Label Switching in Bayesian Mixture Models: Deterministic Relabeling Strategies

Label switching is a well-known problem in the Bayesian analysis of mixture models. On the one hand, it complicates inference, and on the other hand, it has been perceived as a prerequisite to justify Markov chain Monte Carlo (MCMC) convergence. As a result, nonstandard MCMC algorithms that traverse the symmetric copies of the posterior distribution, and possibly genuine modes, have been proposed. To perform component-specific inference, methods to undo the label switching and to recover the interpretation of the components need to be applied. If latent allocations for the design of the MCMC strategy are included, and the sampler has converged, then labels assigned to each component may change from iteration to iteration. However, observations being allocated together must remain similar, and we use this fundamental fact to derive an easy and efficient solution to the label switching problem. We compare our strategy with other relabeling algorithms on univariate and multivariate data examples and demonstrate improvements over alternative strategies. Supplementary materials for this article are available online.     

Markov Chain Monte Carlo With Mixtures of Mutually Singular Distributions

Markov chain Monte Carlo (MCMC) methods for Bayesian computation are mostly used when the dominating measure is the Lebesgue measure, the counting measure, or a product of these. Many Bayesian problems give rise to distributions that are not dominated by the Lebesgue measure or the counting measure alone. In this article we introduce a simple framework for using MCMC algorithms in Bayesian computation with mixtures of mutually singular distributions. The idea is to find a common dominating measure that allows the use of traditional Metropolis-Hastings algorithms. In particular, using our formulation, the Gibbs sampler can be used whenever the full conditionals are available. We compare our formulation with the reversible jump approach and show that the two are closely related. We give results for three examples, involving testing a normal mean, variable selection in regression, and hypothesis testing for differential gene expression under multiple conditions. This allows us to compare the three methods considered: Metropolis-Hastings with mutually singular distributions, Gibbs sampler with mutually singular distributions, and reversible jump. In our examples, we found the Gibbs sampler to be more precise and to need considerably less computer time than the other methods. In addition, the full conditionals used in the Gibbs sampler can be used to further improve the estimates of the model posterior probabilities via Rao-Blackwellization, at no extra cost.     

Markov Chain Monte Carlo Convergence Assessment via Two-Way Analysis of Variance

Abstract

In this article we discuss the problem of assessing the performance of Markov chain Monte Carlo (MCMC) algorithms on the basis of simulation output. In essence, we extend the original ideas of Gelman and Rubin and, more recently, Brooks and Gelman, to problems where we are able to split the variation inherent within the MCMC simulation output into two distinct groups. We show how such a diagnostic may be useful in assessing the performance of MCMC samplers addressing model choice problems, such as the reversible jump MCMC algorithm. In the model choice context, we show how the reversible jump MCMC simulation output for parameters that retain a coherent interpretation throughout the simulation, can be used to assess convergence. By considering various decompositions of the sampling variance of this parameter, we can assess the performance of our MCMC sampler in terms of its mixing properties both within and between models and we illustrate our approach in both the graphical Gaussian models and normal mixtures context. Finally, we provide an example of the application of our diagnostic to the assessment of the influence of different starting values on MCMC simulation output, thereby illustrating the wider utility of our method beyond the Bayesian model choice and reversible jump MCMC context.     

The Generalized Multiset Sampler

The multiset sampler, an MCMC algorithm recently proposed by Leman and coauthors, is an easy-to-implement algorithm which is especially well-suited to drawing samples from a multimodal distribution. We generalize the algorithm by redefining the multiset sampler with an explicit link between target distribution and sampling distribution. The generalized formulation replaces the multiset with a K-tuple, which allows us to use the algorithm on unbounded parameter spaces, improves estimation, and sets up further extensions to adaptive MCMC techniques. Theoretical properties of the algorithm are provided and guidance is given on its implementation. Examples, both simulated and real, confirm that the generalized multiset sampler provides a simple, general and effective approach to sampling from multimodal distributions. Supplementary materials for this article are available online.     

