Bayesian analysis of latent Markov models with non-ignorable missing data

Latent Markov models (LMMs) are widely used in the analysis of heterogeneous longitudinal data. However, most existing LMMs are developed in fully observed data without missing entries. The main objective of this study is to develop a Bayesian approach for analyzing the LMMs with non-ignorable missing data. Bayesian methods for estimation and model comparison are discussed. The empirical performance of the proposed methodology is evaluated through simulation studies. An application to a data set derived from National Longitudinal Survey of Youth 1997 is presented.     

Bayesian multivariate GARCH models with dynamic correlations and asymmetric error distributions

The main goal in this paper is to develop and apply stochastic simulation techniques for GARCH models with multivariate skewed distributions using the Bayesian approach. Both parameter estimation and model comparison are not trivial tasks and several approximate and computationally intensive methods (Markov chain Monte Carlo) will be used to this end. We consider a flexible class of multivariate distributions which can model both skewness and heavy tails. Also, we do not fix tail behaviour when dealing with fat tail distributions but leave it subject to inference.     

Bayesian approach to errors-in-variables in count data regression models with departures from normality and overdispersion

In most practical applications, the quality of count data is often compromised due to errors-in-variables (EIVs). In this paper, we apply Bayesian approach to reduce bias in estimating the parameters of count data regression models that have mismeasured independent variables. Furthermore, the exposure model is misspecified with a flexible distribution, hence our approach remains robust against any departures from normality in its true underlying exposure distribution. The proposed method is also useful in realistic situations as the variance of EIVs is estimated instead of assumed as known, in contrast with other methods of correcting bias especially in count data EIVs regression models. We conduct simulation studies on synthetic data sets using Markov chain Monte Carlo simulation techniques to investigate the performance of our approach. Our findings show that the flexible Bayesian approach is able to estimate the values of the true regression parameters consistently and accurately.     

Bayesian analysis of mortality data

Congdon argued that the use of parametric modelling of mortality data is necessary in many practical demographical problems. In this paper, we focus on a form of model introduced by Heligman and Pollard in 1980, and we adopt a Bayesian analysis, using Markov chain Monte Carlo simulation, to produce the posterior summaries required. This opens the way to richer, more flexible inference summaries and avoids the numerical problems that are encountered with classical methods. Particular methodologies to cope with incomplete life-tables and a derivation of joint lifetimes, median times to death and related quantities of interest are also presented.     

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.     

Bayesian analysis of directed graphs data with applications to social networks

Summary.  A fully Bayesian analysis of directed graphs, with particular emphasis on applica- tions in social networks, is explored. The model is capable of incorporating the effects of covariates, within and between block ties and multiple responses. Inference is straightforward by using software that is based on Markov chain Monte Carlo methods. Examples are provided which highlight the variety of data sets that can be entertained and the ease with which they can be analysed.     

A threshold autoregressive model for wholesale electricity prices

Summary.  We introduce a discrete time model for electricity prices which accounts for both transitory spikes and temperature effects. The model allows for different rates of mean reversion: one for weather events, one around price jumps and another for the remainder of the process. We estimate the model by using a Markov chain Monte Carlo approach with 3 years of daily data from Allegheny County, Pennsylvania. We show that our model outperforms existing stochastic jump diffusion models for this data set. Results also demonstrate the importance of model parameters corresponding to both the temperature effect and the multilevel mean reversion rate.     

Bayesian image analysis with Markov chain Monte Carlo and coloured continuum triangulation models

It is now possible to carry out Bayesian image segmentation from a continuum parametric model with an unknown number of regions. However, few suitable parametric models exist. We set out to model processes which have realizations that are naturally described by coloured planar triangulations. Triangulations are already used, to represent image structure in machine vision, and in finite element analysis, for domain decomposition. However, no normalizable parametric model, with realizations that are coloured triangulations, has been specified to date. We show how this must be done, and in particular we prove that a normalizable measure on the space of triangulations in the interior of a fixed simple polygon derives from a Poisson point process of vertices. We show how such models may be analysed by using Markov chain Monte Carlo methods and we present two case-studies, including convergence analysis.     

Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors

Summary.  We develop an efficient way to select the best subset autoregressive model with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. One main feature of our method is to select important autoregressive and exogenous variables, and at the same time to estimate the unknown parameters. The method proposed uses the stochastic search idea. By adopting Markov chain Monte Carlo techniques, we can identify the best subset model from a large of number of possible choices. A simulation experiment shows that the method is very effective. Misspecification in the mean equation can also be detected by our model selection method. In the application to the stock-market data of seven countries, the lagged 1 US return is found to have a strong influence on the other stock-market returns.     

Bayesian Tobit quantile regression with single-index models

Based on the Bayesian framework of utilizing a Gaussian prior for the univariate nonparametric link function and an asymmetric Laplace distribution (ALD) for the residuals, we develop a Bayesian treatment for the Tobit quantile single-index regression model (TQSIM). With the location-scale mixture representation of the ALD, the posterior inferences of the latent variables and other parameters are achieved via the Markov Chain Monte Carlo computation method. TQSIM broadens the scope of applicability of the Tobit models by accommodating nonlinearity in the data. The proposed method is illustrated by two simulation examples and a labour supply dataset.     

Computational aspects of the EM algorithm for spatial econometric models with missing data

Maximum likelihood (ML) estimation with spatial econometric models is a long-standing problem that finds application in several areas of economic importance. The problem is particularly challenging in the presence of missing data, since there is an implied dependence between all units, irrespective of whether they are observed or not. Out of the several approaches adopted for ML estimation in this context, that of LeSage and Pace [Models for spatially dependent missing data. J Real Estate Financ Econ. 2004;29(2):233–254] stands out as one of the most commonly used with spatial econometric models due to its ability to scale with the number of units. Here, we review their algorithm, and consider several similar alternatives that are also suitable for large datasets. We compare the methods through an extensive empirical study and conclude that, while the approximate approaches are suitable for large sampling ratios, for small sampling ratios the only reliable algorithms are those that yield exact ML or restricted ML estimates.     

Embedding black-box regression techniques into hierarchical Bayesian models

Hierarchical models enable the encoding of a variety of parametric structures. However, when presented with a large number of covariates upon which some component of a model hierarchy depends, the modeller may be unwilling or unable to specify a form for that dependence. Data-mining methods are designed to automatically discover relationships between many covariates and a response surface, easily accommodating non-linearities and higher-order interactions. We present a method of wrapping hierarchical models around data-mining methods, preserving the best qualities of the two paradigms. We fit the resulting semi-parametric models using an approximate Gibbs sampler called HEBBRU. Using a simulated dataset, we show that HEBBRU is useful for exploratory analysis and displays excellent predictive accuracy. Finally, we apply HEBBRU to an ornithological dataset drawn from the eBird database.     

Efficient sampling schemes for Bayesian MARS models with many predictors

Multivariate adaptive regression spline fitting or MARS (Friedman 1991) provides a useful methodology for flexible adaptive regression with many predictors. The MARS methodology produces an estimate of the mean response that is a linear combination of adaptively chosen basis functions. Recently, a Bayesian version of MARS has been proposed (Denison, Mallick and Smith 1998a, Holmes and Denison, 2002) combining the MARS methodology with the benefits of Bayesian methods for accounting for model uncertainty to achieve improvements in predictive performance. In implementation of the Bayesian MARS approach, Markov chain Monte Carlo methods are used for computations, in which at each iteration of the algorithm it is proposed to change the current model by either (a) Adding a basis function (birth step) (b) Deleting a basis function (death step) or (c) Altering an existing basis function (change step). In the algorithm of Denison, Mallick and Smith (1998a), when a birth step is proposed, the type of basis function is determined by simulation from the prior. This works well in problems with a small number of predictors, is simple to program, and leads to a simple form for Metropolis-Hastings acceptance probabilities. However, in problems with very large numbers of predictors where many of the predictors are useless it may be difficult to find interesting interactions with such an approach. In the original MARS algorithm of Friedman (1991) a heuristic is used of building up higher order interactions from lower order ones, which greatly reduces the complexity of the search for good basis functions to add to the model. While we do not exactly follow the intuition of the original MARS algorithm in this paper, we nevertheless suggest a similar idea in which the Metropolis-Hastings proposals of Denison, Mallick and Smith (1998a) are altered to allow dependence on the current model. Our modification allows more rapid identification and exploration of important interactions, especially in problems with very large numbers of predictor variables and many useless predictors. Performance of the algorithms is compared in simulation studies.     

