Judith Rousseau, Clara Grazian, Jeong Eun Lee · 2019
This chapter presents some aspects of Bayesian inference in the context of mixture models. It describes the asymptotic behaviour of the corresponding posterior distributions and explores proposals to construct non-informative or vaguely informative priors in mixture models with a known number of components. The chapter analyses the mixture models can be used for density estimation, for classification or clustering, or for parameter estimation. It also describes the asymptotic behaviour of the posterior distribution in terms of the marginal density of the observations, which is a starting point of most asymptotic analysis. The chapter provides some general results on the concentration of the posterior distribution around the true marginal density of the observations. Then all the usual asymptotic results are valid: namely, the Bernstein–von Mises theorem on the parameters, the Laplace approximation of the marginal density of the observations and 1/n convergence rates of Bayesian estimators such as posterior means.
Cited by 3
Currie, Christine Susan Mary · ePrints Soton (University of Southampton) · 2004
<p>This thesis describes the use of sampling methods in two applications: an epidemic model of tuberculosis (TB) and HIV, and the estimation of the number of components in finite normal mixture models. We use Bayesian statistics for the analysis, which enables us to take into account prior information about parameter values in the case of the epidemic modelling, and smooths the likelihood function when considering finite mixture models. The convergence properties of importance sampling are investigated and methods for diagnosing non-convergence of importance sampling are discussed. We use importance sampling to analyse finite normal mixture models and Marlov Chain Monte Carlo sampling to fit the epidemic model. Results for effectiveness and cost-effectiveness of different interventions against TB and HIV are presented.</p>
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Hisashi Noma, Shigeyuki Matsui, Takashi Omori et al. · Biostatistics · 2009
The main purpose of microarray studies is screening to identify differentially expressed genes as candidates for further investigation. Because of limited resources in this stage, prioritizing or ranking genes is a relevant statistical task in microarray studies. In this article, we develop 3 empirical Bayes methods for gene ranking on the basis of differential expression, using hierarchical mixture models. These methods are based on (i) minimizing mean squared errors of estimation for parameters, (ii) minimizing mean squared errors of estimation for ranks of parameters, and (iii) maximizing sensitivity in selecting prespecified numbers of differential genes, with the largest effect. Our methods incorporate the mixture structures of differential and nondifferential components in empirical Bayes models to allow information borrowing across differential genes, with separation from nuisance, nondifferential genes. The accuracy of our ranking methods is compared with that of conventional methods through simulation studies. An application to a clinical study for breast cancer is provided.
Cited by 20Open access
Walkiria Maria de Oliveira Macerau · LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) · 2023
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
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Paul Amouroux (6475520), Didier Crochard (378225), Margarita Correa (5688806) et al. · Figshare · 2019
<p>The majority-rule consensus tree was calculated from the Bayesian analysis. Bayesian posterior probabilities are shown beyond the nodes (values below 70% are not shown). Taxa are named according to their ABGD group, haplotype code and taxonomic names (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0205475#pone.0205475.t002" target="_blank">Table 2</a>).</p>
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Lynch, Miranda, Thurston, Sally W. · UR Research (University of Rochester)
Thesis (Ph.D.)--University of Rochester. School of Medicine & Dentistry. Dept. of Biostatistics and Computational Biology, 2010.
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W. Holmes Finch, Brian F. French · Journal of Modern Applied Statistical Methods · 2012
The Mixture Item Response Theory (MixIRT) can be used to identify latent classes of examinees in data as well as to estimate item parameters such as difficulty and discrimination for each of the groups. Parameter estimation via maximum likelihood (MLE) and Bayesian estimation based on the Markov Chain Monte Carlo (MCMC) are compared for classification accuracy and parameter estimation bias for difficulty and discrimination. Standard error magnitude and coverage rates were compared across number of items, number of latent groups, group size ratio, total sample size and underlying item response model. Results show that MCMC provides more accurate group membership recovery across conditions and more accurate parameter estimates for smaller samples and fewer items. MLE produces narrower confidence intervals than MCMC and more accurate parameter estimates for larger samples and more items. Implications of these results for research and practice are discussed.