Covariance Decompositions for Accurate Computation in Bayesian Scale-Usage Models

Analyses of multivariate ordinal probit models typically use data augmentation to link the observed (discrete) data to latent (continuous) data via a censoring mechanism defined by a collection of “cutpoints.” Most standard models, for which effective Markov chain Monte Carlo (MCMC) sampling algorithms have been developed, use a separate (and independent) set of cutpoints for each element of the multivariate response. Motivated by the analysis of ratings data, we describe a particular class of multivariate ordinal probit models where it is desirable to use a common set of cutpoints. While this approach is attractive from a data-analytic perspective, we show that the existing efficient MCMC algorithms can no longer be accurately applied. Moreover, we show that attempts to implement these algorithms by numerically approximating required multivariate normal integrals over high-dimensional rectangular regions can result in severely degraded estimates of the posterior distribution. We propose a new data augmentation that is based on a covariance decomposition and that admits a simple and accurate MCMC algorithm. Our data augmentation requires only that univariate normal integrals be evaluated, which can be done quickly and with high accuracy. We provide theoretical results that suggest optimal decompositions within this class of data augmentations, and, based on the theory, recommend default decompositions that we demonstrate work well in practice. This article has supplementary material online.     

Implementations of the Monte Carlo EM Algorithm

The Monte Carlo EM (MCEM) algorithm is a modification of the EM algorithm where the expectation in the E-step is computed numerically through Monte Carlo simulations. The most exible and generally applicable approach to obtaining a Monte Carlo sample in each iteration of an MCEM algorithm is through Markov chain Monte Carlo (MCMC) routines such as the Gibbs and Metropolis–Hastings samplers. Although MCMC estimation presents a tractable solution to problems where the E-step is not available in closed form, two issues arise when implementing this MCEM routine: (1) how do we minimize the computational cost in obtaining an MCMC sample? and (2) how do we choose the Monte Carlo sample size? We address the first question through an application of importance sampling whereby samples drawn during previous EM iterations are recycled rather than running an MCMC sampler each MCEM iteration. The second question is addressed through an application of regenerative simulation. We obtain approximate independent and identical samples by subsampling the generated MCMC sample during different renewal periods. Standard central limit theorems may thus be used to gauge Monte Carlo error. In particular, we apply an automated rule for increasing the Monte Carlo sample size when the Monte Carlo error overwhelms the EM estimate at any given iteration. We illustrate our MCEM algorithm through analyses of two datasets fit by generalized linear mixed models. As a part of these applications, we demonstrate the improvement in computational cost and efficiency of our routine over alternative MCEM strategies.     

Randomized approximation scheme and perfect sampler for closed Jackson networks with multiple servers

In this paper, we propose a fully polynomial-time randomized approximation scheme (FPRAS) for a closed Jackson network. Our algorithm is based on the Markov chain Monte Carlo (MCMC) method. Thus our scheme returns an approximate solution, for which the size of the error satisfies a given bound. To our knowledge, this algorithm is the first polynomial time MCMC algorithm for closed Jackson networks with multiple servers. We propose two new ergodic Markov chains, both of which have a unique stationary distribution that is the product form solution for closed Jackson networks. One of them is for an approximate sampler, and we show that it mixes rapidly. The other is for a perfect sampler based on the monotone coupling from the past (CFTP) algorithm proposed by Propp and Wilson, and we show that it has a monotone update function.     

A Generalized Markov Sampler

A recent development of the Markov chain Monte Carlo (MCMC) technique is the emergence of MCMC samplers that allow transitions between different models. Such samplers make possible a range of computational tasks involving models, including model selection, model evaluation, model averaging and hypothesis testing. An example of this type of sampler is the reversible jump MCMC sampler, which is a generalization of the Metropolis–Hastings algorithm. Here, we present a new MCMC sampler of this type. The new sampler is a generalization of the Gibbs sampler, but somewhat surprisingly, it also turns out to encompass as particular cases all of the well-known MCMC samplers, including those of Metropolis, Barker, and Hastings. Moreover, the new sampler generalizes the reversible jump MCMC. It therefore appears to be a very general framework for MCMC sampling. This paper describes the new sampler and illustrates its use in three applications in Computational Biology, specifically determination of consensus sequences, phylogenetic inference and delineation of isochores via multiple change-point analysis.     

