Bayesian model comparison via parallel model output

Bayesian estimation via MCMC methods opens up new possibilities in estimating complex models. However, there is still considerable debate about how selection among a set of candidate models, or averaging over closely competing models, might be undertaken. This article considers simple approaches for model averaging and choice using predictive and likelihood criteria and associated model weights on the basis of output for models that run in parallel. The operation of such procedures is illustrated with real data sets and a linear regression with simulated data where the true model is known.     

Semiparametric Bayesian hierarchical models for heterogeneous population in nonlinear mixed effect model: application to gastric emptying studies

Gastric emptying studies are frequently used in medical research, both human and animal, when evaluating the effectiveness and determining the unintended side-effects of new and existing medications, diets, and procedures or interventions. It is essential that gastric emptying data be appropriately summarized before making comparisons between study groups of interest and to allow study the comparisons. Since gastric emptying data have a nonlinear emptying curve and are longitudinal data, nonlinear mixed effect (NLME) models can accommodate both the variation among measurements within individuals and the individual-to-individual variation. However, the NLME model requires strong assumptions that are often not satisfied in real applications that involve a relatively small number of subjects, have heterogeneous measurement errors, or have large variation among subjects. Therefore, we propose three semiparametric Bayesian NLMEs constructed with Dirichlet process priors, which automatically cluster sub-populations and estimate heterogeneous measurement errors. To compare three semiparametric models with the parametric model we propose a penalized posterior Bayes factor. We compare the performance of our semiparametric hierarchical Bayesian approaches with that of the parametric Bayesian hierarchical approach. Simulation results suggest that our semiparametric approaches are more robust and flexible. Our gastric emptying studies from equine medicine are used to demonstrate the advantage of our approaches.     

Comparison of Bayesian predictive methods for model selection

The goal of this paper is to compare several widely used Bayesian model selection methods in practical model selection problems, highlight their differences and give recommendations about the preferred approaches. We focus on the variable subset selection for regression and classification and perform several numerical experiments using both simulated and real world data. The results show that the optimization of a utility estimate such as the cross-validation (CV) score is liable to finding overfitted models due to relatively high variance in the utility estimates when the data is scarce. This can also lead to substantial selection induced bias and optimism in the performance evaluation for the selected model. From a predictive viewpoint, best results are obtained by accounting for model uncertainty by forming the full encompassing model, such as the Bayesian model averaging solution over the candidate models. If the encompassing model is too complex, it can be robustly simplified by the projection method, in which the information of the full model is projected onto the submodels. This approach is substantially less prone to overfitting than selection based on CV-score. Overall, the projection method appears to outperform also the maximum a posteriori model and the selection of the most probable variables. The study also demonstrates that the model selection can greatly benefit from using cross-validation outside the searching process both for guiding the model size selection and assessing the predictive performance of the finally selected model.     

Credit risk assessment with Bayesian model averaging

Many credit risk models are based on the selection of a single logistic regression model, on which to base parameter estimation. When many competing models are available, and without enough guidance from economical theory, model averaging represents an appealing alternative to the selection of single models. Despite model averaging approaches have been present in statistics for many years, only recently they are starting to receive attention in economics and finance applications. This contribution shows how Bayesian model averaging can be applied to credit risk estimation, a research area that has received a great deal of attention recently, especially in the light of the global financial crisis of the last few years and the correlated attempts to regulate international finance. The paper considers the use of logistic regression models under the Bayesian Model Averaging paradigm. We argue that Bayesian model averaging is not only more correct from a theoretical viewpoint, but also slightly superior, in terms of predictive performance, with respect to single selected models.     

Bayesian Versus Frequentist Approach of the Frailty Cox Model,Application to Calf Gastroenteritis

In the frailty Cox model, frequentist approaches often present problems of numerical resolution, convergence, and variance calculation. The Bayesian approach offers an alternative. The goal of this study was to compare, using real (calf gastroenteritis) and simulated data, the results obtained with the MCMC method used in the Bayesian approach versus two frequentist approaches: the Newton–Raphson algorithm to solve a penalized likelihood and the EM algorithm. The results obtained showed that when the number of groups in the population decreases, the Bayesian approach gives a less biased estimation of the frailty variance and of the group fixed effect than the frequentist approaches.     

