Extended Bayesian model averaging for heritability in twin studies

Family studies are often conducted to examine the existence of familial aggregation. Particularly, twin studies can model separately the genetic and environmental contribution. Here we estimate the heritability of quantitative traits via variance components of random-effects in linear mixed models (LMMs). The motivating example was a myopia twin study containing complex nesting data structures: twins and siblings in the same family and observations on both eyes for each individual. Three models are considered for this nesting structure. Our proposal takes into account the model uncertainty in both covariates and model structures via an extended Bayesian model averaging (EBMA) procedure. We estimate the heritability using EBMA under three suggested model structures. When compared with the results under the model with the highest posterior model probability, the EBMA estimate has smaller variation and is slightly conservative. Simulation studies are conducted to evaluate the performance of variance-components estimates, as well as the selections of risk factors, under the correct or incorrect structure. The results indicate that EBMA, with consideration of uncertainties in both covariates and model structures, is robust in model misspecification than the usual Bayesian model averaging (BMA) that considers only uncertainty in covariates selection.     

Phase I trial design for drug combinations with Bayesian model averaging

Various statistical models have been proposed for two‐dimensional dose finding in drug‐combination trials. However, it is often a dilemma to decide which model to use when conducting a particular drug‐combination trial. We make a comprehensive comparison of four dose‐finding methods, and for fairness, we apply the same dose‐finding algorithm under the four model structures. Through extensive simulation studies, we compare the operating characteristics of these methods in various practical scenarios. The results show that different models may lead to different design properties and that no single model performs uniformly better in all scenarios. As a result, we propose using Bayesian model averaging to overcome the arbitrariness of the model specification and enhance the robustness of the design. We assign a discrete probability mass to each model as the prior model probability and then estimate the toxicity probabilities of combined doses in the Bayesian model averaging framework. During the trial, we adaptively allocated each new cohort of patients to the most appropriate dose combination by comparing the posterior estimates of the toxicity probabilities with the prespecified toxicity target. The simulation results demonstrate that the Bayesian model averaging approach is robust under various scenarios. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian model averaging for dynamic panels with an application to a trade gravity model

We extend the Bayesian Model Averaging (BMA) framework to dynamic panel data models with endogenous regressors using a Limited Information Bayesian Model Averaging (LIBMA) methodology. Monte Carlo simulations confirm the asymptotic performance of our methodology both in BMA and selection, with high posterior inclusion probabilities for all relevant regressors, and parameter estimates very close to their true values. In addition, we illustrate the use of LIBMA by estimating a dynamic gravity model for bilateral trade. Once model uncertainty, dynamics, and endogeneity are accounted for, we find several factors that are robustly correlated with bilateral trade. We also find that applying methodologies that do not account for either dynamics or endogeneity (or both) results in different sets of robust determinants.     

Bayesian model averaging in longitudinal regression models with AR(1) errors with application to a myopia data set

We propose a new iterative algorithm, called model walking algorithm, to the Bayesian model averaging method on the longitudinal regression models with AR(1) random errors within subjects. The Markov chain Monte Carlo method together with the model walking algorithm are employed. The proposed method is successfully applied to predict the progression rates on a myopia intervention trial in children.     

基于MCMC模拟的贝叶斯分层信用风险评估模型

缺少违约数据与债务人异质性是度量信用风险时面临的重要问题。贝叶斯模型中分层先验信息和马尔可夫链蒙特卡罗（MCMC）模拟方法的应用可以有效缓解数据缺失和测量误差问题,并能对债务人异质性进行评价和比较,从而避免低估风险。针对银行数据的模型拟合与模型诊断均展现了分层估计的适应性和灵活性,相关方法简洁清晰,利于国内风险分析人员采用。同时,涵盖宏观经济协变量的贝叶斯分层模型可以用于更加复杂的风险分析。     