Dynamically Updated Spatially Varying Parameterizations of Hierarchical Bayesian Models for Spatial Data

Fitting hierarchical Bayesian models to spatially correlated datasets using Markov chain Monte Carlo (MCMC) techniques is computationally expensive. Complicated covariance structures of the underlying spatial processes, together with high-dimensional parameter space, mean that the number of calculations required grows cubically with the number of spatial locations at each MCMC iteration. This necessitates the need for efficient model parameterizations that hasten the convergence and improve the mixing of the associated algorithms. We consider partially centred parameterizations (PCPs) which lie on a continuum between what are known as the centered (CP) and noncentered parameterizations (NCP). By introducing a weight matrix we remove the conditional posterior correlation between the fixed and the random effects, and hence construct a PCP which achieves immediate convergence for a three-stage model, based on multiple Gaussian processes with known covariance parameters. When the covariance parameters are unknown we dynamically update the parameterization within the sampler. The PCP outperforms both the CP and the NCP and leads to a fully automated algorithm which has been demonstrated in two simulation examples. The effectiveness of the spatially varying PCP is illustrated with a practical dataset of nitrogen dioxide concentration levels. Supplemental materials consisting of appendices, datasets, and computer code to reproduce the results are available online.     

On the Relationship Between Markov chain Monte Carlo Methods for Model Uncertainty

This article considers Markov chain computational methods for incorporating uncertainty about the dimension of a parameter when performing inference within a Bayesian setting. A general class of methods is proposed for performing such computations, based upon a product space representation of the problem which is similar to that of Carlin and Chib. It is shown that all of the existing algorithms for incorporation of model uncertainty into Markov chain Monte Carlo (MCMC) can be derived as special cases of this general class of methods. In particular, we show that the popular reversible jump method is obtained when a special form of Metropolis–Hastings (M–H) algorithm is applied to the product space. Furthermore, the Gibbs sampling method and the variable selection method are shown to derive straightforwardly from the general framework. We believe that these new relationships between methods, which were until now seen as diverse procedures, are an important aid to the understanding of MCMC model selection procedures and may assist in the future development of improved procedures. Our discussion also sheds some light upon the important issues of “pseudo-prior” selection in the case of the Carlin and Chib sampler and choice of proposal distribution in the case of reversible jump. Finally, we propose efficient reversible jump proposal schemes that take advantage of any analytic structure that may be present in the model. These proposal schemes are compared with a standard reversible jump scheme for the problem of model order uncertainty in autoregressive time series, demonstrating the improvements which can be achieved through careful choice of proposals.     

Precomputing strategy for Hamiltonian Monte Carlo method based on regularity in parameter space

Markov Chain Monte Carlo (MCMC) algorithms play an important role in statistical inference problems dealing with intractable probability distributions. Recently, many MCMC algorithms such as Hamiltonian Monte Carlo (HMC) and Riemannian Manifold HMC have been proposed to provide distant proposals with high acceptance rate. These algorithms, however, tend to be computationally intensive which could limit their usefulness, especially for big data problems due to repetitive evaluations of functions and statistical quantities that depend on the data. This issue occurs in many statistic computing problems. In this paper, we propose a novel strategy that exploits smoothness (regularity) in parameter space to improve computational efficiency of MCMC algorithms. When evaluation of functions or statistical quantities are needed at a point in parameter space, interpolation from precomputed values or previous computed values is used. More specifically, we focus on HMC algorithms that use geometric information for faster exploration of probability distributions. Our proposed method is based on precomputing the required geometric information on a set of grids before running sampling algorithm and approximating the geometric information for the current location of the sampler using the precomputed information at nearby grids at each iteration of HMC. Sparse grid interpolation method is used for high dimensional problems. Tests on computational examples are shown to illustrate the advantages of our method.     