A model averaging approach for the ordered probit and nested logit models with applications

This paper considers model averaging for the ordered probit and nested logit models, which are widely used in empirical research. Within the frameworks of these models, we examine a range of model averaging methods, including the jackknife method, which is proved to have an optimal asymptotic property in this paper. We conduct a large-scale simulation study to examine the behaviour of these model averaging estimators in finite samples, and draw comparisons with model selection estimators. Our results show that while neither averaging nor selection is a consistently better strategy, model selection results in the poorest estimates far more frequently than averaging, and more often than not, averaging yields superior estimates. Among the averaging methods considered, the one based on a smoothed version of the Bayesian Information criterion frequently produces the most accurate estimates. In three real data applications, we demonstrate the usefulness of model averaging in mitigating problems associated with the ‘replication crisis’ that commonly arises with model selection.     

Loss-based approach to two-piece location-scale distributions with applications to dependent data

Two-piece location-scale models are used for modeling data presenting departures from symmetry. In this paper, we propose an objective Bayesian methodology for the tail parameter of two particular distributions of the above family: the skewed exponential power distribution and the skewed generalised logistic distribution. We apply the proposed objective approach to time series models and linear regression models where the error terms follow the distributions object of study. The performance of the proposed approach is illustrated through simulation experiments and real data analysis. The methodology yields improvements in density forecasts, as shown by the analysis we carry out on the electricity prices in Nordpool markets.

     

Inference for the Hyperparameters of Structural Models Under Classical and Bayesian Perspectives: A Comparison Study

Structural models—or dynamic linear models as they are known in the Bayesian literature—have been widely used to model and predict time series using a decomposition in non observable components. Due to the direct interpretation of the parameters, structural models are a powerful and simple methodology to analyze time series in several areas, such as economy, climatology, environmental sciences, among others. The parameters of such models can be estimated either using maximum likelihood or Bayesian procedures, generally implemented using conjugate priors, and there are plenty of works in the literature employing both methods. But are there situations where one of these approaches should be preferred? In this work, instead of conjugate priors for the hyperparameters, the Jeffreys prior is used in the Bayesian approach, along with the uniform prior, and the results are compared to the maximum likelihood method, in an extensive Monte Carlo study. Interval estimation is also evaluated and, to this purpose, bootstrap confidence intervals are introduced in the context of structural models and their performance is compared to the asymptotic and credibility intervals. A real time series of a Brazilian electric company is used as illustration.     

On double hysteretic heteroskedastic model

ABSTRACT

This paper proposes a hysteretic autoregressive model with GARCH specification and a skew Student's t-error distribution for financial time series. With an integrated hysteresis zone, this model allows both the conditional mean and conditional volatility switching in a regime to be delayed when the hysteresis variable lies in a hysteresis zone. We perform Bayesian estimation via an adaptive Markov Chain Monte Carlo sampling scheme. The proposed Bayesian method allows simultaneous inferences for all unknown parameters, including threshold values and a delay parameter. To implement model selection, we propose a numerical approximation of the marginal likelihoods to posterior odds. The proposed methodology is illustrated using simulation studies and two major Asia stock basis series. We conduct a model comparison for variant hysteresis and threshold GARCH models based on the posterior odds ratios, finding strong evidence of the hysteretic effect and some asymmetric heavy-tailness. Versus multi-regime threshold GARCH models, this new collection of models is more suitable to describe real data sets. Finally, we employ Bayesian forecasting methods in a Value-at-Risk study of the return series.     

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.     

