Stochastic approximation Monte Carlo EM for change-point analysis

In the expectation–maximization (EM) algorithm for maximum likelihood estimation from incomplete data, Markov chain Monte Carlo (MCMC) methods have been used in change-point inference for a long time when the expectation step is intractable. However, the conventional MCMC algorithms tend to get trapped in local mode in simulating from the posterior distribution of change points. To overcome this problem, in this paper we propose a stochastic approximation Monte Carlo version of EM (SAMCEM), which is a combination of adaptive Markov chain Monte Carlo and EM utilizing a maximum likelihood method. SAMCEM is compared with the stochastic approximation version of EM and reversible jump Markov chain Monte Carlo version of EM on simulated and real datasets. The numerical results indicate that SAMCEM can outperform among the three methods by producing much more accurate parameter estimates and the ability to achieve change-point positions and estimates simultaneously.     

A computational framework for empirical Bayes inference

In empirical Bayes inference one is typically interested in sampling from the posterior distribution of a parameter with a hyper-parameter set to its maximum likelihood estimate. This is often problematic particularly when the likelihood function of the hyper-parameter is not available in closed form and the posterior distribution is intractable. Previous works have dealt with this problem using a multi-step approach based on the EM algorithm and Markov Chain Monte Carlo (MCMC). We propose a framework based on recent developments in adaptive MCMC, where this problem is addressed more efficiently using a single Monte Carlo run. We discuss the convergence of the algorithm and its connection with the EM algorithm. We apply our algorithm to the Bayesian Lasso of Park and Casella (J. Am. Stat. Assoc. 103:681–686, 2008) and on the empirical Bayes variable selection of George and Foster (J. Am. Stat. Assoc. 87:731–747, 2000).     

Estimating the Burr XII Parameters in Constant-Stress Partially Accelerated Life Tests Under Multiple Censored Data

In this article, we present the performance of the maximum likelihood estimates of the Burr XII parameters for constant-stress partially accelerated life tests under multiple censored data. Two maximum likelihood estimation methods are considered. One method is based on observed-data likelihood function and the maximum likelihood estimates are obtained by using the quasi-Newton algorithm. The other method is based on complete-data likelihood function and the maximum likelihood estimates are derived by using the expectation-maximization (EM) algorithm. The variance–covariance matrices are derived to construct the confidence intervals of the parameters. The performance of these two algorithms is compared with each other by a simulation study. The simulation results show that the maximum likelihood estimation via the EM algorithm outperforms the quasi-Newton algorithm in terms of the absolute relative bias, the bias, the root mean square error and the coverage rate. Finally, a numerical example is given to illustrate the performance of the proposed methods.     

Likelihood-Based Inference in Autoregressive Models with Scaled t-Distributed Innovations by Means of EM-Based Algorithms

This article applies the EM-based (ECM and ECME) algorithms to find the maximum likelihood estimates of model parameters in general AR models with independent scaled t-distributed innovations whenever the degrees of freedom are unknown. The ECME, sharing advantages with both EM and Newton–Raphson algorithms, is an extension of ECM, which itself is an extension of the EM algorithm. The ECM and ECME algorithms, which are analytically quite simple to use, are then compared based on the computational running time and the accuracy of estimation via a simulation study. The results demonstrate that the ECME is efficient and usable in practice. We also show how our method can be applied to the Wolfer's sunspot data.     

Probabilistic relabelling strategies for the label switching problem in Bayesian mixture models

The label switching problem is caused by the likelihood of a Bayesian mixture model being invariant to permutations of the labels. The permutation can change multiple times between Markov Chain Monte Carlo (MCMC) iterations making it difficult to infer component-specific parameters of the model. Various so-called ‘relabelling’ strategies exist with the goal to ‘undo’ the label switches that have occurred to enable estimation of functions that depend on component-specific parameters. Existing deterministic relabelling algorithms rely upon specifying a loss function, and relabelling by minimising its posterior expected loss. In this paper we develop probabilistic approaches to relabelling that allow for estimation and incorporation of the uncertainty in the relabelling process. Variants of the probabilistic relabelling algorithm are introduced and compared to existing deterministic relabelling algorithms. We demonstrate that the idea of probabilistic relabelling can be expressed in a rigorous framework based on the EM algorithm.     

A comparison between the simulated annealing and the EM algorithms in normal mixture decompositions

We compare the performances of the simulated annealing and the EM algorithms in problems of decomposition of normal mixtures according to the likelihood approach. In this case the likelihood function has multiple maxima and singularities, and we consider a suitable reformulation of the problem which yields an optimization problem having a global solution and at least a smaller number of spurious maxima. The results are compared considering some distance measures between the estimated distributions and the true ones. No overwhelming superiority of either method has been demonstrated, though in one of our cases simulated annealing achieved better results.     

Fitting mixtures of linear regressions

In most applications, the parameters of a mixture of linear regression models are estimated by maximum likelihood using the expectation maximization (EM) algorithm. In this article, we propose the comparison of three algorithms to compute maximum likelihood estimates of the parameters of these models: the EM algorithm, the classification EM algorithm and the stochastic EM algorithm. The comparison of the three procedures was done through a simulation study of the performance (computational effort, statistical properties of estimators and goodness of fit) of these approaches on simulated data sets.

