Clustering time-course microarray data using functional Bayesian infinite mixture model

This paper presents a new Bayesian, infinite mixture model based, clustering approach, specifically designed for time-course microarray data. The problem is to group together genes which have “similar” expression profiles, given the set of noisy measurements of their expression levels over a specific time interval. In order to capture temporal variations of each curve, a non-parametric regression approach is used. Each expression profile is expanded over a set of basis functions and the sets of coefficients of each curve are subsequently modeled through a Bayesian infinite mixture of Gaussian distributions. Therefore, the task of finding clusters of genes with similar expression profiles is then reduced to the problem of grouping together genes whose coefficients are sampled from the same distribution in the mixture. Dirichlet processes prior is naturally employed in such kinds of models, since it allows one to deal automatically with the uncertainty about the number of clusters. The posterior inference is carried out by a split and merge MCMC sampling scheme which integrates out parameters of the component distributions and updates only the latent vector of the cluster membership. The final configuration is obtained via the maximum a posteriori estimator. The performance of the method is studied using synthetic and real microarray data and is compared with the performances of competitive techniques.     

Clustering Gene Expression Data using a Posterior Split‐Merge‐Birth Procedure

Abstract. DNA array technology is an important tool for genomic research due to its capa‐city of measuring simultaneously the expression levels of a great number of genes or fragments of genes in different experimental conditions. An important point in gene expression data analysis is to identify clusters of genes which present similar expression levels. We propose a new procedure for estimating the mixture model for clustering of gene expression data. The proposed method is a posterior split‐merge‐birth MCMC procedure which does not require the specification of the number of components, since it is estimated jointly with component parameters. The strategy for splitting is based on data and on posterior distribution from the previously allocated observations. This procedure defines a quick split proposal in contrary to other split procedures, which require substantial computational effort. The performance of the method is verified using real and simulated datasets.     

Stochastic volatility modelling in continuous time with general marginal distributions: Inference,prediction and model selection

We compare results for stochastic volatility models where the underlying volatility process having generalized inverse Gaussian (GIG) and tempered stable marginal laws. We use a continuous time stochastic volatility model where the volatility follows an Ornstein–Uhlenbeck stochastic differential equation driven by a Lévy process. A model for long-range dependence is also considered, its merit and practical relevance discussed. We find that the full GIG and a special case, the inverse gamma, marginal distributions accurately fit real data. Inference is carried out in a Bayesian framework, with computation using Markov chain Monte Carlo (MCMC). We develop an MCMC algorithm that can be used for a general marginal model.     

Relabelling in Bayesian mixture models by pivotal units

Label switching is a well-known and fundamental problem in Bayesian estimation of finite mixture models. It arises when exploring complex posterior distributions by Markov Chain Monte Carlo (MCMC) algorithms, because the likelihood of the model is invariant to the relabelling of mixture components. If the MCMC sampler randomly switches labels, then it is unsuitable for exploring the posterior distributions for component-related parameters. In this paper, a new procedure based on the post-MCMC relabelling of the chains is proposed. The main idea of the method is to perform a clustering technique on the similarity matrix, obtained through the MCMC sample, whose elements are the probabilities that any two units in the observed sample are drawn from the same component. Although it cannot be generalized to any situation, it may be handy in many applications because of its simplicity and very low computational burden.     

A sequential clustering algorithm with applications to gene expression data

Clustering algorithms are used in the analysis of gene expression data to identify groups of genes with similar expression patterns. These algorithms group genes with respect to a predefined dissimilarity measure without using any prior classification of the data. Most of the clustering algorithms require the number of clusters as input, and all the objects in the dataset are usually assigned to one of the clusters. We propose a clustering algorithm that finds clusters sequentially, and allows for sporadic objects, so there are objects that are not assigned to any cluster. The proposed sequential clustering algorithm has two steps. First it finds candidates for centers of clusters. Multiple candidates are used to make the search for clusters more efficient. Secondly, it conducts a local search around the candidate centers to find the set of objects that defines a cluster. The candidate clusters are compared using a predefined score, the best cluster is removed from data, and the procedure is repeated. We investigate the performance of this algorithm using simulated data and we apply this method to analyze gene expression profiles in a study on the plasticity of the dendritic cells.     

