A Spectral Estimator of Arma Parameters from Thresholded Data

We consider computationally-fast methods for estimating parameters in ARMA processes from binary time series data, obtained by thresholding the latent ARMA process. All methods involve matching estimated and expected autocorrelations of the binary series. In particular, we focus on the spectral representation of the likelihood of an ARMA process and derive a restricted form of this likelihood, which uses correlations at only the first few lags. We contrast these methods with an efficient but computationally-intensive Markov chain Monte Carlo (MCMC) method. In a simulation study we show that, for a range of ARMA processes, the spectral method is more efficient than variants of least squares and much faster than MCMC. We illustrate by fitting an ARMA(2,1) model to a binary time series of cow feeding data.     

Exact Bayesian Inference and Model Selection for Stochastic Models of Epidemics Among a Community of Households

Abstract.  Much recent methodological progress in the analysis of infectious disease data has been due to Markov chain Monte Carlo (MCMC) methodology. In this paper, it is illustrated that rejection sampling can also be applied to a family of inference problems in the context of epidemic models, avoiding the issues of convergence associated with MCMC methods. Specifically, we consider models for epidemic data arising from a population divided into households. The models allow individuals to be potentially infected both from outside and from within the household. We develop methodology for selection between competing models via the computation of Bayes factors. We also demonstrate how an initial sample can be used to adjust the algorithm and improve efficiency. The data are assumed to consist of the final numbers ultimately infected within a sample of households in some community. The methods are applied to data taken from outbreaks of influenza.     

Parallel multivariate slice sampling

Slice sampling provides an easily implemented method for constructing a Markov chain Monte Carlo (MCMC) algorithm. However, slice sampling has two major drawbacks: (i) it requires repeated evaluation of likelihoods for each update, which can make it impractical when evaluations are expensive or as the number of evaluations grows (geometrically) with the dimension of the slice sampler, and (ii) since it can be challenging to construct multivariate updates, the updates are typically univariate, which often results in slow mixing samplers. We propose an approach to multivariate slice sampling that naturally lends itself to a parallel implementation. Our approach takes advantage of recent advances in computer architectures, for instance, the newest generation of graphics cards can execute roughly 30,000 threads simultaneously. We demonstrate that it is possible to construct a multivariate slice sampler that has good mixing properties and is efficient in terms of computing time. The contributions of this article are therefore twofold. We study approaches for constructing a multivariate slice sampler, and we show how parallel computing can be useful for making MCMC algorithms computationally efficient. We study various implementations of our algorithm in the context of real and simulated data.     

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 Block Updating in Markov Random Field Models for Disease Mapping

Gaussian Markov random field (GMRF) models are commonly used to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly single-site updating algorithms have been considered. However, convergence and mixing properties of such algorithms can be extremely poor due to strong dependencies of parameters in the posterior distribution. In this paper, we propose various block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for non-standard full conditionals, and can be applied in a modular fashion in a large number of different scenarios. For illustration we consider three different applications: two formulations for spatial modelling of a single disease (with and without additional unstructured parameters respectively), and one formulation for the joint analysis of two diseases. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. Implementation of such block algorithms is relatively easy using methods for fast sampling of Gaussian Markov random fields ( Rue, 2001 ). By comparison, Monte Carlo estimates based on single-site updating can be rather misleading, even for very long runs. Our results may have wider relevance for efficient MCMC simulation in hierarchical models with Markov random field components.     

A general method to determine sampling windows for nonlinear mixed effects models with an application to population pharmacokinetic studies

Optimal design methods have been proposed to determine the best sampling times when sparse blood sampling is required in clinical pharmacokinetic studies. However, the optimal blood sampling time points may not be feasible in clinical practice. Sampling windows, a time interval for blood sample collection, have been proposed to provide flexibility in blood sampling times while preserving efficient parameter estimation. Because of the complexity of the population pharmacokinetic models, which are generally nonlinear mixed effects models, there is no analytical solution available to determine sampling windows. We propose a method for determination of sampling windows based on MCMC sampling techniques. The proposed method attains a stationary distribution rapidly and provides time-sensitive windows around the optimal design points. The proposed method is applicable to determine sampling windows for any nonlinear mixed effects model although our work focuses on an application to population pharmacokinetic models.     

Bayesian weighted inference from surveys

Data from large surveys are often supplemented with sampling weights that are designed to reflect unequal probabilities of response and selection inherent in complex survey sampling methods. We propose two methods for Bayesian estimation of parametric models in a setting where the survey data and the weights are available, but where information on how the weights were constructed is unavailable. The first approach is to simply replace the likelihood with the pseudo likelihood in the formulation of Bayes theorem. This is proven to lead to a consistent estimator but also leads to credible intervals that suffer from systematic undercoverage. Our second approach involves using the weights to generate a representative sample which is integrated into a Markov chain Monte Carlo (MCMC) or other simulation algorithms designed to estimate the parameters of the model. In the extensive simulation studies, the latter methodology is shown to achieve performance comparable to the standard frequentist solution of pseudo maximum likelihood, with the added advantage of being applicable to models that require inference via MCMC. The methodology is demonstrated further by fitting a mixture of gamma densities to a sample of Australian household income.     