Simulation results show that the choice of the approach depends essentially on the configuration of the true regression lines and the initialization of the algorithms.     

An automated (Markov chain) Monte Carlo EM algorithm

We present an automated Monte Carlo EM (MCEM) algorithm which efficiently assesses Monte Carlo error in the presence of dependent Monte Carlo, particularly Markov chain Monte Carlo, E-step samples and chooses an appropriate Monte Carlo sample size to minimize this Monte Carlo error with respect to progressive EM step estimates. Monte Carlo error is gauged though an application of the central limit theorem during renewal periods of the MCMC sampler used in the E-step. The resulting normal approximation allows us to construct a rigorous and adaptive rule for updating the Monte Carlo sample size each iteration of the MCEM algorithm. We illustrate our automated routine and compare the performance with competing MCEM algorithms in an analysis of a data set fit by a generalized linear mixed model.     

Standard errors for EM estimation

The EM algorithm is a popular method for computing maximum likelihood estimates. One of its drawbacks is that it does not produce standard errors as a by-product. We consider obtaining standard errors by numerical differentiation. Two approaches are considered. The first differentiates the Fisher score vector to yield the Hessian of the log-likelihood. The second differentiates the EM operator and uses an identity that relates its derivative to the Hessian of the log-likelihood. The well-known SEM algorithm uses the second approach. We consider three additional algorithms: one that uses the first approach and two that use the second. We evaluate the complexity and precision of these three and the SEM in algorithm seven examples. The first is a single-parameter example used to give insight. The others are three examples in each of two areas of EM application: Poisson mixture models and the estimation of covariance from incomplete data. The examples show that there are algorithms that are much simpler and more accurate than the SEM algorithm. Hopefully their simplicity will increase the availability of standard error estimates in EM applications. It is shown that, as previously conjectured, a symmetry diagnostic can accurately estimate errors arising from numerical differentiation. Some issues related to the speed of the EM algorithm and algorithms that differentiate the EM operator are identified.     

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 Analysis of the Censored Regression Model with an AEPD Error Term

We present the censored regression model with the error term following the asymmetric exponential power distribution. We propose three Markov chain Monte Carlo (MCMC) algorithms: the first one uses the probability integral transformation; the second one uses a combination of the probability integral transformation and random walk draws; while the third one uses random walk draws. Using simulated data we compare the performance of the three MCMC algorithms. Then we compare the posterior means, or Bayes estimates, with maximum likelihood estimates. We estimate the stock option portion of executive compensation as an example of the empirical application.     

The Cox-Aalen model for left-truncated and mixed interval-censored data

Scheike and Zhang [An additive-multiplicative Cox-Aalen regression model. Scand J Stat. 2002;29:75–88] proposed a flexible additive-multiplicative hazard model, called the Cox-Aalen model, by replacing the baseline hazard function in the well-known Cox model with a covariate-dependent Aalen model, which allows for both fixed and dynamic covariate effects. In this paper, based on left-truncated and mixed interval-censored (LT-MIC) data, we consider maximum likelihood estimation for the Cox-Aalen model with fixed covariates. We propose expectation-maximization (EM) algorithms for obtaining the conditional maximum likelihood estimators (cMLE) of the regression coefficients for the Cox-Aalen model. We establish the consistency of the cMLE. Numerical studies show that estimation via the EM algorithms performs well.     

On parallelizable Markov chain Monte Carlo algorithms with waste-recycling

Parallelizable Markov chain Monte Carlo (MCMC) generates multiple proposals and parallelizes the evaluations of the likelihood function on different cores at each MCMC iteration. Inspired by Calderhead (Proc Natl Acad Sci 111(49):17408–17413, 2014), we introduce a general ‘waste-recycling’ framework for parallelizable MCMC, under which we show that using weighted samples from waste-recycling is preferable to resampling in terms of both statistical and computational efficiencies. We also provide a simple-to-use criteria, the generalized effective sample size, for evaluating efficiencies of parallelizable MCMC algorithms, which applies to both the waste-recycling and the vanilla versions. A moment estimator of the generalized effective sample size is provided and shown to be reasonably accurate by simulations.     

Strategies for Fitting Large,Geostatistical Data in MCMC Simulation

Models for geostatistical data introduce spatial dependence in the covariance matrix of location-specific random effects. This is usually defined to be a parametric function of the distances between locations. Bayesian formulations of such models overcome asymptotic inference and estimation problems involved in maximum likelihood-based approaches and can be fitted using Markov chain Monte Carlo (MCMC) simulation. The MCMC implementation, however, requires repeated inversions of the covariance matrix which makes the problem computationally intensive, especially for large number of locations. In the present work, we propose to convert the spatial covariance matrix to a sparse matrix and compare a number of numerical algorithms especially suited within the MCMC framework in order to accelerate large matrix inversion. The algorithms are assessed empirically on simulated datasets of different size and sparsity. We conclude that the band solver applied after ordering the distance matrix reduces the computational time in inverting covariance matrices substantially.     