An online Bayesian mixture labelling method by minimizing deviance of classification probabilities to reference labels

Solving label switching is crucial for interpreting the results of fitting Bayesian mixture models. The label switching originates from the invariance of posterior distribution to permutation of component labels. As a result, the component labels in Markov chain simulation may switch to another equivalent permutation, and the marginal posterior distribution associated with all labels may be similar and useless for inferring quantities relating to each individual component. In this article, we propose a new simple labelling method by minimizing the deviance of the class probabilities to a fixed reference labels. The reference labels can be chosen before running Markov chain Monte Carlo (MCMC) using optimization methods, such as expectation-maximization algorithms, and therefore the new labelling method can be implemented by an online algorithm, which can reduce the storage requirements and save much computation time. Using the Acid data set and Galaxy data set, we demonstrate the success of the proposed labelling method for removing the labelling switching in the raw MCMC samples.     

Bayesian clustering for continuous-time hidden Markov models

We develop clustering procedures for longitudinal trajectories based on a continuous-time hidden Markov model (CTHMM) and a generalized linear observation model. Specifically, in this article we carry out finite and infinite mixture model-based clustering for a CTHMM and achieve inference using Markov chain Monte Carlo (MCMC). For a finite mixture model with a prior on the number of components, we implement reversible-jump MCMC to facilitate the trans-dimensional move between models with different numbers of clusters. For a Dirichlet process mixture model, we utilize restricted Gibbs sampling split–merge proposals to improve the performance of the MCMC algorithm. We apply our proposed algorithms to simulated data as well as a real-data example, and the results demonstrate the desired performance of the new sampler.     

A two-state regime switching autoregressive model with an application to river flow analysis

We propose a regime switching autoregressive model and apply it to analyze daily water discharge series of River Tisza in Hungary. The dynamics is governed by two regimes, along which both the autoregressive coefficients and the innovation distributions are altering, moreover, the hidden regime indicator process is allowed to be non-Markovian. After examining stationarity and basic properties of the model, we turn to its estimation by Markov Chain Monte Carlo (MCMC) methods and propose two algorithms. The values of the latent process serve as auxiliary parameters in the first one, while the change points of the regimes do the same in the second one in a reversible jump MCMC setting. After comparing the mixing performance of the two methods, the model is fitted to the water discharge data. Simulations show that it reproduces the important features of the water discharge series such as the highly skewed marginal distribution and the asymmetric shape of the hydrograph.     

Bayesian semiparametric inference for the accelerated failure-time model

Bayesian semiparametric inference is considered for a loglinear model. This model consists of a parametric component for the regression coefficients and a nonparametric component for the unknown error distribution. Bayesian analysis is studied for the case of a parametric prior on the regression coefficients and a mixture-of-Dirichlet-processes prior on the unknown error distribution. A Markov-chain Monte Carlo (MCMC) method is developed to compute the features of the posterior distribution. A model selection method for obtaining a more parsimonious set of predictors is studied. The method adds indicator variables to the regression equation. The set of indicator variables represents all the possible subsets to be considered. A MCMC method is developed to search stochastically for the best subset. These procedures are applied to two examples, one with censored data.     

Estimation,prediction and interpretation of NGG random effects models: an application to Kevlar fibre failure times

We propose a class of Bayesian semiparametric mixed-effects models; its distinctive feature is the randomness of the grouping of observations, which can be inferred from the data. The model can be viewed under a more natural perspective, as a Bayesian semiparametric regression model on the log-scale; hence, in the original scale, the error is a mixture of Weibull densities mixed on both parameters by a normalized generalized gamma random measure, encompassing the Dirichlet process. As an estimate of the posterior distribution of the clustering of the random-effects parameters, we consider the partition minimizing the posterior expectation of a suitable class of loss functions. As a merely illustrative application of our model we consider a Kevlar fibre lifetime dataset (with censoring). We implement an MCMC scheme, obtaining posterior credibility intervals for the predictive distributions and for the quantiles of the failure times under different stress levels. Compared to a previous parametric Bayesian analysis, we obtain narrower credibility intervals and a better fit to the data. We found that there are three main clusters among the random-effects parameters, in accordance with previous frequentist analysis.     