Parallel Markov chain Monte Carlo for Bayesian hierarchical models with big data,in two stages

Due to the escalating growth of big data sets in recent years, new Bayesian Markov chain Monte Carlo (MCMC) parallel computing methods have been developed. These methods partition large data sets by observations into subsets. However, for Bayesian nested hierarchical models, typically only a few parameters are common for the full data set, with most parameters being group specific. Thus, parallel Bayesian MCMC methods that take into account the structure of the model and split the full data set by groups rather than by observations are a more natural approach for analysis. Here, we adapt and extend a recently introduced two-stage Bayesian hierarchical modeling approach, and we partition complete data sets by groups. In stage 1, the group-specific parameters are estimated independently in parallel. The stage 1 posteriors are used as proposal distributions in stage 2, where the target distribution is the full model. Using three-level and four-level models, we show in both simulation and real data studies that results of our method agree closely with the full data analysis, with greatly increased MCMC efficiency and greatly reduced computation times. The advantages of our method versus existing parallel MCMC computing methods are also described.     

A comparison of Bayesian model selection based on MCMC with an application to GARCH-type models

This paper presents a comprehensive review and comparison of five computational methods for Bayesian model selection, based on MCMC simulations from posterior model parameter distributions. We apply these methods to a well-known and important class of models in financial time series analysis, namely GARCH and GARCH-t models for conditional return distributions (assuming normal and t-distributions). We compare their performance with the more common maximum likelihood-based model selection for simulated and real market data. All five MCMC methods proved reliable in the simulation study, although differing in their computational demands. Results on simulated data also show that for large degrees of freedom (where the t-distribution becomes more similar to a normal one), Bayesian model selection results in better decisions in favor of the true model than maximum likelihood. Results on market data show the instability of the harmonic mean estimator and reliability of the advanced model selection methods.     

Efficient Bayesian analysis of multivariate aggregate choices

In estimating individual choice behaviour using multivariate aggregate choice data, the method of data augmentation requires the imputation of individual choices given their partial sums. This article proposes and develops an efficient procedure of simulating multivariate individual choices given their aggregate sums, capitalizing on a sequence of auxiliary distributions. In this framework, a joint distribution of multiple binary vectors given their sums is approximated as a sequence of conditional Bernoulli distributions. The proposed approach is evaluated through a simulation study and is applied to a political science study.     

A Bayesian approach for misclassified ordinal response data

ABSTRACT

Motivated by a longitudinal oral health study, the Signal-Tandmobiel^® study, a Bayesian approach has been developed to model misclassified ordinal response data. Two regression models have been considered to incorporate misclassification in the categorical response. Specifically, probit and logit models have been developed. The computational difficulties have been avoided by using data augmentation. This idea is exploited to derive efficient Markov chain Monte Carlo methods. Although the method is proposed for ordered categories, it can also be implemented for unordered ones in a simple way. The model performance is shown through a simulation-based example and the analysis of the motivating study.     

Equilibrium dynamics of ice streams: a Bayesian statistical analysis

Studies of the behaviors of glaciers, ice sheets, and ice streams rely heavily on both observations and physical models. Data acquired via remote sensing provide critical information on geometry and movement of ice over large sections of Antarctica and Greenland. However, uncertainties are present in both the observations and the models. Hence, there is a need for combining these information sources in a fashion that incorporates uncertainty and quantifies its impact on conclusions. We present a hierarchical Bayesian approach to modeling ice-stream velocities incorporating physical models and observations regarding velocity, ice thickness, and surface elevation from the North East Ice Stream in Greenland. The Bayesian model leads to interesting issues in model assessment and computation.     

On cross‐validation of Bayesian models

The authors examine several aspects of cross‐validation for Bayesian models. In particular, they propose a computational scheme which does not require a separate posterior sample for each training sample.