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12
Estimating the integrated likelihood via posterior simulation using the harmonic mean identity
 Bayesian Statistics
, 2007
"... The integrated likelihood (also called the marginal likelihood or the normalizing constant) is a central quantity in Bayesian model selection and model averaging. It is defined as the integral over the parameter space of the likelihood times the prior density. The Bayes factor for model comparison a ..."
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Cited by 24 (2 self)
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The integrated likelihood (also called the marginal likelihood or the normalizing constant) is a central quantity in Bayesian model selection and model averaging. It is defined as the integral over the parameter space of the likelihood times the prior density. The Bayes factor for model comparison and Bayesian testing is a ratio of integrated likelihoods, and the model weights in Bayesian model averaging are proportional to the integrated likelihoods. We consider the estimation of the integrated likelihood from posterior simulation output, aiming at a generic method that uses only the likelihoods from the posterior simulation iterations. The key is the harmonic mean identity, which says that the reciprocal of the integrated likelihood is equal to the posterior harmonic mean of the likelihood. The simplest estimator based on the identity is thus the harmonic mean of the likelihoods. While this is an unbiased and simulationconsistent estimator, its reciprocal can have infinite variance and so it is unstable in general. We describe two methods for stabilizing the harmonic mean estimator. In the first one, the parameter space is reduced in such a way that the modified estimator involves a harmonic mean of heaviertailed densities, thus resulting in a finite variance estimator. The resulting
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.
Bayesian Inference on Mixtures of Distributions
, 2008
"... This survey covers stateoftheart Bayesian techniques for the estimation of mixtures. It complements the earlier Marin et al. (2005) by studying new types of distributions, the multinomial, latent class and t distributions. It also exhibits closed form solutions for Bayesian inference in some disc ..."
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Cited by 6 (5 self)
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This survey covers stateoftheart Bayesian techniques for the estimation of mixtures. It complements the earlier Marin et al. (2005) by studying new types of distributions, the multinomial, latent class and t distributions. It also exhibits closed form solutions for Bayesian inference in some discrete setups. At last, it sheds a new light on the computation of Bayes factors via the approximation of Chib (1995).
Estimating and Projecting Trends in HIV/AIDS Generalized Epidemics Using Incremental Mixture Importance Sampling
"... The Joint United Nations Programme on HIV/AIDS (UNAIDS) has decided to use Bayesian melding as the basis for its probabilistic projections of HIV prevalence in countries with generalized epidemics. This combines a mechanistic epidemiological model, prevalence data and expert opinion. Initially, the ..."
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Cited by 3 (2 self)
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The Joint United Nations Programme on HIV/AIDS (UNAIDS) has decided to use Bayesian melding as the basis for its probabilistic projections of HIV prevalence in countries with generalized epidemics. This combines a mechanistic epidemiological model, prevalence data and expert opinion. Initially, the posterior distribution was approximated by samplingimportanceresampling, which is simple to implement, easy to interpret, transparent to users and gave acceptable results for most countries. For some countries, however, this is not computationally efficient because the posterior distribution tends to be concentrated around nonlinear ridges and can also be multimodal. We propose instead Incremental Mixture Importance Sampling (IMIS), which iteratively builds up a better importance sampling function. This retains the simplicity and transparency of sampling importance resampling, but is much more efficient computationally. It also leads to a simple estimator of the integrated likelihood that is the basis for Bayesian model comparison and model averaging. In simulation experiments and on real data it outperformed both sampling importance resampling and three publicly available generic Markov chain Monte Carlo algorithms for this
Modelbased Clustering of nonGaussian Panel Data
"... In this paper we propose a modelbased method to cluster units within a panel. The underlying model is autoregressive and nonGaussian, allowing for both skewness and fat tails, and the units are clustered according to their dynamic behaviour and equilibrium level. Inference is addressed from a Baye ..."
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Cited by 2 (1 self)
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In this paper we propose a modelbased method to cluster units within a panel. The underlying model is autoregressive and nonGaussian, allowing for both skewness and fat tails, and the units are clustered according to their dynamic behaviour and equilibrium level. Inference is addressed from a Bayesian perspective and model comparison is conducted using the formal tool of Bayes factors. Particular attention is paid to prior elicitation and posterior propriety. We suggest priors that require little subjective input from the user and possess hierarchical structures that enhance the robustness of the inference. Two examples illustrate the methodology: one analyses economic growth of OECD countries and the second one investigates employment growth of Spanish manufacturing firms.
Estimates of AgeSpecific Reductions in HIV Prevalence in Uganda: Bayesian Melding Estimation and Probabilistic Population Forecast with an HIVenabled Cohort Component Projection Model
, 2010
"... We estimate agespecific HIV incidence and prevalence in Tanzania and Uganda in the late 1990s and forecast forward assuming no change in incidence. Comparisons between our forecasts of HIV prevalence and direct measures from the HIV/AIDS Indicator and Demographic and Health Surveys in the mid2000s ..."
