log n) b / √ n) (a,b> 0), respectively. 1. Introduction. Order