Bayesian Analysis Of Mixtures Research Articles

Bayesian Mixture Analysis of a global database to improve unit weight prediction from CPTu

Empirical correlations between different geotechnical parameters are frequently sought after by exploiting multivariate databases. An example are estimations of total unit weight from cone penetration tests (CPTu), which are very useful in earlier design stages. If the underlying soil database includes data from many different soils a single correlation may lack precision. Precision is gained when the underlying database is narrowed down to some specific soil type, but the applicability of a soil-specific correlation is also limited. A way out of this dilemma is to apply clustering techniques to a general database before developing separate correlations for different clusters. Projecting the clustered data back to a convenient classification space (e.g., one spanned by normalized CPTu metrics) new data can be easily assigned to different clusters and the appropriate correlation used. This idea is illustrated here using Bayesian Mixture Analysis (BMA) to identify hidden soil classes within a general geotechnical database that supports correlations between soil total unit weight and CPTu readings. It is shown that BMA supported clustering improves the accuracy of previous regressions, and, more importantly, facilitates the formulation of novel and more accurate regressions. A simple discriminant criterion is developed to facilitate application of cluster-based regressions to new sites. The good performance of the method is illustrated with application to a deltaic site.

Bayesian Mixture Analysis of a Global Database to Improve Unit Weight Prediction from Cptu

Bayesian Mixture Analysis of a Global Database to Improve Unit Weight Prediction from Cptu

Bayesian MISE convergence rates of Polya urn based density estimators: asymptotic comparisons and choice of prior parameters

ABSTRACT Bhattacharya [Gibbs sampling based Bayesian analysis of mixtures with unknown number of components. Sankhya B. 2008;70:133–155] introduced a mixture model based on the Dirichlet process, where an upper bound on the unknown number of components is to be specified. Defining a Bayesian analogue of the mean integrated squared error (Bayesian MISE), here we consider a Bayesian asymptotic density estimation framework forobjectively specifying the upper bound, as well as the precision parameter of the Dirichlet process, such that the Bayesian MISE converges at a desired rate. As a byproduct of our approach, we also investigate Bayesian MISE convergence rate of the traditional Dirichlet process mixture model, which leads to asymptotic specification of the precision parameter. Various asymptotic issues related to the two aforementioned mixtures, including comparative performances, are also investigated. The theoretical studies, supplemented with simulation experiments, bring out the superiority of the approach of Bhattacharya (2008).

Mixing rates in weakly differentiated stocks of greater amberjack (Seriola dumerili) in the Gulf of Mexico.

The greater amberjack (Seriola dumerili) is a commercially and recreationally important marine fish species in the southeastern United States, where it has been historically managed as two non-mixing stocks (Gulf of Mexico and Atlantic). Mark-recapture studies and analysis of mitochondrial DNA have suggested the two stocks are demographically independent; however, little is currently known about when and where spawning occurs in Gulf of Mexico amberjack, and whether stock mixture occurs on breeding grounds. The primary objective of this study was to quantify stock mixture among breeding populations of amberjack collected from the Atlantic and Gulf of Mexico. Genetic data based on 11 loci identified very low, though statistically significant differentiation among Gulf of Mexico samples (GST=0.007, [Formula: see text]=0.009; all P = 0.001) and between reproductive adults collected from two spawning areas (GST=0.007, [Formula: see text]=0.014; all P = 0.001). Naïve Bayesian mixture analysis supported a single genetic cluster [p(S|data) = 0.734] whereas trained clustering (using Atlantic and Gulf spawning fish) gave the highest support to a two-cluster model (p(S|data) = 1.0). Our results support the argument that the genetic structuring of greater amberjack is more complex than the previously assumed two, non-mixing stock model. Although our data provide evidence of limited population structure, we argue in favour of non-panmixia among reproductive fish collected from the Gulf of Mexico and Florida Keys.