Cited by 35Open access
Eleftherios Meletis, Juan Carlos Arango‐Sabogal, J. Trenton McClure et al. · Journal of Dairy Science · 2025
Ensuring high-quality colostrum for newborn calves is essential for their health and future productivity. We applied Bayesian finite mixture models to estimate optimal cutoff values and evaluate the diagnostic accuracy of 3 methods-radial immunodiffusion (RID) assay, transmission infrared (TIR) spectroscopy, and digital Brix (dBrix) refractometry-measured on a continuous scale for assessing bovine colostrum quality, using 591 colostrum samples from 42 Holstein dairy herds in Atlantic Canada. The mean and standard deviation of IgG concentrations for high-quality colostrum were 61.07 ± 39.8 g/L, 51.28 ± 27.38 g/L, and 24.32 ± 4.13% Brix for RID assay, TIR spectroscopy, and dBrix refractometry, respectively, compared with 19.93 ± 15.54 g/L, 7.78 ± 37.4 g/L, and 15.87 ± 3.45% Brix for low-quality samples. The prevalence of high-quality colostrum was estimated at 83% (95% credible interval [CrI]: 0.79-0.88). The dBrix refractometer demonstrated the highest discriminatory power, with an area under the curve (AUC) of 0.94 (95% CrI: 0.91-0.97), followed by RID assay (AUC: 0.92; 95% CrI: 0.88-0.96) and TIR spectroscopy (AUC: 0.82; 95% CrI: 0.76-0.88). Optimal cutoff values were determined using Youden's index: 34.15 g/L for RID assay (sensitivity [Se] = 0.86, specificity [Sp] = 0.83), 22.74 g/L for TIR spectroscopy (Se = 0.88, Sp = 0.66), and 19.62% Brix for dBrix refractometry (Se = 0.90, Sp = 0.85). Correlation between RID assay and TIR spectroscopy was stronger for high-quality colostrum samples (0.80; 95% CrI: 0.77-0.84) than for low-quality samples (0.36; 95% CrI: 0.16-0.55), indicating that these methods are not perfectly correlated and justifying the need for multiple diagnostic approaches. Among individual methods, dBrix refractometry showed the highest positive predictive value (PPV = 1.00), and all methods demonstrated moderate negative predictive values (NPV = 0.46-0.57). Combining methods in series interpretation increased PPV up to 1.00 when all 3 methods were used together, though with reduced NPV. Conversely, parallel interpretation substantially improved NPV, reaching 0.98 when all 3 methods were combined. By modeling continuous measurements instead of dichotomized test results, our analysis produced refined cutoff values for assessing colostrum quality. The findings indicate that existing thresholds remain largely adequate, offering only minor performance improvements, and emphasize the need to balance diagnostic refinements with their potential effects on calf management and passive immunity. Furthermore, our findings suggest that, although individual assessment methods offer valuable diagnostic information, combining multiple methods can optimize either Se or Sp, depending on the interpretation approach, thereby further enhancing the accuracy of colostrum quality evaluation.
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Porteous, Ian
This study brings together Bayesian networks, topic models, hierarchical Bayes modeling and nonparametric Bayesian methods to build a framework for efficiently designing and implementing a family of (non)parametric Bayesian mixture models. Bayesian mixture models, including Bayesian topic models, have shown themselves to be a useful tool for modeling and discovering latent structure in a number of domains. We introduce a modeling framework, networks of mixture blocks, that brings together these developments in a way that facilitates the definition and implementation of complex (non)parametric Bayesian networks for data with partitioned structure. Networks of mixture blocks can be viewed as Bayesian networks that have been factored into a network of sub-models, mixture blocks, which are conditionally independent of each other given the introduction of auxiliary partition variables. We use this framework to develop several novel nonparametric Bayesian models for collaborative filtering and text modeling.
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Lin Lin, Boris P. Hejblum · Wiley Interdisciplinary Reviews Computational Statistics · 2020
Abstract Bayesian mixture models are increasingly used for model‐based clustering and the follow‐up analysis on the clusters identified. As such, they are of particular interest for analyzing cytometry data where unsupervised clustering and association studies are often part of the scientific questions. Cytometry data are large quantitative data measured in a multidimensional space that typically ranges from a few dimensions to several dozens, and which keeps increasing due to innovative high‐throughput biotechonologies. We present several recent parametric and nonparametric Bayesian mixture modeling approaches, and describe advantages and limitations of these models under different research context for cytometry data analysis. We also acknowledge current computational challenges associated with the use of Bayesian mixture models for analyzing cytometry data, and we draw attention to recent developments in advanced numerical algorithms for estimating large Bayesian mixture models, which we believe have the potential to make Bayesian mixture model more applicable to new types of single‐cell data with higher dimensions. This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Knowledge Discovery Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory
Cited by 5Open access
Jack D. Center · AIP conference proceedings · 2001
This paper relates Bayesian mixture-model classifiers to other popular pattern classification algorithms including Parzen kernel, radial-basis-function neural network, and support vector machine algorithms. It compares both the training and operation modes of the different algorithms. It shows that the models underlying the other methods can be viewed as subsets of mixture models. In particular, it shows that support vector machine methods can be used to establish starting points for Bayesian mixture model training methods.