Nonparametric Bayes Stochastically Ordered Latent Class Models

Latent class models (LCMs) are used increasingly for addressing a broad variety of problems, including sparse modeling of multivariate and longitudinal data, model-based clustering, and flexible inferences on predictor effects. Typical frequentist LCMs require estimation of a single finite number of classes, which does not increase with the sample size, and have a well-known sensitivity to parametric assumptions on the distributions within a class. Bayesian nonparametric methods have been developed to allow an infinite number of classes in the general population, with the number represented in a sample increasing with sample size. In this article, we propose a new nonparametric Bayes model that allows predictors to flexibly impact the allocation to latent classes, while limiting sensitivity to parametric assumptions by allowing class-specific distributions to be unknown subject to a stochastic ordering constraint. An efficient MCMC algorithm is developed for posterior computation. The methods are validated using simulation studies and applied to the problem of ranking medical procedures in terms of the distribution of patient morbidity.     

Forward Simulation Markov Chain Monte Carlo with Applications to Stochastic Epidemic Models

For many stochastic models, it is difficult to make inference about the model parameters because it is impossible to write down a tractable likelihood given the observed data. A common solution is data augmentation in a Markov chain Monte Carlo (MCMC) framework. However, there are statistical problems where this approach has proved infeasible but where simulation from the model is straightforward leading to the popularity of the approximate Bayesian computation algorithm. We introduce a forward simulation MCMC (fsMCMC) algorithm, which is primarily based upon simulation from the model. The fsMCMC algorithm formulates the simulation of the process explicitly as a data augmentation problem. By exploiting non‐centred parameterizations, an efficient MCMC updating schema for the parameters and augmented data is introduced, whilst maintaining straightforward simulation from the model. The fsMCMC algorithm is successfully applied to two distinct epidemic models including a birth–death–mutation model that has only previously been analysed using approximate Bayesian computation methods.     

SVM-like decision theoretical classification of high-dimensional vectors

In this paper, we consider the classification of high-dimensional vectors based on a small number of training samples from each class. The proposed method follows the Bayesian paradigm, and it is based on a small vector which can be viewed as the regression of the new observation on the space spanned by the training samples. The classification method provides posterior probabilities that the new vector belongs to each of the classes, hence it adapts naturally to any number of classes. Furthermore, we show a direct similarity between the proposed method and the multicategory linear support vector machine introduced in Lee et al. [2004. Multicategory support vector machines: theory and applications to the classification of microarray data and satellite radiance data. Journal of the American Statistical Association 99 (465), 67–81]. We compare the performance of the technique proposed in this paper with the SVM classifier using real-life military and microarray datasets. The study shows that the misclassification errors of both methods are very similar, and that the posterior probabilities assigned to each class are fairly accurate.     

A comparative study of the K-means algorithm and the normal mixture model for clustering: Univariate case

This paper gives a comparative study of the K-means algorithm and the mixture model (MM) method for clustering normal data. The EM algorithm is used to compute the maximum likelihood estimators (MLEs) of the parameters of the MM model. These parameters include mixing proportions, which may be thought of as the prior probabilities of different clusters; the maximum posterior (Bayes) rule is used for clustering. Hence, asymptotically the MM method approaches the Bayes rule for known parameters, which is optimal in terms of minimizing the expected misclassification rate (EMCR).     

Fully Bayesian logistic regression with hyper-LASSO priors for high-dimensional feature selection

Feature selection arises in many areas of modern science. For example, in genomic research, we want to find the genes that can be used to separate tissues of different classes (e.g. cancer and normal). One approach is to fit regression/classification models with certain penalization. In the past decade, hyper-LASSO penalization (priors) have received increasing attention in the literature. However, fully Bayesian methods that use Markov chain Monte Carlo (MCMC) for regression/classification with hyper-LASSO priors are still in lack of development. In this paper, we introduce an MCMC method for learning multinomial logistic regression with hyper-LASSO priors. Our MCMC algorithm uses Hamiltonian Monte Carlo in a restricted Gibbs sampling framework. We have used simulation studies and real data to demonstrate the superior performance of hyper-LASSO priors compared to LASSO, and to investigate the issues of choosing heaviness and scale of hyper-LASSO priors.     