Investigation about a screening step in model selection

In many studies a large number of variables is measured and the identification of relevant variables influencing an outcome is an important task. For variable selection several procedures are available. However, focusing on one model only neglects that there usually exist other equally appropriate models. Bayesian or frequentist model averaging approaches have been proposed to improve the development of a predictor. With a larger number of variables (say more than ten variables) the resulting class of models can be very large. For Bayesian model averaging Occam’s window is a popular approach to reduce the model space. As this approach may not eliminate any variables, a variable screening step was proposed for a frequentist model averaging procedure. Based on the results of selected models in bootstrap samples, variables are eliminated before deriving a model averaging predictor. As a simple alternative screening procedure backward elimination can be used. Through two examples and by means of simulation we investigate some properties of the screening step. In the simulation study we consider situations with fifteen and 25 variables, respectively, of which seven have an influence on the outcome. With the screening step most of the uninfluential variables will be eliminated, but also some variables with a weak effect. Variable screening leads to more applicable models without eliminating models, which are more strongly supported by the data. Furthermore, we give recommendations for important parameters of the screening step.     

Bayesian analysis of covariance under inverse Gaussian model

This paper considers the problem of analysis of covariance (ANCOVA) under the assumption of inverse Gaussian distribution for response variable from the Bayesian point of view. We develop a fully Bayesian model for ANCOVA based on the conjugate prior distributions for parameters contained in the model. The Bayes estimator of parameters, ANCOVA model and adjusted effects for both treatments and covariates along with predictive distribution of future observations are developed. We also provide the essentials for comparing adjusted treatments effects and adjusted factor effects. A simulation study and a real world application are also performed to illustrate and evaluate the proposed Bayesian model.     

A general approach to heteroscedastic linear regression

Our article presents a general treatment of the linear regression model, in which the error distribution is modelled nonparametrically and the error variances may be heteroscedastic, thus eliminating the need to transform the dependent variable in many data sets. The mean and variance components of the model may be either parametric or nonparametric, with parsimony achieved through variable selection and model averaging. A Bayesian approach is used for inference with priors that are data-based so that estimation can be carried out automatically with minimal input by the user. A Dirichlet process mixture prior is used to model the error distribution nonparametrically; when there are no regressors in the model, the method reduces to Bayesian density estimation, and we show that in this case the estimator compares favourably with a well-regarded plug-in density estimator. We also consider a method for checking the fit of the full model. The methodology is applied to a number of simulated and real examples and is shown to work well.     

Assessing safety at the end of clinical trials using system organ classes: A case and comparative study

Recent approaches to the statistical analysis of adverse event (AE) data in clinical trials have proposed the use of groupings of related AEs, such as by system organ class (SOC). These methods have opened up the possibility of scanning large numbers of AEs while controlling for multiple comparisons, making the comparative performance of the different methods in terms of AE detection and error rates of interest to investigators. We apply two Bayesian models and two procedures for controlling the false discovery rate (FDR), which use groupings of AEs, to real clinical trial safety data. We find that while the Bayesian models are appropriate for the full data set, the error controlling methods only give similar results to the Bayesian methods when low incidence AEs are removed. A simulation study is used to compare the relative performances of the methods. We investigate the differences between the methods over full trial data sets, and over data sets with low incidence AEs and SOCs removed. We find that while the removal of low incidence AEs increases the power of the error controlling procedures, the estimated power of the Bayesian methods remains relatively constant over all data sizes. Automatic removal of low-incidence AEs however does have an effect on the error rates of all the methods, and a clinically guided approach to their removal is needed. Overall we found that the Bayesian approaches are particularly useful for scanning the large amounts of AE data gathered.     

Using large data sets to forecast sectoral employment

We use several models using classical and Bayesian methods to forecast employment for eight sectors of the US economy. In addition to using standard vector-autoregressive and Bayesian vector autoregressive models, we also augment these models to include the information content of 143 additional monthly series in some models. Several approaches exist for incorporating information from a large number of series. We consider two multivariate approaches—extracting common factors (principal components) and Bayesian shrinkage. After extracting the common factors, we use Bayesian factor-augmented vector autoregressive and vector error-correction models, as well as Bayesian shrinkage in a large-scale Bayesian vector autoregressive models. For an in-sample period of January 1972 to December 1989 and an out-of-sample period of January 1990 to March 2010, we compare the forecast performance of the alternative models. More specifically, we perform ex-post and ex-ante out-of-sample forecasts from January 1990 through March 2009 and from April 2009 through March 2010, respectively. We find that factor augmented models, especially error-correction versions, generally prove the best in out-of-sample forecast performance, implying that in addition to macroeconomic variables, incorporating long-run relationships along with short-run dynamics play an important role in forecasting employment. Forecast combination models, however, based on the simple average forecasts of the various models used, outperform the best performing individual models for six of the eight sectoral employment series.     

