An EM-type algorithm for multivariate mixture models

This paper introduces a new approach, based on dependent univariate GLMs, for fitting multivariate mixture models. This approach is a multivariate generalization of the method for univariate mixtures presented by Hinde (1982). Its accuracy and efficiency are compared with direct maximization of the log-likelihood. Using a simulation study, we also compare the efficiency of Monte Carlo and Gaussian quadrature methods for approximating the mixture distribution. The new approach with Gaussian quadrature outperforms the alternative methods considered. The work is motivated by the multivariate mixture models which have been proposed for modelling changes of employment states at an individual level. Similar formulations are of interest for modelling movement between other social and economic states and multivariate mixture models also occur in biostatistics and epidemiology.     

Generalized Self-Consistency: Multinomial logit model and Poisson likelihood

A generalized self-consistency approach to maximum likelihood estimation (MLE) and model building was developed in Tsodikov [2003. Semiparametric models: a generalized self-consistency approach. J. Roy. Statist. Soc. Ser. B Statist. Methodology 65(3), 759–774] and applied to a survival analysis problem. We extend the framework to obtain second-order results such as information matrix and properties of the variance. Multinomial model motivates the paper and is used throughout as an example. Computational challenges with the multinomial likelihood motivated Baker [1994. The Multinomial–Poisson transformation. The Statist. 43, 495–504] to develop the Multinomial–Poisson (MP) transformation for a large variety of regression models with multinomial likelihood kernel. Multinomial regression is transformed into a Poisson regression at the cost of augmenting model parameters and restricting the problem to discrete covariates. Imposing normalization restrictions by means of Lagrange multipliers [Lang, J., 1996. On the comparison of multinomial and Poisson log-linear models. J. Roy. Statist. Soc. Ser. B Statist. Methodology 58, 253–266] justifies the approach. Using the self-consistency framework we develop an alternative solution to multinomial model fitting that does not require augmenting parameters while allowing for a Poisson likelihood and arbitrary covariate structures. Normalization restrictions are imposed by averaging over artificial “missing data” (fake mixture). Lack of probabilistic interpretation at the “complete-data” level makes the use of the generalized self-consistency machinery essential.     

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.     

Robust Estimation for Zero‐Inflated Poisson Regression

Abstract. The zero‐inflated Poisson regression model is a special case of finite mixture models that is useful for count data containing many zeros. Typically, maximum likelihood (ML) estimation is used for fitting such models. However, it is well known that the ML estimator is highly sensitive to the presence of outliers and can become unstable when mixture components are poorly separated. In this paper, we propose an alternative robust estimation approach, robust expectation‐solution (RES) estimation. We compare the RES approach with an existing robust approach, minimum Hellinger distance (MHD) estimation. Simulation results indicate that both methods improve on ML when outliers are present and/or when the mixture components are poorly separated. However, the RES approach is more efficient in all the scenarios we considered. In addition, the RES method is shown to yield consistent and asymptotically normal estimators and, in contrast to MHD, can be applied quite generally.     

A case-base sampling method for estimating recurrent event intensities

Case-base sampling provides an alternative to risk set sampling based methods to estimate hazard regression models, in particular when absolute hazards are also of interest in addition to hazard ratios. The case-base sampling approach results in a likelihood expression of the logistic regression form, but instead of categorized time, such an expression is obtained through sampling of a discrete set of person-time coordinates from all follow-up data. In this paper, in the context of a time-dependent exposure such as vaccination, and a potentially recurrent adverse event outcome, we show that the resulting partial likelihood for the outcome event intensity has the asymptotic properties of a likelihood. We contrast this approach to self-matched case-base sampling, which involves only within-individual comparisons. The efficiency of the case-base methods is compared to that of standard methods through simulations, suggesting that the information loss due to sampling is minimal.     

Finite mixture of regression models for a stratified sample

Despite the popularity and importance, there is limited work on modelling data which come from complex survey design using finite mixture models. In this work, we explored the use of finite mixture regression models when the samples were drawn using a complex survey design. In particular, we considered modelling data collected based on stratified sampling design. We developed a new design-based inference where we integrated sampling weights in the complete-data log-likelihood function. The expectation–maximisation algorithm was developed accordingly. A simulation study was conducted to compare the new methodology with the usual finite mixture of a regression model. The comparison was done using bias-variance components of mean square error. Additionally, a simulation study was conducted to assess the ability of the Bayesian information criterion to select the optimal number of components under the proposed modelling approach. The methodology was implemented on real data with good results.     

A practical sampling approach for a Bayesian mixture model with unknown number of components

Recently, mixture distribution becomes more and more popular in many scientific fields. Statistical computation and analysis of mixture models, however, are extremely complex due to the large number of parameters involved. Both EM algorithms for likelihood inference and MCMC procedures for Bayesian analysis have various difficulties in dealing with mixtures with unknown number of components. In this paper, we propose a direct sampling approach to the computation of Bayesian finite mixture models with varying number of components. This approach requires only the knowledge of the density function up to a multiplicative constant. It is easy to implement, numerically efficient and very practical in real applications. A simulation study shows that it performs quite satisfactorily on relatively high dimensional distributions. A well-known genetic data set is used to demonstrate the simplicity of this method and its power for the computation of high dimensional Bayesian mixture models.     