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Carcinogen Mixture Experiment, Jing-shiang Hwang, Tzuu-huei Ueng · 2001
this paper, we propose trinomial/lognormal mixture models for modeling two dichotomous and one continuous outcomes simultaneously. Monte Carlo methods based on Markov chains are used for the models which are built in a Bayesian framework. Bayesian inferences on odds of these multiple events and also the survivorship of these events for the CA mixture exposure groups are presented. Key words: Trinomial/lognormal mixture; Carcinogen mixture; MCMC 1
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Shipeng Yu, Kai Yu, Volker Tresp et al. · 2005
Abstract. This paper studies a Bayesian framework for density modeling with mixture of exponential family distributions. Variational Bayesian Dirichlet-Multinomial allocation (VBDMA) is introduced, which performs inference and learning efficiently using variational Bayesian methods and performs automatic model selection. The model is closely related to Dirichlet process mixture models and demonstrates similar automatic model selection in the variational Bayesian context. 1
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Lipovec, Thomas · reposiTUm (TU Wien) · 2023
Bayesian nonparametric models have emerged as a flexible tool for learning patterns and structures in complex data. A well-known Bayesian nonparametric model is the Dirichlet process mixture (DPM) model. The DPM model extends the Bayesian mixture model with a finite number of mixture components to a Bayesian mixture model with a countably infinite number of mixture components while permitting the use of efficient Bayesian inference methods. Thereby, it becomes possible to cluster data and estimate unknown parameters without specifying the number of clusters and unknown parameters a priori. Practical Bayesian inference methods for DPM models include Markov chain Monte Carlo (MCMC) and variational inference (VI) methods. In this thesis, we focus on the VI methodology and provide a detailed derivation of the coordinate ascent variational inference (CAVI) algorithm for DPM models. Subsequently, we apply the CAVI algorithm to a Gaussian estimation problem that involves noisy observations of object features modeled by a DPM with Gaussian mixture components. We present simulation results that compare the improvement of the estimation accuracy of the object features due to clustering achieved by our CAVI algorithm and by a previously proposed MCMC algorithm. Our simulation results demonstrate to which extent we have to sacrifice estimation accuracy when opting for the less accurate but more efficient CAVI algorithm as opposed to the MCMC algorithm.
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Abdelbaset A.Sh Abdalla, Nasir Ibrahim Elmesmari, Mohammed M. Elnazzal · Zenodo (CERN European Organization for Nuclear Research) · 2024
Mixture distributions are widely used to model data with distinct groups, providing a flexible approach to estimating density. However, Bayesian approaches for mixture models pose challenges, such as label switching in the Gibbs sampler output due to the non-identifiability of component parameters. We review advanced methods for Bayesian analysis, including the Markov chain Monte Carlo (MCMC) reversible jump algorithm and model comparison based on joint measures of fit and complexity. We also present a Bayesian regression model based on a two-component mixture model, implemented using the Gibbs sampler algorithm and applied to a dataset of time measurement differences between two clocks. Our theoretical investigation highlights the importance of latent variables in implementing the Bayesian normal mixture model with two components. When applied to the dataset, our model effectively assigned probabilities to the two states of the phenomenon under study and identified two processes with identical slopes, intercepts, and variances. Our findings demonstrate the power of Bayesian mixture models in uncovering hidden structures within complex datasets. In general, our review and application provide insight into the challenges and potential solutions for Bayesian mixture modeling and highlight the usefulness of these methods in various fields.
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A.A. Abd-Alla, Nasir Ibrahim Elmesmari, Mohammed Elnazzal · World Journal of Advanced Research and Reviews · 2024
Mixture distributions are widely used to model data with distinct groups, providing a flexible approach to estimating density. However, Bayesian approaches for mixture models pose challenges, such as label switching in the Gibbs sampler output due to the non-identifiability of component parameters. We review advanced methods for Bayesian analysis, including the Markov chain Monte Carlo (MCMC) reversible jump algorithm and model comparison based on joint measures of fit and complexity. We also present a Bayesian regression model based on a two-component mixture model, implemented using the Gibbs sampler algorithm and applied to a dataset of time measurement differences between two clocks. Our theoretical investigation highlights the importance of latent variables in implementing the Bayesian normal mixture model with two components. When applied to the dataset, our model effectively assigned probabilities to the two states of the phenomenon under study and identified two processes with identical slopes, intercepts, and variances. Our findings demonstrate the power of Bayesian mixture models in uncovering hidden structures within complex datasets. In general, our review and application provide insight into the challenges and potential solutions for Bayesian mixture modeling and highlight the usefulness of these methods in various fields.