On MCMC sampling in hierarchical longitudinal models

Markov chain Monte Carlo (MCMC) algorithms have revolutionized Bayesian practice. In their simplest form (i.e., when parameters are updated one at a time) they are, however, often slow to converge when applied to high-dimensional statistical models. A remedy for this problem is to block the parameters into groups, which are then updated simultaneously using either a Gibbs or Metropolis-Hastings step. In this paper we construct several (partially and fully blocked) MCMC algorithms for minimizing the autocorrelation in MCMC samples arising from important classes of longitudinal data models. We exploit an identity used by Chib (1995) in the context of Bayes factor computation to show how the parameters in a general linear mixed model may be updated in a single block, improving convergence and producing essentially independent draws from the posterior of the parameters of interest. We also investigate the value of blocking in non-Gaussian mixed models, as well as in a class of binary response data longitudinal models. We illustrate the approaches in detail with three real-data examples.     

Markov chain Monte Carlo with the Integrated Nested Laplace Approximation

The Integrated Nested Laplace Approximation (INLA) has established itself as a widely used method for approximate inference on Bayesian hierarchical models which can be represented as a latent Gaussian model (LGM). INLA is based on producing an accurate approximation to the posterior marginal distributions of the parameters in the model and some other quantities of interest by using repeated approximations to intermediate distributions and integrals that appear in the computation of the posterior marginals. INLA focuses on models whose latent effects are a Gaussian Markov random field. For this reason, we have explored alternative ways of expanding the number of possible models that can be fitted using the INLA methodology. In this paper, we present a novel approach that combines INLA and Markov chain Monte Carlo (MCMC). The aim is to consider a wider range of models that can be fitted with INLA only when some of the parameters of the model have been fixed. We show how new values of these parameters can be drawn from their posterior by using conditional models fitted with INLA and standard MCMC algorithms, such as Metropolis–Hastings. Hence, this will extend the use of INLA to fit models that can be expressed as a conditional LGM. Also, this new approach can be used to build simpler MCMC samplers for complex models as it allows sampling only on a limited number of parameters in the model. We will demonstrate how our approach can extend the class of models that could benefit from INLA, and how the R-INLA package will ease its implementation. We will go through simple examples of this new approach before we discuss more advanced applications with datasets taken from the relevant literature. In particular, INLA within MCMC will be used to fit models with Laplace priors in a Bayesian Lasso model, imputation of missing covariates in linear models, fitting spatial econometrics models with complex nonlinear terms in the linear predictor and classification of data with mixture models. Furthermore, in some of the examples we could exploit INLA within MCMC to make joint inference on an ensemble of model parameters.     

Computation for intrinsic variable selection in normal regression models via expected-posterior prior

In this paper, we focus on the variable selection problem in normal regression models using the expected-posterior prior methodology. We provide a straightforward MCMC scheme for the derivation of the posterior distribution, as well as Monte Carlo estimates for the computation of the marginal likelihood and posterior model probabilities. Additionally, for large spaces, a model search algorithm based on $\mathit{MC}^{3}$ is constructed. The proposed methodology is applied in two real life examples, already used in the relevant literature of objective variable selection. In both examples, uncertainty over different training samples is taken into consideration.     

Bayesian inference for dynamic social network data

We consider a continuous-time model for the evolution of social networks. A social network is here conceived as a (di-) graph on a set of vertices, representing actors, and the changes of interest are creation and disappearance over time of (arcs) edges in the graph. Hence we model a collection of random edge indicators that are not, in general, independent. We explicitly model the interdependencies between edge indicators that arise from interaction between social entities. A Markov chain is defined in terms of an embedded chain with holding times and transition probabilities. Data are observed at fixed points in time and hence we are not able to observe the embedded chain directly. Introducing a prior distribution for the parameters we may implement an MCMC algorithm for exploring the posterior distribution of the parameters by simulating the evolution of the embedded process between observations.     

Variable selection in quantile regression via Gibbs sampling

Due to computational challenges and non-availability of conjugate prior distributions, Bayesian variable selection in quantile regression models is often a difficult task. In this paper, we address these two issues for quantile regression models. In particular, we develop an informative stochastic search variable selection (ISSVS) for quantile regression models that introduces an informative prior distribution. We adopt prior structures which incorporate historical data into the current data by quantifying them with a suitable prior distribution on the model parameters. This allows ISSVS to search more efficiently in the model space and choose the more likely models. In addition, a Gibbs sampler is derived to facilitate the computation of the posterior probabilities. A major advantage of ISSVS is that it avoids instability in the posterior estimates for the Gibbs sampler as well as convergence problems that may arise from choosing vague priors. Finally, the proposed methods are illustrated with both simulation and real data.