Bayesian temporal density estimation with autoregressive species sampling models

We propose a novel Bayesian nonparametric (BNP) model, which is built on a class of species sampling models, for estimating density functions of temporal data. In particular, we introduce species sampling mixture models with temporal dependence. To accommodate temporal dependence, we define dependent species sampling models by modeling random support points and weights through an autoregressive model, and then we construct the mixture models based on the collection of these dependent species sampling models. We propose an algorithm to generate posterior samples and present simulation studies to compare the performance of the proposed models with competitors that are based on Dirichlet process mixture models. We apply our method to the estimation of densities for the price of apartment in Seoul, the closing price in Korea Composite Stock Price Index (KOSPI), and climate variables (daily maximum temperature and precipitation) of around the Korean peninsula.     

Estimating finite mixture of continuous trees using penalized mutual information

Abstract

In this paper we introduce continuous tree mixture model that is the mixture of undirected graphical models with tree structured graphs and is considered as multivariate analysis with a non parametric approach. We estimate its parameters, the component edge sets and mixture proportions through regularized maximum likalihood procedure. Our new algorithm, which uses expectation maximization algorithm and the modified version of Kruskal algorithm, simultaneosly estimates and prunes the mixture component trees. Simulation studies indicate this method performs better than the alternative Gaussian graphical mixture model. The proposed method is also applied to water-level data set and is compared with the results of Gaussian mixture model.     

An Alternative Estimation Approach for the Heterogeneity Linear Mixed Model

In this article, an alternative estimation approach is proposed to fit linear mixed effects models where the random effects follow a finite mixture of normal distributions. This heterogeneity linear mixed model is an interesting tool since it relaxes the classical normality assumption and is also perfectly suitable for classification purposes, based on longitudinal profiles. Instead of fitting directly the heterogeneity linear mixed model, we propose to fit an equivalent mixture of linear mixed models under some restrictions which is computationally simpler. Unlike the former model, the latter can be maximized analytically using an EM-algorithm and the obtained parameter estimates can be easily used to compute the parameter estimates of interest.     

A Simulation Comparison of Approximate Tests for Fixed Effects in Random Coefficients Growth Curve Models

Often, the response variables on sampling units are observed repeatedly over time. The sampling units may come from different populations, such as treatment groups. This setting is routinely modeled by a random coefficients growth curve model, and the techniques of general linear mixed models are applied to address the primary research aim. An alternative approach is to reduce each subject’s data to summary measures, such as within-subject averages or regression coefficients. One may then test for equality of means of the summary measures (or functions of them) among treatment groups. Here, we compare by simulation the performance characteristics of three approximate tests based on summary measures and one based on the full data, focusing mainly on accuracy of p-values. We find that performances of these procedures can be quite different for small samples in several different configurations of parameter values. The summary-measures approach performed at least as well as the full-data mixed models approach.     

Alternative modeling techniques for the quantal response data in mixture experiments

Mixture experiments are commonly encountered in many fields including chemical, pharmaceutical and consumer product industries. Due to their wide applications, mixture experiments, a special study of response surface methodology, have been given greater attention in both model building and determination of designs compared with other experimental studies. In this paper, some new approaches are suggested on model building and selection for the analysis of the data in mixture experiments by using a special generalized linear models, logistic regression model, proposed by Chen et al. [7]. Generally, the special mixture models, which do not have a constant term, are highly affected by collinearity in modeling the mixture experiments. For this reason, in order to alleviate the undesired effects of collinearity in the analysis of mixture experiments with logistic regression, a new mixture model is defined with an alternative ratio variable. The deviance analysis table is given for standard mixture polynomial models defined by transformations and special mixture models used as linear predictors. The effects of components on the response in the restricted experimental region are given by using an alternative representation of Cox's direction approach. In addition, odds ratio and the confidence intervals of odds ratio are identified according to the chosen reference and control groups. To compare the suggested models, some model selection criteria, graphical odds ratio and the confidence intervals of the odds ratio are used. The advantage of the suggested approaches is illustrated on tumor incidence data set.     

Bayesian analysis of mixture modelling using the multivariate t distribution

A finite mixture model using the multivariate t distribution has been shown as a robust extension of normal mixtures. In this paper, we present a Bayesian approach for inference about parameters of t-mixture models. The specifications of prior distributions are weakly informative to avoid causing nonintegrable posterior distributions. We present two efficient EM-type algorithms for computing the joint posterior mode with the observed data and an incomplete future vector as the sample. Markov chain Monte Carlo sampling schemes are also developed to obtain the target posterior distribution of parameters. The advantages of Bayesian approach over the maximum likelihood method are demonstrated via a set of real data.     