A class of improved informative priors for bayesian analysis of two-component mixture of failure time distributions from doubly censored data

The Bayesian analysis of the mixture models has received a sizable attention of the analysts during recent years. However, most of the contributions have been discussed under singly type I censored samples. This paper aims to discuss the Bayesian analysis of the two-component mixture of lifetime distribution, with a particular case for Rayleigh distribution, under doubly censored samples. A class of improved informative priors has been assumed for posterior estimation. The squared error and k-loss functions have been proposed to derive the Bayes estimators and the corresponding posterior risks. The prior elicitation has been discussed via prior predictive approach. The comparisons among the performance of different estimators have been made in terms of posterior risks based on analysis of simulated and real life data sets.

Cone penetration data classification with Bayesian Mixture Analysis

ABSTRACTThis paper presents an application of the Bayesian Mixture Analysis (BMA) to deal with the classification of spatially variable soil data from the cone penetration test. The cone penetration data classification postulates a problem where a set of cone penetration measurements is used to identify “hidden or unobserved” soil classes. The problem is formulated as an incomplete-data Gaussian mixture where the observed data are defined by the natural logarithm-transformed values of the normalized friction and the normalized cone resistance, while the soil classes to be identified are considered as hidden data. The solution for the incomplete-data problem which consists of class-dependent mixture probabilities and Gaussian distribution parameters is defined in a Bayesian framework. The implementation of conjugate priors for the Gaussian mixtures enables an efficient sampling of the posterior parameters by the Gibbs algorithm of the Markov Chain Monte Carlo method. When compared to the well-established Robertson classification charts, the BMA formulation has an advantage due to the Bayesian framework which enables the definition of soil classes through mixture priors, class-dependent posterior parameter estimates, and a probabilistic soil classification. The presented approach is applied to the cone penetration data from the Sheringham Shoal Offshore Wind Farm site.

Bayesian mixture analysis for metagenomic community profiling.

Motivation: Deep sequencing of clinical samples is now an established tool for the detection of infectious pathogens, with direct medical applications. The large amount of data generated produces an opportunity to detect species even at very low levels, provided that computational tools can effectively profile the relevant metagenomic communities. Data interpretation is complicated by the fact that short sequencing reads can match multiple organisms and by the lack of completeness of existing databases, in particular for viral pathogens. Here we present metaMix, a Bayesian mixture model framework for resolving complex metagenomic mixtures. We show that the use of parallel Monte Carlo Markov chains for the exploration of the species space enables the identification of the set of species most likely to contribute to the mixture. Results: We demonstrate the greater accuracy of metaMix compared with relevant methods, particularly for profiling complex communities consisting of several related species. We designed metaMix specifically for the analysis of deep transcriptome sequencing datasets, with a focus on viral pathogen detection; however, the principles are generally applicable to all types of metagenomic mixtures. Availability and implementation: metaMix is implemented as a user friendly R package, freely available on CRAN: http://cran.r-project.org/web/packages/metaMix Contact: sofia.morfopoulou.10@ucl.ac.uk Supplementary information: Supplementary data are available at Bionformatics online.

Bayesian Analysis of Mixture of Normal Factor Analysis Model

Factor analysis model belongs to a family of linear models and because the multivariate data sets do not vacate outliers . For this reason , this paper is concerned with Bayesian analysis of mixed normal factor analysis model . The mixture normal distribution treats the observations of a sample as they are taken from two different populations and none from single population . The model is analyzed when the common factor is (stochastic and non-stochastic) variable . We suppose that all the parameters of the model are unknown , some of them have prior distributions belong to the conjugate family.

A Simple Solution to Bayesian Mixture Labeling

The label-switching problem is one of the fundamental problems in Bayesian mixture analysis. Using all the Markov chain Monte Carlo samples as the initials for the expectation-maximization (EM) algorithm, we propose to label the samples based on the modes they converge to. Our method is based on the assumption that the samples converged to the same mode have the same labels. If a relative noninformative prior is used or the sample size is large, the posterior will be close to the likelihood and then the posterior modes can be located approximately by the EM algorithm for mixture likelihood, without assuming the availability of the closed form of the posterior. In order to speed up the computation of this labeling method, we also propose to first cluster the samples by K-means with a large number of clusters K. Then, by assuming that the samples within each cluster have the same labels, we only need to find one converged mode for each cluster. Using a Monte Carlo simulation study and a real dataset, we demonstrate the success of our new method in dealing with the label-switching problem.