GENERALIZED EXTREME VALUE ADDITIVE MODEL ANALYSIS VIA MEAN FIELD VARIATIONAL BAYES

We develop Mean Field Variational Bayes methodology for fast approximate inference in Bayesian Generalized Extreme Value additive model analysis. Such models are useful for flexibly assessing the impact of continuous predictor variables on sample extremes. The new methodology allows large Bayesian models to be fitted and assessed without the significant computing costs of Markov Chain Monte Carlo methods. We illustrate our new methodology with maximum rainfall data from the Sydney, Australia, hinterland. Comparisons are made between the Mean Field Variational Bayes and Markov Chain Monte Carlo approaches.     

Model averaging for multivariate multiple regression models

The theories and applications of model averaging have been developed comprehensively in the past two decades. In this paper, we consider model averaging for multivariate multiple regression models. In order to make use of the correlation information of the dependent variables sufficiently, we propose a model averaging method based on Mahalanobis distance which is related to the correlation of the dependent variables. We prove the asymptotic optimality of the resulting Mahalanobis Mallows model averaging (MMMA) estimators under certain assumptions. In the simulation study, we show that the proposed MMMA estimators compare favourably with model averaging estimators based on AIC and BIC weights and the Mallows model averaging estimators from the single dependent variable regression models. We further apply our method to the real data on urbanization rate and the proportion of non-agricultural population in ethnic minority areas of China.     

Analysis of the amended Davidson model with order effect for paired comparison in the Bayesian paradigm

We commonly observe many types of paired nature of competitions in which the objects are compared by the respondents pairwise in a subjective manner. The Bayesian statistics, contrary to the classical statistics, presents a generic tool to incorporate new experimental evidence and update the existing information. These and other properties have ushered the statisticians to focus their attention on the Bayesian analysis of different paired comparison models. The present article focuses on the amended Davidson model for paired comparison in which an amendment has been introduced that accommodates the option of not distinguishing the effects of two treatments when they are compared pairwise. However, Bayesian analysis of the amended Davidson model is performed using the noninformative priors after making another small modification of incorporating the parameter of order effect factor. The joint and marginal posterior distributions of the parameters, their posterior estimates, predictive and posterior probabilities to compare the treatment parameters are obtained.     

Variational Bayes for estimating the parameters of a hidden Potts model

Hidden Markov random field models provide an appealing representation of images and other spatial problems. The drawback is that inference is not straightforward for these models as the normalisation constant for the likelihood is generally intractable except for very small observation sets. Variational methods are an emerging tool for Bayesian inference and they have already been successfully applied in other contexts. Focusing on the particular case of a hidden Potts model with Gaussian noise, we show how variational Bayesian methods can be applied to hidden Markov random field inference. To tackle the obstacle of the intractable normalising constant for the likelihood, we explore alternative estimation approaches for incorporation into the variational Bayes algorithm. We consider a pseudo-likelihood approach as well as the more recent reduced dependence approximation of the normalisation constant. To illustrate the effectiveness of these approaches we present empirical results from the analysis of simulated datasets. We also analyse a real dataset and compare results with those of previous analyses as well as those obtained from the recently developed auxiliary variable MCMC method and the recursive MCMC method. Our results show that the variational Bayesian analyses can be carried out much faster than the MCMC analyses and produce good estimates of model parameters. We also found that the reduced dependence approximation of the normalisation constant outperformed the pseudo-likelihood approximation in our analysis of real and synthetic datasets.     

Bayesian nonparametric survival analysis using mixture of Burr XII distributions

ABSTRACT

Recently, the Bayesian nonparametric approaches in survival studies attract much more attentions. Because of multimodality in survival data, the mixture models are very common. We introduce a Bayesian nonparametric mixture model with Burr distribution (Burr type XII) as the kernel. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time datasets, we show the preference of this model and compare it with Dirichlet process mixture models with different kernels. The Markov chain Monte Carlo (MCMC) simulation methods to calculate the posterior distribution are used.