Slice sampling mixture models

We propose a more efficient version of the slice sampler for Dirichlet process mixture models described by Walker (Commun. Stat., Simul. Comput. 36:45–54, 2007). This new sampler allows for the fitting of infinite mixture models with a wide-range of prior specifications. To illustrate this flexibility we consider priors defined through infinite sequences of independent positive random variables. Two applications are considered: density estimation using mixture models and hazard function estimation. In each case we show how the slice efficient sampler can be applied to make inference in the models. In the mixture case, two submodels are studied in detail. The first one assumes that the positive random variables are Gamma distributed and the second assumes that they are inverse-Gaussian distributed. Both priors have two hyperparameters and we consider their effect on the prior distribution of the number of occupied clusters in a sample. Extensive computational comparisons with alternative “conditional” simulation techniques for mixture models using the standard Dirichlet process prior and our new priors are made. The properties of the new priors are illustrated on a density estimation problem.     

Reversible jump and the label switching problem in hidden Markov models

Reversible jump Markov chain Monte Carlo (RJMCMC) algorithms can be efficiently applied in Bayesian inference for hidden Markov models (HMMs), when the number of latent regimes is unknown. As for finite mixture models, when priors are invariant to the relabelling of the regimes, HMMs are unidentifiable in data fitting, because multiple ways to label the regimes can alternate during the MCMC iterations; this is the so-called label switching problem. HMMs with an unknown number of regimes are considered here and the goal of this paper is the comparison, both applied and theoretical, of five methods used for tackling label switching within a RJMCMC algorithm; they are: post-processing, partial reordering, permutation sampling, sampling from a Markov prior and rejection sampling. The five strategies we compare have been proposed mostly in the literature of finite mixture models and only two of them, i.e. rejection sampling and partial reordering, have been presented in RJMCMC algorithms for HMMs. We consider RJMCMC algorithms in which the parameters are updated by Gibbs sampling and the dimension of the model changes in split-and-merge and birth-and-death moves. Finally, an example illustrates and compares the five different methodologies.     

Joint tests of non-nested models and general error specifications

This paper is concerned with joint tests of non-nested models and simultaneous departures from homoskedasticity, serial independence and normality of the disturbance terms. Locally equivalent alternative models are used to construct joint tests since they provide a convenient way to incorporate more than one type of departure from the classical conditions. The joint tests represent a simple asymptotic solution to the “pre-testing” problem in the context of non-nested linear regression models. Our simulation results indicate that the proposed tests have good finite sample properties.     

Posterior Analysis for Normalized Random Measures with Independent Increments

Abstract.  One of the main research areas in Bayesian Nonparametrics is the proposal and study of priors which generalize the Dirichlet process. In this paper, we provide a comprehensive Bayesian non-parametric analysis of random probabilities which are obtained by normalizing random measures with independent increments (NRMI). Special cases of these priors have already shown to be useful for statistical applications such as mixture models and species sampling problems. However, in order to fully exploit these priors, the derivation of the posterior distribution of NRMIs is crucial: here we achieve this goal and, indeed, provide explicit and tractable expressions suitable for practical implementation. The posterior distribution of an NRMI turns out to be a mixture with respect to the distribution of a specific latent variable. The analysis is completed by the derivation of the corresponding predictive distributions and by a thorough investigation of the marginal structure. These results allow to derive a generalized Blackwell–MacQueen sampling scheme, which is then adapted to cover also mixture models driven by general NRMIs.     

Estimation of Dynamic Bivariate Mixture Models

This note compares a Bayesian Markov chain Monte Carlo approach implemented by Watanabe with a maximum likelihood ML approach based on an efficient importance sampling procedure to estimate dynamic bivariate mixture models. In these models, stock price volatility and trading volume are jointly directed by the unobservable number of price-relevant information arrivals, which is specified as a serially correlated random variable. It is shown that the efficient importance sampling technique is extremely accurate and that it produces results that differ significantly from those reported by Watanabe.     

Gibbs sampling for Bayesian non-conjugate and hierarchical models by using auxiliary variables

We demonstrate the use of auxiliary (or latent) variables for sampling non-standard densities which arise in the context of the Bayesian analysis of non-conjugate and hierarchical models by using a Gibbs sampler. Their strategic use can result in a Gibbs sampler having easily sampled full conditionals. We propose such a procedure to simplify or speed up the Markov chain Monte Carlo algorithm. The strength of this approach lies in its generality and its ease of implementation. The aim of the paper, therefore, is to provide an alternative sampling algorithm to rejection-based methods and other sampling approaches such as the Metropolis–Hastings algorithm.     

An algorithm with applications in ranked-set sampling

A mixture of order statistics is a random variable whose distribution is a finite mixture of the distributions for order statistics. Such mixtures show up in the literature on ranked-set sampling and related sampling schemes as models for imperfect rankings. In this paper, we derive an algorithm for computing the probability that independent mixtures of order statistics come in a particular order. The algorithm is far faster than previous proposals from the literature. As an application, we show that the algorithm can be used to create Kolmogorov–Smirnov-type confidence bands that adjust for the presence of imperfect rankings.