Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation

We propose a two-stage algorithm for computing maximum likelihood estimates for a class of spatial models. The algorithm combines Markov chain Monte Carlo methods such as the Metropolis–Hastings–Green algorithm and the Gibbs sampler, and stochastic approximation methods such as the off-line average and adaptive search direction. A new criterion is built into the algorithm so stopping is automatic once the desired precision has been set. Simulation studies and applications to some real data sets have been conducted with three spatial models. We compared the algorithm proposed with a direct application of the classical Robbins–Monro algorithm using Wiebe's wheat data and found that our procedure is at least 15 times faster.     

Exact inference in contingency tables via stochastic approximation Monte Carlo

Monte Carlo methods for the exact inference have received much attention recently in complete or incomplete contingency table analysis. However, conventional Markov chain Monte Carlo, such as the Metropolis–Hastings algorithm, and importance sampling methods sometimes generate the poor performance by failing to produce valid tables. In this paper, we apply an adaptive Monte Carlo algorithm, the stochastic approximation Monte Carlo algorithm (SAMC; Liang, Liu, & Carroll, 2007), to the exact test of the goodness-of-fit of the model in complete or incomplete contingency tables containing some structural zero cells. The numerical results are in favor of our method in terms of quality of estimates.     

Automatic choice of driving values in Monte Carlo likelihood approximation via posterior simulations

For models with random effects or missing data, the likelihood function is sometimes intractable analytically but amenable to Monte Carlo approximation. To get a good approximation, the parameter value that drives the simulations should be sufficiently close to the maximum likelihood estimate (MLE) which unfortunately is unknown. Introducing a working prior distribution, we express the likelihood function as a posterior expectation and approximate it using posterior simulations. If the sample size is large, the sample information is likely to outweigh the prior specification and the posterior simulations will be concentrated around the MLE automatically, leading to good approximation of the likelihood near the MLE. For smaller samples, we propose to use the current posterior as the next prior distribution to make the posterior simulations closer to the MLE and hence improve the likelihood approximation. By using the technique of data duplication, we can simulate from the sharpened posterior distribution without actually updating the prior distribution. The suggested method works well in several test cases. A more complex example involving censored spatial data is also discussed.     

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.     

Bayesian inference for generalized additive mixed models based on Markov random field priors

Most regression problems in practice require flexible semiparametric forms of the predictor for modelling the dependence of responses on covariates. Moreover, it is often necessary to add random effects accounting for overdispersion caused by unobserved heterogeneity or for correlation in longitudinal or spatial data. We present a unified approach for Bayesian inference via Markov chain Monte Carlo simulation in generalized additive and semiparametric mixed models. Different types of covariates, such as the usual covariates with fixed effects, metrical covariates with non-linear effects, unstructured random effects, trend and seasonal components in longitudinal data and spatial covariates, are all treated within the same general framework by assigning appropriate Markov random field priors with different forms and degrees of smoothness. We applied the approach in several case-studies and consulting cases, showing that the methods are also computationally feasible in problems with many covariates and large data sets. In this paper, we choose two typical applications.     

Bayesian inference for stochastic multitype epidemics in structured populations via random graphs

Summary.  The paper is concerned with new methodology for statistical inference for final outcome infectious disease data using certain structured population stochastic epidemic models. A major obstacle to inference for such models is that the likelihood is both analytically and numerically intractable. The approach that is taken here is to impute missing information in the form of a random graph that describes the potential infectious contacts between individuals. This level of imputation overcomes various constraints of existing methodologies and yields more detailed information about the spread of disease. The methods are illustrated with both real and test data.     

Consistency of stochastic approximation algorithm with quasi-associated random errors

ABSTRACT

Many mathematical and physical problems are led to find a root of a real function f. This kind of equation is an inverse problem and it is difficult to solve it. Especially in engineering sciences, the analytical expression of the function f is unknown to the experimenter, but it can be measured at each point x_k with M(x_k) as expected value and induced error ξ_k. The aim is to approximate the unique root θ under some assumptions on the function f and errors ξ_k. We use a stochastic approximation algorithm that constructs a sequence (x_k)_{k ? 1}. We establish the almost complete convergence of the sequence (x_k)_k to the exact root θ by considering the errors (ξ_k)_k quasi-associated and we illustrate the method by numerical examples to show its efficiency.     

A random cluster algorithm for some continuous Markov random fields

Based on a random cluster representation, the Swendsen–Wang algorithm for the Ising and Potts distributions is extended to a class of continuous Markov random fields. The algorithm can be described briefly as follows. A given configuration is decomposed into clusters. Probabilities for flipping the values of the random variables in each cluster are calculated. According to these probabilities, values of all the random variables in each cluster will be either updated or kept unchanged and this is done independently across the clusters. A new configuration is then obtained. We will show through a simulation study that, like the Swendsen–Wang algorithm in the case of Ising and Potts distributions, the cluster algorithm here also outperforms the Gibbs sampler in beating the critical slowing down for some strongly correlated Markov random fields.     