Bayes model averaging with selection of regressors

Summary. When a number of distinct models contend for use in prediction, the choice of a single model can offer rather unstable predictions. In regression, stochastic search variable selection with Bayesian model averaging offers a cure for this robustness issue but at the expense of requiring very many predictors. Here we look at Bayes model averaging incorporating variable selection for prediction. This offers similar mean-square errors of prediction but with a vastly reduced predictor space. This can greatly aid the interpretation of the model. It also reduces the cost if measured variables have costs. The development here uses decision theory in the context of the multivariate general linear model. In passing, this reduced predictor space Bayes model averaging is contrasted with single-model approximations. A fast algorithm for updating regressions in the Markov chain Monte Carlo searches for posterior inference is developed, allowing many more variables than observations to be contemplated. We discuss the merits of absolute rather than proportionate shrinkage in regression, especially when there are more variables than observations. The methodology is illustrated on a set of spectroscopic data used for measuring the amounts of different sugars in an aqueous solution.     

A comparison of scientific and engineering criteria for Bayesian model selection

Given a set of possible models for variables X and a set of possible parameters for each model, the Bayesian estimate of the probability distribution for X given observed data is obtained by averaging over the possible models and their parameters. An often-used approximation for this estimate is obtained by selecting a single model and averaging over its parameters. The approximation is useful because it is computationally efficient, and because it provides a model that facilitates understanding of the domain. A common criterion for model selection is the posterior probability of the model. Another criterion for model selection, proposed by San Martini and Spezzafari (1984), is the predictive performance of a model for the next observation to be seen. From the standpoint of domain understanding, both criteria are useful, because one identifies the model that is most likely, whereas the other identifies the model that is the best predictor of the next observation. To highlight the difference, we refer to the posterior-probability and alternative criteria as the scientific criterion (SC) and engineering criterion (EC), respectively. When we are interested in predicting the next observation, the model-averaged estimate is at least as good as that produced by EC, which itself is at least as good as the estimate produced by SC. We show experimentally that, for Bayesian-network models containing discrete variables only, the predictive performance of the model average can be significantly better than those of single models selected by either criterion, and that differences between models selected by the two criterion can be substantial.     

A quasi-Bayesian model averaging approach for conditional quantile models

The value at risk (VaR) is a risk measure that is widely used by financial institutions to allocate risk. VaR forecast estimation involves the evaluation of conditional quantiles based on the currently available information. Recent advances in VaR evaluation incorporate conditional variance into the quantile estimation, which yields the conditional autoregressive VaR (CAViaR) models. However, uncertainty with regard to model selection in CAViaR model estimators raises the issue of identifying the better quantile predictor via averaging. In this study, we propose a quasi-Bayesian model averaging method that generates combinations of conditional VaR estimators based on single CAViaR models. This approach provides us a basis for comparing single CAViaR models against averaged ones for their ability to forecast VaR. We illustrate this method using simulated and financial daily return data series. The results demonstrate significant findings with regard to the use of averaged conditional VaR estimates when forecasting quantile risk.     

Model weights for model choice and averaging

A method is suggested to estimate posterior model probabilities and model averaged parameters via MCMC sampling under a Bayesian approach. The estimates use pooled output for J models (J>1) whereby all models are updated at each iteration. Posterior probabilities are based on averages of continuous weights obtained for each model at each iteration, while samples of averaged parameters are obtained from iteration specific averages that are based on these weights. Parallel sampling of models assists in deriving posterior densities for parameter contrasts between models and in assessing hypotheses regarding model averaged parameters. Four worked examples illustrate application of the approach, two involving fixed effect regression, and two involving random effects.     

Bayesian model comparison for compartmental models with applications in positron emission tomography

We develop strategies for Bayesian modelling as well as model comparison, averaging and selection for compartmental models with particular emphasis on those that occur in the analysis of positron emission tomography (PET) data. Both modelling and computational issues are considered. Biophysically inspired informative priors are developed for the problem at hand, and by comparison with default vague priors it is shown that the proposed modelling is not overly sensitive to prior specification. It is also shown that an additive normal error structure does not describe measured PET data well, despite being very widely used, and that within a simple Bayesian framework simultaneous parameter estimation and model comparison can be performed with a more general noise model. The proposed approach is compared with standard techniques using both simulated and real data. In addition to good, robust estimation performance, the proposed technique provides, automatically, a characterisation of the uncertainty in the resulting estimates which can be considerable in applications such as PET.     