Cited by 0Open access
Holmes, Chris, Stephens, David, Jasra, Ajay · OpenGrey (Institut de l'Information Scientifique et Technique) · 2006
In the past fifteen years there has been a dramatic increase of interest in Bayesian mixture models. This is because we are now able to apply Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) methods. This thesis is concerned with Bayesian mixture modelling via such approaches. The following topics are considered. Firstly, an important problem with Bayesian mixture models is that of label switch- ing. This is caused by the nonidentifiability of the components under symmetric priors. Therefore, Monte Carlo estimates of component specific quantities will be identical and thus useless for inference. We review the existing solutions to the label switching problem and develop a probabilistic method to deal with it. Secondly, another difficulty is actually simulating from trans-dimensional measures induced by mixtures. To solve this problem, we present an extension of population-based MCMC. We provide a result related to the uniform ergodicity of a population Markov kernel. We also give an example of a population algorithm for simulating from a Bayesian multivariate mixture model. Thirdly, a problem for SMC samplers (Del Moral et al.; 2005) is that of low particle diversity for multimodal targets. In such situations, even under reasonable Markov kernels, poor Monte Carlo estimates may be derived. We present an interacting SMC method which seeks to maintain a diverse set of particles. We establish some convergence results. Additionally, we show how the methodology may be used for a Bayesian mixture model and, using adaptive methods, a mixture problem in population genetics. Finally, we develop statistical methodology for geochronological data. The main characteristics of geochronological data are heterogeneity and measurement error. The former property means that mixture models are appropriate for its analysis. We establish that current methods for analyzing such data are not always suitable. Therefore, we develop Bayesian mixture models for this task.
Cited by 0
Ajay Jasra · Spiral (Imperial College London) · 2006
In the past fifteen years there has been a dramatic increase of interest in Bayesian mixture models. This is because we are now able to apply Markov chain Monte Carlo (MCMC) and sequential Monte Carlo (SMC) methods. This thesis is concerned with Bayesian mixture modelling via such approaches. The following topics are considered. Firstly, an important problem with Bayesian mixture models is that of label switch- ing. This is caused by the nonidentifiability of the components under symmetric priors. Therefore, Monte Carlo estimates of component specific quantities will be identical and thus useless for inference. We review the existing solutions to the label switching problem and develop a probabilistic method to deal with it. Secondly, another difficulty is actually simulating from trans-dimensional measures induced by mixtures. To solve this problem, we present an extension of population-based MCMC. We provide a result related to the uniform ergodicity of a population Markov kernel. We also give an example of a population algorithm for simulating from a Bayesian multivariate mixture model. Thirdly, a problem for SMC samplers (Del Moral et al.; 2005) is that of low particle diversity for multimodal targets. In such situations, even under reasonable Markov kernels, poor Monte Carlo estimates may be derived. We present an interacting SMC method which seeks to maintain a diverse set of particles. We establish some convergence results. Additionally, we show how the methodology may be used for a Bayesian mixture model and, using adaptive methods, a mixture problem in population genetics. Finally, we develop statistical methodology for geochronological data. The main characteristics of geochronological data are heterogeneity and measurement error. The former property means that mixture models are appropriate for its analysis. We establish that current methods for analyzing such data are not always suitable. Therefore, we develop Bayesian mixture models for this task.
Cited by 6Open access
The Tien Mai, Zhi Zhao · 2026
Methods for maximum likelihood and Bayesian estimation for the Wishart mixture model and the mixture-of-experts Wishart (MoE-Wishart) model. The package provides four inference algorithms for these models, each implemented using the expectation–maximization (EM) algorithm for maximum likelihood estimation and a fully Bayesian approach via Gibbs-within-Metropolis–Hastings sampling.
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Merrilee Hurn, Ana Justel, Christian P. Robert · Journal of Computational and Graphical Statistics · 2003
AbstractThis article shows how Bayesian inference for switching regression models and their generalizations can be achieved by the specification of loss functions which overcome the label switching problem common to all mixture models. We also derive an extension to models where the number of components in the mixture is unknown, based on the birthand-death technique developed in recent literature. The methods are illustrated on various real datasets.Key Words: Bayesian inferenceBirth-and-death processLabel switchingLogistic regressionLoss functionsMCMC algorithmsPoisson regressionSwitching regression
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