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We estimate agespecific HIV incidence and prevalence in Tanzania and Uganda in the late 1990s and forecast forward assuming no change in incidence. Comparisons between our forecasts of HIV prevalence and direct measures from the HIV/AIDS Indicator and Demographic and Health Surveys in the mid2000s provide an agespecific measure of changes in HIV prevalence. In Tanzania our forecast accurately predicts agespecific HIV prevalence, suggesting little change in HIV incidence in Tanzania over the intervening decade. In Uganda our forecasts significantly overstate HIV prevalence. The age pattern of our forecast errors reflects the agespecific reductions in HIV prevalence and incidence in Uganda. Our estimates and forecasts are produced using an HIVenabled cohort component model of population projection first proposed by Heuveline (2003). We refine that model (Thomas and Clark, 2008) and implement the Bayesian melding with IMIS estimation method (Raftery and Bao, 2010). This method allows us to estimate the parameters of the Heuveline model with robust measures of uncertainty and to quantify uncertainty in the model outputs, e.g. forecasts. We validate
Performance of Bayesian Model Selection Criteria for Gaussian Mixture Models
, 2009
"... Bayesian methods are widely used for selecting the number of components in a mixture models, in part because frequentist methods have difficulty in addressing this problem in general. Here we compare some of the Bayesianly motivated or justifiable methods for choosing the number of components in a o ..."
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Bayesian methods are widely used for selecting the number of components in a mixture models, in part because frequentist methods have difficulty in addressing this problem in general. Here we compare some of the Bayesianly motivated or justifiable methods for choosing the number of components in a onedimensional Gaussian mixture model: posterior probabilities for a wellknown proper prior, BIC, ICL, DIC and AIC. We also introduce a new explicit unitinformation prior for mixture models, analogous to the prior to which BIC corresponds in regular statistical models. We base the comparison on a simulation study, designed to reflect published estimates of mixture model parameters from the scientific literature across a range of disciplines. We found that BIC clearly outperformed the five other
Biometrics DOI: 10.1111/j.15410420.2010.01399.x Estimating and Projecting Trends in HIV/AIDS Generalized Epidemics Using Incremental Mixture Importance Sampling
"... Summary. The Joint United Nations Programme on HIV/AIDS (UNAIDS) has decided to use Bayesian melding as the basis for its probabilistic projections of HIV prevalence in countries with generalized epidemics. This combines a mechanistic epidemiological model, prevalence data, and expert opinion. Initi ..."
Abstract
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Summary. The Joint United Nations Programme on HIV/AIDS (UNAIDS) has decided to use Bayesian melding as the basis for its probabilistic projections of HIV prevalence in countries with generalized epidemics. This combines a mechanistic epidemiological model, prevalence data, and expert opinion. Initially, the posterior distribution was approximated by samplingimportanceresampling, which is simple to implement, easy to interpret, transparent to users, and gave acceptable results for most countries. For some countries, however, this is not computationally efficient because the posterior distribution tends to be concentrated around nonlinear ridges and can also be multimodal. We propose instead incremental mixture importance sampling (IMIS), which iteratively builds up a better importance sampling function. This retains the simplicity and transparency of sampling importance resampling, but is much more efficient computationally. It also leads to a simple estimator of the integrated likelihood that is the basis for Bayesian model comparison and model averaging. In simulation experiments and on real data, it outperformed both sampling importance resampling and three publicly available generic Markov chain Monte Carlo algorithms for this kind of problem.
Testing for the existence of clusters
"... Detecting and determining clusters present in a certain sample has been an important concern, among researchers from different fields, for a long time. In particular, assessing whether the clusters are statistically significant, is a question that has been asked by a number of experimenters. Recentl ..."
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Detecting and determining clusters present in a certain sample has been an important concern, among researchers from different fields, for a long time. In particular, assessing whether the clusters are statistically significant, is a question that has been asked by a number of experimenters. Recently, this question arose again in a study in maize genetics, where determining the significance of clusters is crucial as a primary step in the identification of a genomewide collection of mutants that may affect the kernel composition. Although several efforts have been made in this direction, not much has been done with the aim of developing an actual hypothesis test in order to assess the significance of clusters. In this paper, we propose a new methodology that allows the examination of the hypothesis test H 0: κ=1 vs. H 1:κ=k, whereκdenotes the number of clusters present in a certain population. Our procedure, based on Bayesian tools, permits us to obtain closed form expressions for the posterior probabilities corresponding to the null hypothesis. From here, we calibrate our results by estimating the frequentist null distribution of the posterior probabilities in order to obtain the pvalues associated with the observed posterior probabilities. In most cases, actual evaluation of the posterior probabilities is computationally intensive and several algorithms have been discussed in the literature. Here, we propose a simple estimation procedure, based on MCMC techniques, that permits an efficient and easily implementable evaluation of the test. Finally, we present simulation studies that support our conclusions, and we apply our method to the analysis of NIR spectroscopy data coming from the genetic study that motivated this work. MSC:?????????