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

The article considers Bayesian analysis of hierarchical models for count, binomial and multinomial data using efficient MCMC sampling procedures. To this end, an improved method of auxiliary mixture sampling is proposed. In contrast to previously proposed samplers the method uses a bounded number of latent variables per observation, independent of the intensity of the underlying Poisson process in the case of count data, or of the number of experiments in the case of binomial and multinomial data. The bounded number of latent variables results in a more general error distribution, which is a negative log-Gamma distribution with arbitrary integer shape parameter. The required approximations of these distributions by Gaussian mixtures have been computed. Overall, the improvement leads to a substantial increase in efficiency of auxiliary mixture sampling for highly structured models. The method is illustrated for finite mixtures of generalized linear models and an epidemiological case study.     

Bayesian quantile regression for longitudinal data models

In this paper, we discuss a fully Bayesian quantile inference using Markov Chain Monte Carlo (MCMC) method for longitudinal data models with random effects. Under the assumption of error term subject to asymmetric Laplace distribution, we establish a hierarchical Bayesian model and obtain the posterior distribution of unknown parameters at τ-th level. We overcome the current computational limitations using two approaches. One is the general MCMC technique with Metropolis–Hastings algorithm and another is the Gibbs sampling from the full conditional distribution. These two methods outperform the traditional frequentist methods under a wide array of simulated data models and are flexible enough to easily accommodate changes in the number of random effects and in their assumed distribution. We apply the Gibbs sampling method to analyse a mouse growth data and some different conclusions from those in the literatures are obtained.     

Efficient Classification-Based Relabeling in Mixture Models

Effective component relabeling in Bayesian analyses of mixture models is critical to the routine use of mixtures in classification with analysis based on Markov chain Monte Carlo methods. The classification-based relabeling approach here is computationally attractive and statistically effective, and scales well with sample size and number of mixture components concordant with enabling routine analyses of increasingly large data sets. Building on the best of existing methods, practical relabeling aims to match data:component classification indicators in MCMC iterates with those of a defined reference mixture distribution. The method performs as well as or better than existing methods in small dimensional problems, while being practically superior in problems with larger data sets as the approach is scalable. We describe examples and computational benchmarks, and provide supporting code with efficient computational implementation of the algorithm that will be of use to others in practical applications of mixture models.     

Inference for Lévy‐Driven Stochastic Volatility Models via Adaptive Sequential Monte Carlo

Abstract. We investigate simulation methodology for Bayesian inference in Lévy‐driven stochastic volatility (SV) models. Typically, Bayesian inference from such models is performed using Markov chain Monte Carlo (MCMC); this is often a challenging task. Sequential Monte Carlo (SMC) samplers are methods that can improve over MCMC; however, there are many user‐set parameters to specify. We develop a fully automated SMC algorithm, which substantially improves over the standard MCMC methods in the literature. To illustrate our methodology, we look at a model comprised of a Heston model with an independent, additive, variance gamma process in the returns equation. The driving gamma process can capture the stylized behaviour of many financial time series and a discretized version, fit in a Bayesian manner, has been found to be very useful for modelling equity data. We demonstrate that it is possible to draw exact inference, in the sense of no time‐discretization error, from the Bayesian SV model.     

Bayesian estimation with integrated nested Laplace approximation for binary logit mixed models

In multilevel models for binary responses, estimation is computationally challenging due to the need to evaluate intractable integrals. In this paper, we investigate the performance of integrated nested Laplace approximation (INLA), a fast deterministic method for Bayesian inference. In particular, we conduct an extensive simulation study to compare the results obtained with INLA to the results obtained with a traditional stochastic method for Bayesian inference (MCMC Gibbs sampling), and with maximum likelihood through adaptive quadrature. Particular attention is devoted to the case of small number of clusters. The specification of the prior distribution for the cluster variance plays a crucial role and it turns out to be more relevant than the choice of the estimation method. The simulations show that INLA has an excellent performance as it achieves good accuracy (similar to MCMC) with reduced computational times (similar to adaptive quadrature).     

Some asymptotic results for semiparametric nonlinear mixed-effects models with incomplete data

In modeling complex longitudinal data, semiparametric nonlinear mixed-effects (SNLME) models are very flexible and useful. Covariates are often introduced in the models to partially explain the inter-individual variations. In practice, data are often incomplete in the sense that there are often measurement errors and missing data in longitudinal studies. The likelihood method is a standard approach for inference for these models but it can be computationally very challenging, so computationally efficient approximate methods are quite valuable. However, the performance of these approximate methods is often based on limited simulation studies, and theoretical results are unavailable for many approximate methods. In this article, we consider a computationally efficient approximate method for a class of SNLME models with incomplete data and investigate its theoretical properties. We show that the estimates based on the approximate method are consistent and asymptotically normally distributed.     