Efficient estimation for incomplete multivariate data

We review the Fisher scoring and EM algorithms for incomplete multivariate data from an estimating function point of view, and examine the corresponding quasi-score functions under second-moment assumptions. A bias-corrected REML-type estimator for the covariance matrix is derived, and the Fisher, Godambe and empirical sandwich information matrices are compared. We make a numerical investigation of the two algorithms, and compare with a hybrid algorithm, where Fisher scoring is used for the mean vector and the EM algorithm for the covariance matrix.     

An augmented data scoring algorithm for maximum likelihood

The expectation-maximization (EM) method facilitates computation of max¬imum likelihood (ML) and maximum penalized likelihood (MPL) solutions. The procedure requires specification of unobservabie complete data which augment the measured or incomplete data. This specification defines a conditional expectation of the complete data log-likelihood function which is computed in the E-stcp. The EM algorithm is most effective when maximizing the iunction Q{0) denned in the F-stnp is easier than maximizing the likelihood function.

The Monte Carlo EM (MCEM) algorithm of Wei & Tanner (1990) was introduced for problems where computation of Q is difficult or intractable. However Monte Carlo can he computationally expensive, e.g. in signal processing applications involving large numbers of parameters. We provide another approach: a modification of thc standard EM algorithm avoiding computation of conditional expectations.     

The EM algorithm with gradient function update for discrete mixtures with known (fixed) number of components

The paper is focussing on some recent developments in nonparametric mixture distributions. It discusses nonparametric maximum likelihood estimation of the mixing distribution and will emphasize gradient type results, especially in terms of global results and global convergence of algorithms such as vertex direction or vertex exchange method. However, the NPMLE (or the algorithms constructing it) provides also an estimate of the number of components of the mixing distribution which might be not desirable for theoretical reasons or might be not allowed from the physical interpretation of the mixture model. When the number of components is fixed in advance, the before mentioned algorithms can not be used and globally convergent algorithms do not exist up to now. Instead, the EM algorithm is often used to find maximum likelihood estimates. However, in this case multiple maxima are often occuring. An example from a meta-analyis of vitamin A and childhood mortality is used to illustrate the considerable, inferential importance of identifying the correct global likelihood. To improve the behavior of the EM algorithm we suggest a combination of gradient function steps and EM steps to achieve global convergence leading to the EM algorithm with gradient function update (EMGFU). This algorithms retains the number of components to be exactly k and typically converges to the global maximum. The behavior of the algorithm is highlighted at hand of several examples.     

Likelihood-Based Inference for Multivariate Skew-Normal Regression Models

In this article, we present EM algorithms for performing maximum likelihood estimation for three multivariate skew-normal regression models of considerable practical interest. We also consider the restricted estimation of the parameters of certain important special cases of two models. The methodology developed is applied in the analysis of longitudinal data on dental plaque and cholesterol levels.     

Markov chain Monte Carlo estimation of a mixture item response theory model

Markov chain Monte Carlo (MCMC) algorithms have been shown to be useful for estimation of complex item response theory (IRT) models. Although an MCMC algorithm can be very useful, it also requires care in use and interpretation of results. In particular, MCMC algorithms generally make extensive use of priors on model parameters. In this paper, MCMC estimation is illustrated using a simple mixture IRT model, a mixture Rasch model (MRM), to demonstrate how the algorithm operates and how results may be affected by some commonly used priors. Priors on the probabilities of mixtures, label switching, model selection, metric anchoring, and implementation of the MCMC algorithm using WinBUGS are described, and their effects illustrated on parameter recovery in practical testing situations. In addition, an example is presented in which an MRM is fitted to a set of educational test data using the MCMC algorithm and a comparison is illustrated with results from three existing maximum likelihood estimation methods.     

The EM Algorithm—an Old Folk-song Sung to a Fast New Tune

Celebrating the 20th anniversary of the presentation of the paper by Dempster, Laird and Rubin which popularized the EM algorithm, we investigate, after a brief historical account, strategies that aim to make the EM algorithm converge faster while maintaining its simplicity and stability (e.g. automatic monotone convergence in likelihood). First we introduce the idea of a 'working parameter' to facilitate the search for efficient data augmentation schemes and thus fast EM implementations. Second, summarizing various recent extensions of the EM algorithm, we formulate a general alternating expectation–conditional maximization algorithm AECM that couples flexible data augmentation schemes with model reduction schemes to achieve efficient computations. We illustrate these methods using multivariate t -models with known or unknown degrees of freedom and Poisson models for image reconstruction. We show, through both empirical and theoretical evidence, the potential for a dramatic reduction in computational time with little increase in human effort. We also discuss the intrinsic connection between EM-type algorithms and the Gibbs sampler, and the possibility of using the techniques presented here to speed up the latter. The main conclusion of the paper is that, with the help of statistical considerations, it is possible to construct algorithms that are simple, stable and fast.