Acceleration Tools for Diagonal Information Global Optimization Algorithms

In this paper we face a classical global optimization problem—minimization of a multiextremal multidimensional Lipschitz function over a hyperinterval. We introduce two new diagonal global optimization algorithms unifying the power of the following three approaches: efficient univariate information global optimization methods, diagonal approach for generalizing univariate algorithms to the multidimensional case, and local tuning on the behaviour of the objective function (estimates of the local Lipschitz constants over different subregions) during the global search. Global convergence conditions of a new type are established for the diagonal information methods. The new algorithms demonstrate quite satisfactory performance in comparison with the diagonal methods using only global information about the Lipschitz constant.     

Efficient MCMC estimation of inflated beta regression models

This paper introduces a new and computationally efficient Markov chain Monte Carlo (MCMC) estimation algorithm for the Bayesian analysis of zero, one, and zero and one inflated beta regression models. The algorithm is computationally efficient in the sense that it has low MCMC autocorrelations and computational time. A simulation study shows that the proposed algorithm outperforms the slice sampling and random walk Metropolis–Hastings algorithms in both small and large sample settings. An empirical illustration on a loss given default banking model demonstrates the usefulness of the proposed algorithm.     

Variational Approximation for Mixtures of Linear Mixed Models

Mixtures of linear mixed models (MLMMs) are useful for clustering grouped data and can be estimated by likelihood maximization through the Expectation–Maximization algorithm. A suitable number of components is then determined conventionally by comparing different mixture models using penalized log-likelihood criteria such as Bayesian information criterion. We propose fitting MLMMs with variational methods, which can perform parameter estimation and model selection simultaneously. We describe a variational approximation for MLMMs where the variational lower bound is in closed form, allowing for fast evaluation and develop a novel variational greedy algorithm for model selection and learning of the mixture components. This approach handles algorithm initialization and returns a plausible number of mixture components automatically. In cases of weak identifiability of certain model parameters, we use hierarchical centering to reparameterize the model and show empirically that there is a gain in efficiency in variational algorithms similar to that in Markov chain Monte Carlo (MCMC) algorithms. Related to this, we prove that the approximate rate of convergence of variational algorithms by Gaussian approximation is equal to that of the corresponding Gibbs sampler, which suggests that reparameterizations can lead to improved convergence in variational algorithms just as in MCMC algorithms. Supplementary materials for the article are available online.     

Minimizing variable selection criteria by Markov chain Monte Carlo

Regression models with a large number of predictors arise in diverse fields of social sciences and natural sciences. For proper interpretation, we often would like to identify a smaller subset of the variables that shows the strongest information. In such a large size of candidate predictors setting, one would encounter a computationally cumbersome search in practice by optimizing some criteria for selecting variables, such as AIC, \(C_{P}\) and BIC, through all possible subsets. In this paper, we present two efficient optimization algorithms vis Markov chain Monte Carlo (MCMC) approach for searching the global optimal subset. Simulated examples as well as one real data set exhibit that our proposed MCMC algorithms did find better solutions than other popular search methods in terms of minimizing a given criterion.     

A hierarchical model for the joint mortality analysis of pension scheme data with missing covariates

A hierarchical model is developed for the joint mortality analysis of pension scheme datasets. The proposed model allows for a rigorous statistical treatment of missing data. While our approach works for any missing data pattern, we are particularly interested in a scenario where some covariates are observed for members of one pension scheme but not the other. Therefore, our approach allows for the joint modelling of datasets which contain different information about individual lives. The proposed model generalizes the specification of parametric models when accounting for covariates. We consider parameter uncertainty using Bayesian techniques. Model parametrization is analysed in order to obtain an efficient MCMC sampler, and address model selection. The inferential framework described here accommodates any missing-data pattern, and turns out to be useful to analyse statistical relationships among covariates. Finally, we assess the financial impact of using the covariates, and of the optimal use of the whole available sample when combining data from different mortality experiences.     