Monte Carlo Likelihood Estimation for Three Multivariate Stochastic Volatility Models

Estimating parameters in a stochastic volatility (SV) model is a challenging task. Among other estimation methods and approaches, efficient simulation methods based on importance sampling have been developed for the Monte Carlo maximum likelihood estimation of univariate SV models. This paper shows that importance sampling methods can be used in a general multivariate SV setting. The sampling methods are computationally efficient. To illustrate the versatility of this approach, three different multivariate stochastic volatility models are estimated for a standard data set. The empirical results are compared to those from earlier studies in the literature. Monte Carlo simulation experiments, based on parameter estimates from the standard data set, are used to show the effectiveness of the importance sampling methods.     

Computational methods for complex stochastic systems: a review of some alternatives to MCMC

We consider analysis of complex stochastic models based upon partial information. MCMC and reversible jump MCMC are often the methods of choice for such problems, but in some situations they can be difficult to implement; and suffer from problems such as poor mixing, and the difficulty of diagnosing convergence. Here we review three alternatives to MCMC methods: importance sampling, the forward-backward algorithm, and sequential Monte Carlo (SMC). We discuss how to design good proposal densities for importance sampling, show some of the range of models for which the forward-backward algorithm can be applied, and show how resampling ideas from SMC can be used to improve the efficiency of the other two methods. We demonstrate these methods on a range of examples, including estimating the transition density of a diffusion and of a discrete-state continuous-time Markov chain; inferring structure in population genetics; and segmenting genetic divergence data.     

Monte Carlo Likelihood Estimation for Three Multivariate Stochastic Volatility Models

Estimating parameters in a stochastic volatility (SV) model is a challenging task. Among other estimation methods and approaches, efficient simulation methods based on importance sampling have been developed for the Monte Carlo maximum likelihood estimation of univariate SV models. This paper shows that importance sampling methods can be used in a general multivariate SV setting. The sampling methods are computationally efficient. To illustrate the versatility of this approach, three different multivariate stochastic volatility models are estimated for a standard data set. The empirical results are compared to those from earlier studies in the literature. Monte Carlo simulation experiments, based on parameter estimates from the standard data set, are used to show the effectiveness of the importance sampling methods.     

Sampling schemes for generalized linear Dirichlet process random effects models

We evaluate MCMC sampling schemes for a variety of link functions in generalized linear models with Dirichlet process random effects. First, we find that there is a large amount of variability in the performance of MCMC algorithms, with the slice sampler typically being less desirable than either a Kolmogorov–Smirnov mixture representation or a Metropolis–Hastings algorithm. Second, in fitting the Dirichlet process, dealing with the precision parameter has troubled model specifications in the past. Here we find that incorporating this parameter into the MCMC sampling scheme is not only computationally feasible, but also results in a more robust set of estimates, in that they are marginalized-over rather than conditioned-upon. Applications are provided with social science problems in areas where the data can be difficult to model, and we find that the nonparametric nature of the Dirichlet process priors for the random effects leads to improved analyses with more reasonable inferences.     

Bayesian regression analysis of data with censored initiating and terminating times: applications to aids

Data with censored initiating and terminating times arises quite frequently in acquired immunodeficiency syndrome (AIDS) epidemiologic studies. Analysis of such data involves a complicated bivariate likelihood, which is difficult to deal with computationally. Bayesian analysis, op the other hand, presents added complexities that have yet to be resolved. By exploiting the simple form of a complete data likelihood and utilizing the power of a Markov Chain Monte Carlo (MCMC) algorithm, this paper presents a methodology for fitting Bayesian regression models to such data. The proposed methods extend the work of Sinha (1997), who considered non-parametric Bayesian analysis of this type of data. The methodology is illustiated with an application to a cohort of HIV infected hemophiliac patients.     

Efficient calculation of the normalizing constant of the autologistic and related models on the cylinder and lattice

Summary. Motivated by the autologistic model for the analysis of spatial binary data on the two-dimensional lattice, we develop efficient computational methods for calculating the normalizing constant for models for discrete data defined on the cylinder and lattice. Because the normalizing constant is generally unknown analytically, statisticians have developed various ad hoc methods to overcome this difficulty. Our aim is to provide computationally and statistically efficient methods for calculating the normalizing constant so that efficient likelihood-based statistical methods are then available for inference. We extend the so-called transition method to find a feasible computational method of obtaining the normalizing constant for the cylinder boundary condition. To extend the result to the free-boundary condition on the lattice we use an efficient path sampling Markov chain Monte Carlo scheme. The methods are generally applicable to association patterns other than spatial, such as clustered binary data, and to variables taking three or more values described by, for example, Potts models.     

Markov chain Monte Carlo methods for parameter estimation of the modified Weibull distribution

In this paper, the Markov chain Monte Carlo (MCMC) method is used to estimate the parameters of a modified Weibull distribution based on a complete sample. While maximum-likelihood estimation (MLE) is the most used method for parameter estimation, MCMC has recently emerged as a good alternative. When applied to parameter estimation, MCMC methods have been shown to be easy to implement computationally, the estimates always exist and are statistically consistent, and their probability intervals are convenient to construct. Details of applying MCMC to parameter estimation for the modified Weibull model are elaborated and a numerical example is presented to illustrate the methods of inference discussed in this paper. To compare MCMC with MLE, a simulation study is provided, and the differences between the estimates obtained by the two algorithms are examined.