Particle methods for maximum likelihood estimation in latent variable models

Standard methods for maximum likelihood parameter estimation in latent variable models rely on the Expectation-Maximization algorithm and its Monte Carlo variants. Our approach is different and motivated by similar considerations to simulated annealing; that is we build a sequence of artificial distributions whose support concentrates itself on the set of maximum likelihood estimates. We sample from these distributions using a sequential Monte Carlo approach. We demonstrate state-of-the-art performance for several applications of the proposed approach.     

On-line expectation–maximization algorithm for latent data models

Summary.  We propose a generic on-line (also sometimes called adaptive or recursive) version of the expectation–maximization (EM) algorithm applicable to latent variable models of independent observations. Compared with the algorithm of Titterington, this approach is more directly connected to the usual EM algorithm and does not rely on integration with respect to the complete-data distribution. The resulting algorithm is usually simpler and is shown to achieve convergence to the stationary points of the Kullback–Leibler divergence between the marginal distribution of the observation and the model distribution at the optimal rate, i.e. that of the maximum likelihood estimator. In addition, the approach proposed is also suitable for conditional (or regression) models, as illustrated in the case of the mixture of linear regressions model.     

Fitting Bayesian multiple random effects models

Bayesian random effects models may be fitted using Gibbs sampling, but the Gibbs sampler can be slow mixing due to what might be regarded as lack of model identifiability. This slow mixing substantially increases the number of iterations required during Gibbs sampling. We present an analysis of data on immunity after Rubella vaccinations which results in a slow-mixing Gibbs sampler. We show that this problem of slow mixing can be resolved by transforming the random effects and then, if desired, expressing their joint prior distribution as a sequence of univariate conditional distributions. The resulting analysis shows that the decline in antibodies after Rubella vaccination is relatively shallow compared to the decline in antibodies which has been shown after Hepatitis B vaccination.     

Hierarchical random effect models for coastal erosion of cliffs in the Holderness coast

Prediction of possible cliff erosion at some future date is fundamental to coastal planning and shoreline management, for example to avoid development in vulnerable areas. Historically, to predict cliff recession rates deterministic methods were used. More recently, recession predictions have been expressed in probabilistic terms. However, to date, only simplistic models have been developed. We consider the cliff erosion along the Holderness Coast. Since 1951 a monitoring program has been started in 118 stations along the coast, providing an invaluable, but often missing, source of information. We build hierarchical random effect models, taking account of the known dynamics of the process and including the missing information.     

Analysis of generalized linear mixed models via a stochastic approximation algorithm with Markov chain Monte-Carlo method

In recent years much effort has been devoted to maximum likelihood estimation of generalized linear mixed models. Most of the existing methods use the EM algorithm, with various techniques in handling the intractable E-step. In this paper, a new implementation of a stochastic approximation algorithm with Markov chain Monte Carlo method is investigated. The proposed algorithm is computationally straightforward and its convergence is guaranteed. A simulation and three real data sets, including the challenging salamander data, are used to illustrate the procedure and to compare it with some existing methods. The results indicate that the proposed algorithm is an attractive alternative for problems with a large number of random effects or with high dimensional intractable integrals in the likelihood function.     

Maximum likelihood estimates in the multivariate normal with patterned mean and covariance via the em algorithm

The maximum likelihood equations for a multivariate normal model with structured mean and structured covariance matrix may not have an explicit solution. In some cases the model's error term may be decomposed as the sum of two independent error terms, each having a patterned covariance matrix, such that if one of the unobservable error terms is artificially treated as "missing data", the EM algorithm can be used to compute the maximum likelihood estimates for the original problem. Some decompositions produce likelihood equations which do not have an explicit solution at each iteration of the EM algorithm, but within-iteration explicit solutions are shown for two general classes of models including covariance component models used for analysis of longitudinal data.     

Sampling from the posterior distribution in generalized linear mixed models

Generalized linear mixed models provide a unified framework for treatment of exponential family regression models, overdispersed data and longitudinal studies. These problems typically involve the presence of random effects and this paper presents a new methodology for making Bayesian inference about them. The approach is simulation-based and involves the use of Markov chain Monte Carlo techniques. The usual iterative weighted least squares algorithm is extended to include a sampling step based on the Metropolis–Hastings algorithm thus providing a unified iterative scheme. Non-normal prior distributions for the regression coefficients and for the random effects distribution are considered. Random effect structures with nesting required by longitudinal studies are also considered. Particular interests concern the significance of regression coefficients and assessment of the form of the random effects. Extensions to unknown scale parameters, unknown link functions, survival and frailty models are outlined.     