An Adaptive Parallel Tempering Algorithm

Parallel tempering is a generic Markov chain Monte Carlo sampling method which allows good mixing with multimodal target distributions, where conventional Metropolis-Hastings algorithms often fail. The mixing properties of the sampler depend strongly on the choice of tuning parameters, such as the temperature schedule and the proposal distribution used for local exploration. We propose an adaptive algorithm with fixed number of temperatures which tunes both the temperature schedule and the parameters of the random-walk Metropolis kernel automatically. We prove the convergence of the adaptation and a strong law of large numbers for the algorithm under general conditions. We also prove as a side result the geometric ergodicity of the parallel tempering algorithm. We illustrate the performance of our method with examples. Our empirical findings indicate that the algorithm can cope well with different kinds of scenarios without prior tuning. Supplementary materials including the proofs and the Matlab implementation are available online.     

The Gibbs Cloner for Combinatorial Optimization, Counting and Sampling

We present a randomized algorithm, called the cloning algorithm, for approximating the solutions of quite general NP-hard combinatorial optimization problems, counting, rare-event estimation and uniform sampling on complex regions. Similar to the algorithms of Diaconis–Holmes–Ross and Botev–Kroese the cloning algorithm is based on the MCMC (Gibbs) sampler equipped with an importance sampling pdf and, as usual for randomized algorithms, it uses a sequential sampling plan to decompose a “difficult” problem into a sequence of “easy” ones. The cloning algorithm combines the best features of the Diaconis–Holmes–Ross and the Botev–Kroese. In addition to some other enhancements, it has a special mechanism, called the “cloning” device, which makes the cloning algorithm, also called the Gibbs cloner fast and accurate. We believe that it is the fastest and the most accurate randomized algorithm for counting known so far. In addition it is well suited for solving problems associated with the Boltzmann distribution, like estimating the partition functions in an Ising model. We also present a combined version of the cloning and cross-entropy (CE) algorithms. We prove the polynomial complexity of a particular version of the Gibbs cloner for counting. We finally present efficient numerical results with the Gibbs cloner and the combined version, while solving quite general integer and combinatorial optimization problems as well as counting ones, like SAT.     

Bayesian estimation of NIG models via Markov chain Monte Carlo methods

The normal inverse Gaussian (NIG) distribution is a promising alternative for modelling financial data since it is a continuous distribution that allows for skewness and fat tails. There is an increasing number of applications of the NIG distribution to financial problems. Due to the complicated nature of its density, estimation procedures are not simple. In this paper we propose Bayesian estimation for the parameters of the NIG distribution via an MCMC scheme based on the Gibbs sampler. Our approach makes use of the data augmentation provided by the mixture representation of the distribution. We also extend the model to allow for modelling heteroscedastic regression situations. Examples with financial and simulated data are provided. Copyright © 2004 John Wiley & Sons, Ltd.     

Bayesian analysis of multivariate t linear mixed models using a combination of IBF and Gibbs samplers

The multivariate linear mixed model (MLMM) has become the most widely used tool for analyzing multi-outcome longitudinal data. Although it offers great flexibility for modeling the between- and within-subject correlation among multi-outcome repeated measures, the underlying normality assumption is vulnerable to potential atypical observations. We present a fully Bayesian approach to the multivariate t linear mixed model (MtLMM), which is a robust extension of MLMM with the random effects and errors jointly distributed as a multivariate t distribution. Owing to the introduction of too many hidden variables in the model, the conventional Markov chain Monte Carlo (MCMC) method may converge painfully slowly and thus fails to provide valid inference. To alleviate this problem, a computationally efficient inverse Bayes formulas (IBF) sampler coupled with the Gibbs scheme, called the IBF-Gibbs sampler, is developed and shown to be effective in drawing samples from the target distributions. The issues related to model determination and Bayesian predictive inference for future values are also investigated. The proposed methodologies are illustrated with a real example from an AIDS clinical trial and a careful simulation study.