Bayesian analysis of mixtures in structural equation models with non‐ignorable missing data

Structural equation models (SEMs) have become widely used to determine the interrelationships between latent and observed variables in social, psychological, and behavioural sciences. As heterogeneous data are very common in practical research in these fields, the analysis of mixture models has received a lot of attention in the literature. An important issue in the analysis of mixture SEMs is the presence of missing data, in particular of data missing with a non-ignorable mechanism. However, only a limited amount of work has been done in analysing mixture SEMs with non-ignorable missing data. The main objective of this paper is to develop a Bayesian approach for analysing mixture SEMs with an unknown number of components and non-ignorable missing data. A simulation study shows that Bayesian estimates obtained by the proposed Markov chain Monte Carlo methods are accurate and the Bayes factor computed via a path sampling procedure is useful for identifying the correct number of components, selecting an appropriate missingness mechanism, and investigating various effects of latent variables in the mixture SEMs. A real data set on a study of job satisfaction is used to demonstrate the methodology.

An Artificial Allocations Based Solution to the Label Switching Problem in Bayesian Analysis of Mixtures of Distributions

Label switching is a well-known problem occurring in MCMC outputs in Bayesian mixture modeling. In this article we propose a formal solution to this problem by considering the space of the artificial allocation variables. We show that there exist certain subsets of the allocation space leading to a class of nonsymmetric distributions that have the same support with the symmetric posterior distribution and can reproduce it by simply permuting the labels. Moreover, we select one of these distributions as a solution to the label switching problem using the simple matching distance between the artificial allocation variables. The proposed algorithm can be used in any mixture model and its computational cost depends on the length of the simulated chain but not on the parameter space dimension. Real and simulated data examples are provided in both univariate and multivariate settings. Supplemental material for this article is available online.

Flexible modelling of random effects in linear mixed models—A Bayesian approach

Flexible modelling of random effects in linear mixed models has attracted some attention recently. In this paper, we propose the use of finite Gaussian mixtures as in Verbeke and Lesaffre [A linear mixed model with heterogeneity in the random-effects population, J. Amu. Statist. Assoc. 91, 217–221]. We adopt a fully Bayesian hierarchical framework that allows simultaneous estimation of the number of mixture components together with other model parameters. The technique employed is the Reversible Jump MCMC algorithm (Richardson and Green [On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion). J. Roy. Statist. Soc. Ser. B 59, 731–792]). This approach has the advantage of producing a direct comparison of different mixture models through posterior probabilities from a single run of the MCMC algorithm. Moreover, the Bayesian setting allows us to integrate over different mixture models to obtain a more robust density estimate of the random effects. We focus on linear mixed models with a random intercept and a random slope. Numerical results on simulated data sets and a real data set are provided to demonstrate the usefulness of the proposed method.

Bayesian analysis of finite mixtures of multinomial and negative-multinomial distributions

The Bayesian implementation of finite mixtures of distributions has been an area of considerable interest within the literature. Computational advances on approximation techniques such as Markov chain Monte Carlo (MCMC) methods have been a keystone to Bayesian analysis of mixture models. This paper deals with the Bayesian analysis of finite mixtures of two particular types of multidimensional distributions: the multinomial and the negative-multinomial ones. A unified framework addressing the main topics in a Bayesian analysis is developed for the case with a known number of component distributions. In particular, theoretical results and algorithms to solve the label-switching problem are provided. An illustrative example is presented to show that the proposed techniques are easily applied in practice.

Variational approximations in Bayesian model selection for finite mixture distributions

Variational methods, which have become popular in the neural computing/machine learning literature, are applied to the Bayesian analysis of mixtures of Gaussian distributions. It is also shown how the deviance information criterion, (DIC), can be extended to these types of model by exploiting the use of variational approximations. The use of variational methods for model selection and the calculation of a DIC are illustrated with real and simulated data. The variational approach allows the simultaneous estimation of the component parameters and the model complexity. It is found that initial selection of a large number of components results in superfluous components being eliminated as the method converges to a solution. This corresponds to an automatic choice of model complexity. The appropriateness of this is reflected in the DIC values.