Adaptive Gaussian Markov random fields with applications in human brain mapping

Summary.  Functional magnetic resonance imaging has become a standard technology in human brain mapping. Analyses of the massive spatiotemporal functional magnetic resonance imaging data sets often focus on parametric or non-parametric modelling of the temporal component, whereas spatial smoothing is based on Gaussian kernels or random fields. A weakness of Gaussian spatial smoothing is underestimation of activation peaks or blurring of high curvature transitions between activated and non-activated regions of the brain. To improve spatial adaptivity, we introduce a class of inhomogeneous Markov random fields with stochastic interaction weights in a space-varying coefficient model. For given weights, the random field is conditionally Gaussian, but marginally it is non-Gaussian. Fully Bayesian inference, including estimation of weights and variance parameters, can be carried out through efficient Markov chain Monte Carlo simulation. Although motivated by the analysis of functional magnetic resonance imaging data, the methodological development is general and can also be used for spatial smoothing and regression analysis of areal data on irregular lattices. An application to stylized artificial data and to real functional magnetic resonance imaging data from a visual stimulation experiment demonstrates the performance of our approach in comparison with Gaussian and robustified non-Gaussian Markov random-field models.     

Modified linear projection for large spatial datasets

Recent developments in engineering techniques for spatial data collection such as geographic information systems have resulted in an increasing need for methods to analyze large spatial datasets. These sorts of datasets can be found in various fields of the natural and social sciences. However, model fitting and spatial prediction using these large spatial datasets are impractically time-consuming, because of the necessary matrix inversions. Various methods have been developed to deal with this problem, including a reduced rank approach and a sparse matrix approximation. In this article, we propose a modification to an existing reduced rank approach to capture both the large- and small-scale spatial variations effectively. We have used simulated examples and an empirical data analysis to demonstrate that our proposed approach consistently performs well when compared with other methods. In particular, the performance of our new method does not depend on the dependence properties of the spatial covariance functions.     

Bayesian local influence analysis of skew-normal spatial dynamic panel data models

The existing studies on spatial dynamic panel data model (SDPDM) mainly focus on the normality assumption of response variables and random effects. This assumption may be inappropriate in some applications. This paper proposes a new SDPDM by assuming that response variables and random effects follow the multivariate skew-normal distribution. A Markov chain Monte Carlo algorithm is developed to evaluate Bayesian estimates of unknown parameters and random effects in skew-normal SDPDM by combining the Gibbs sampler and the Metropolis–Hastings algorithm. A Bayesian local influence analysis method is developed to simultaneously assess the effect of minor perturbations to the data, priors and sampling distributions. Simulation studies are conducted to investigate the finite-sample performance of the proposed methodologies. An example is illustrated by the proposed methodologies.     

Approximating hidden Gaussian Markov random fields

Summary.  Gaussian Markov random-field (GMRF) models are frequently used in a wide variety of applications. In most cases parts of the GMRF are observed through mutually independent data; hence the full conditional of the GMRF, a hidden GMRF (HGMRF), is of interest. We are concerned with the case where the likelihood is non-Gaussian, leading to non-Gaussian HGMRF models. Several researchers have constructed block sampling Markov chain Monte Carlo schemes based on approximations of the HGMRF by a GMRF, using a second-order expansion of the log-density at or near the mode. This is possible as the GMRF approximation can be sampled exactly with a known normalizing constant. The Markov property of the GMRF approximation yields computational efficiency.The main contribution in the paper is to go beyond the GMRF approximation and to construct a class of non-Gaussian approximations which adapt automatically to the particular HGMRF that is under study. The accuracy can be tuned by intuitive parameters to nearly any precision. These non-Gaussian approximations share the same computational complexity as those which are based on GMRFs and can be sampled exactly with computable normalizing constants. We apply our approximations in spatial disease mapping and model-based geostatistical models with different likelihoods, obtain procedures for block updating and construct Metropolized independence samplers.     

On the Value of derivative evaluations and random walk suppression in Markov Chain Monte Carlo algorithms

Two strategies that can potentially improve Markov Chain Monte Carlo algorithms are to use derivative evaluations of the target density, and to suppress random walk behaviour in the chain. The use of one or both of these strategies has been investigated in a few specific applications, but neither is used routinely. We undertake a broader evaluation of these techniques, with a view to assessing their utility for routine use. In addition to comparing different algorithms, we also compare two different ways in which the algorithms can be applied to a multivariate target distribution. Specifically, the univariate version of an algorithm can be applied repeatedly to one-dimensional conditional distributions, or the multivariate version can be applied directly to the target distribution.