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Dealing with label switching in mixture models
 Journal of the Royal Statistical Society, Series B
, 2000
"... In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions ..."
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Cited by 124 (0 self)
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In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward that might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarising joint posterior distributions by marginal distributions, often leads to nonsensical answers. This is due to the socalled “labelswitching” problem, which is caused by symmetry in the likelihood of the model parameters. A frequent response to this problem is to remove the symmetry using artificial identifiability constraints. We demonstrate that this fails in general to solve the problem, and describe an alternative class of approaches, relabelling algorithms, which arise from attempting to minimise the posterior expected loss under a class of loss functions. We describe in detail one particularly simple and general relabelling algorithm, and illustrate its success in dealing with the labelswitching problem on two examples.
A computational approach for full nonparametric Bayesian inference under Dirichlet process mixture models
 Journal of Computational and Graphical Statistics
, 2002
"... Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci � cations. Nonparametric aspects are often introduced to enrich this class, resultingin semiparametricmodels. Focusing on single or ksample problems,many classical nonparametricapproachesare limited ..."
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Cited by 28 (8 self)
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Widely used parametric generalizedlinear models are, unfortunately,a somewhat limited class of speci � cations. Nonparametric aspects are often introduced to enrich this class, resultingin semiparametricmodels. Focusing on single or ksample problems,many classical nonparametricapproachesare limited to hypothesistesting. Those that allow estimation are limited to certain functionals of the underlying distributions. Moreover, the associated inference often relies upon asymptotics when nonparametric speci � cations are often most appealing for smaller sample sizes. Bayesian nonparametricapproachesavoid asymptotics but have, to date, been limited in the range of inference. Working with Dirichlet process priors, we overcome the limitations of existing simulationbasedmodel � tting approaches which yield inference that is con � ned to posterior moments of linear functionals of the population distribution.This article provides a computationalapproach to obtain the entire posterior distribution for more general functionals. We illustrate with three applications: investigation of extreme value distributions associated with a single population, comparison of medians in a ksample problem, and comparison of survival times from different populations under fairly heavy censoring.
Bayesian Model Selection in Finite Mixtures by Marginal Density Decompositions
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 2001
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MCMC methods in wavelet shrinkage: Nonequally spaced regression, density and spectral density estimation
 BAYESIAN INFERENCE IN WAVELET BASED MODELS, VOLUME 141 OF LECTURE NOTES IN STATISTICS
, 1999
"... We consider posterior inference in wavelet based models for nonparametric regression with unequally spaced data, density estimation and spectral density estimation. The common theme in all three applications is the lack of posterior independence for the wavelet coe cientsdjk. In contrast, most comm ..."
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Cited by 5 (1 self)
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We consider posterior inference in wavelet based models for nonparametric regression with unequally spaced data, density estimation and spectral density estimation. The common theme in all three applications is the lack of posterior independence for the wavelet coe cientsdjk. In contrast, most commonly considered applications of wavelet decompositions in Statistics are based on a setup which implies a posteriori independent coe cients, essentially reducing the inference problem to a series of univariate problems. This is generally true for regression with equally spaced data, image reconstruction, density estimation based on smoothing the empirical distribution, time series applications and deconvolution problems. We propose a hierarchical mixture model as prior probability model on the wavelet coe cients. The model includes a leveldependent positive prior probability mass at zero, i.e., for vanishing coe cients. This implements wavelet coe cient thresholding as a formal Bayes rule. For nonzero coecients weintroduce shrinkage by assuming normal priors. Allowing di erent prior variance at each level of detail we obtain leveldependent shrinkage for nonzero coe cients. We implement inference in all three proposed models by a Markov chain Monte Carlo scheme which requires only minor modi cations for the di erent applications. Allowing zero coe cients requires simulation over variable dimension parameter space (Green 1995). We use a pseudoprior mechanism (Carlin and Chib 1995) to achieve this.
Galaxy Data
"... Roeder (1990) gives the data set and uses it to exemplify her theoretical results. Here is a brief description. According to the Big Bang theory, matter in the universe expanded at a tremendous rate. Gravitational forces caused the formation of galaxies. Astronomers speculate that gravitational pull ..."
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Roeder (1990) gives the data set and uses it to exemplify her theoretical results. Here is a brief description. According to the Big Bang theory, matter in the universe expanded at a tremendous rate. Gravitational forces caused the formation of galaxies. Astronomers speculate that gravitational pull led to clustering of galaxies and the research indicates the presence of superclusters of galaxies surrounded by large voids (stringandlament pattern). Measurements have recently become available for the distances between our galaxy and others. The distance is estimated by the red shift in the light spectrum in a fashion similar to how the Doppler e ect measures the changes in speed via changes in sound. Under the expansionuniverse paradigm, the furthest (from our galaxy) galaxies must be moving at greater velocities, because the distances and velocities are proportional. If, in reality, the galaxies are clumped, the velocities should have amultimodal distribution, each mode corresponding to a cluster. In the region of Corona Borealis the velocities of 82 galaxies were measured. The relative measurement
Maximum a Posteriori Parameter Estimation for Hidden Markov Models
"... this paper is a simulated annealing algorithm in the sense that we consider increasing powers of the posterior density ..."
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this paper is a simulated annealing algorithm in the sense that we consider increasing powers of the posterior density