Bayesian analysis of mixture of autoregressive components with an application to financial market volatility

In this paper, we present a fully Bayesian analysis of a finite mixture of autoregressive components. Neither the number of mixture components nor the autoregressive order of each component have to be fixed, since we treat them as stochastic variables. Parameter estimation and model selection are performed using Markov chain Monte Carlo methods. This analysis allows us to take into account the stationarity conditions on the model parameters, which are often ignored by Bayesian approaches. Finally, the application to return volatility of financial markets will be illustrated. Our model seems to be consistent with some empirical facts concerning volatility such as persistence, clustering effects, nonsymmetrical dependencies. Copyright © 2006 John Wiley & Sons, Ltd.

Likelihood and Bayesian analysis of mixtures

This paper compares likelihood and Bayesian analyses of finite mixture distributions, and expresses reservations about the latter. In particular, the role of prior assumptions in the full Monte Carlo Markov chain Bayes analysis is obscure, yet these assumptions clearly play a major role in the conclusions. These issues are illustrated with a detailed discussion of the well-known galaxy data.

Likelihood and Bayesian analysis of mixtures

Likelihood and Bayesian analysis of mixtures

Corrigendum: On Bayesian Analysis of Mixtures With An Unknown Number of Components

We regret to report that there is an error in equation (12) on p. 740, concerning the birth± death move for empty components. The correct expression is A ˆ p k‡ 1† p k† 1 B k , † w y1 j* 1y wj*† yk k‡ 1† dk‡1 k0 ‡ 1†bk 1 g1,k wj*† 1y wj*†: In the paper as printed, the power of 1y wj* in the ®nal factor, the Jacobian term, was given as k instead of ky 1. The source of the error was neglect of the condition j wj ˆ 1 in computing the partial derivatives of wj with respect to wj. Note that expression (11) for the split±combine move acceptance ratio is correct as printed. Having made the correction, we repeated the calculations leading to all the numerical results reported in the paper. As might be expected, the e€ects of the error are noticeable but small. The maximum changes to any of the posterior probabilities p kjy† presented in Tables 1 and 2 and Fig. 6 are 0.015, 0.011 and 0.020 respectively; in each case the maximum discrepancy occurs near the mode of the distributions, and so has little impact. The error in Fig. 2 is within plotting accuracy. In none of the ®gures is the visual impression altered, and none of our qualitative conclusions are a€ected. We are grateful to Tobias Ryden of Lund University for discovering this error, and we apologize for any confusion that it has caused.

On Bayesian Analysis of Mixtures with an Unknown Number of Components (with discussion)

Summary New methodology for fully Bayesian mixture analysis is developed, making use of reversible jump Markov chain Monte Carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture. A sample from the full joint distribution of all unknown variables is thereby generated, and this can be used as a basis for a thorough presentation of many aspects of the posterior distribution. The methodology is applied here to the analysis of univariate normal mixtures, using a hierarchical prior model that offers an approach to dealing with weak prior information while avoiding the mathematical pitfalls of using improper priors in the mixture context.

Bayesian analysis of mixtures applied to post-synaptic potential fluctuations

Bayesian inference techniques have been applied to the analysis of fluctuation of post-synaptic potentials in the hippocampus. The underlying statistical model assumes that the varying synaptic signals are characterized by mixtures of (unknown) numbers of individual gaussian, or normal, component distributions. Each solution consists of a group of individual components with unique mean values and relative probabilities of occurrence and a predictive probability density. The advantages of bayesian inference techniques over the alternative method of maximum likelihood estimation (MLE) of the parameters of an unknown mixture distribution include the following: (1) prior information may be incorporated in the estimation of model parameters; (2) conditional probability estimates of the number of individual components in the mixture are calculated; (3) flexibility exists in the extent to which the estimated noise standard deviation indicates the width of each component; (4) posterior distributions for component means are calculated, including measures of uncertainty about the means; and (5) probability density functions of the component distributions and the overall mixture distribution are estimated in relation to the raw grouped data, together with measures of uncertainty about these estimates. This expository report describes this novel approach to the unconstrained identification of components within a mixture, and provides demonstration of the usefulness of the technique in the context of both simulations and the analysis of distributions of synaptic potential signals.