Bayesian hierarchical regression models for detecting QTLs in plant experiments

Quantitative trait loci (QTL) mapping is a growing field in statistical genetics. In plants, QTL detection experiments often feature replicates or clones within a specific genetic line. In this work, a Bayesian hierarchical regression model is applied to simulated QTL data and to a dataset from the Arabidopsis thaliana plants for locating the QTL mapping associated with cotyledon opening. A conditional model search strategy based on Bayesian model averaging is utilized to reduce the computational burden.     

Bayesian Nonlinear Regression for the Air Pollution Effects on Daily Clinic Visits in Small Areas of Taiwan

We consider the complete clinic visit records and environmental monitoring data at 50 townships and city districts where ambient air monitoring stations of Taiwan Air Quality Monitoring Stations are located. A Bayesian analysis is carried out using regression spline model on principal components obtained from several pollutant covariables. The appropriate model is selected using Bayesian model averaging. A brief account of our results is provided for the elderly patients group.     

Frequentist model averaging for envelope models

The envelope method produces efficient estimation in multivariate linear regression, and is widely applied in biology, psychology, and economics. This paper estimates parameters through a model averaging methodology and promotes the predicting abilities of the envelope models. We propose a frequentist model averaging method by minimizing a cross-validation criterion. When all the candidate models are misspecified, the proposed model averaging estimator is proved to be asymptotically optimal. When correct candidate models exist, the coefficient estimator is proved to be consistent, and the sum of the weights assigned to the correct models, in probability, converges to one. Simulations and an empirical application demonstrate the effectiveness of the proposed method.     

Wavelet-based Bayesian nonlinear regression for the air pollution effects on clinic visits in small areas of Taiwan

We revisit the complete clinic visit records and environmental monitoring data at 50 townships and city districts of Taiwan. Extending the earlier analyses, here we consider a Bayesian analysis using Daubechies wavelet. Appropriate model selection is also considered using Bayesian model averaging. Temperature, dew point, and NO₂ and CO of the current day and the previous day are identified as the pollutants in different areas of the island following some spatial pattern.     

Bayesian regression with multivariate linear splines

We present a Bayesian analysis of a piecewise linear model constructed by using basis functions which generalizes the univariate linear spline to higher dimensions. Prior distributions are adopted on both the number and the locations of the splines, which leads to a model averaging approach to prediction with predictive distributions that take into account model uncertainty. Conditioning on the data produces a Bayes local linear model with distributions on both predictions and local linear parameters. The method is spatially adaptive and covariate selection is achieved by using splines of lower dimension than the data.     

Bayesian Model Averaging in Proportional Hazard Models: Assessing the Risk of a Stroke

In the context of the Cardiovascular Health Study, a comprehensive investigation into the risk factors for strokes, we apply Bayesian model averaging to the selection of variables in Cox proportional hazard models. We use an extension of the leaps-and-bounds algorithm for locating the models that are to be averaged over and make available S-PLUS software to implement the methods. Bayesian model averaging provides a posterior probability that each variable belongs in the model, a more directly interpretable measure of variable importance than a P -value. P -values from models preferred by stepwise methods tend to overstate the evidence for the predictive value of a variable and do not account for model uncertainty. We introduce the partial predictive score to evaluate predictive performance. For the Cardiovascular Health Study, Bayesian model averaging predictively outperforms standard model selection and does a better job of assessing who is at high risk for a stroke.     

Bayesian forecasting for AR(1) models with normal coefficients

This paper presents a Bayesian solution to the problem of time series forecasting, for the case in which the generating process is an autoregressive of order one, with a normal random coefficient. The proposed procedure is based on the predictive density of the future observation. Conjugate priors are used for some parameters, while improper vague priors are used for others.     

Bayesian estimation for first-order autoregressive model with explanatory variables

In this article, we develop a Bayesian analysis in autoregressive model with explanatory variables. When σ² is known, we consider a normal prior and give the Bayesian estimator for the regression coefficients of the model. For the case σ² is unknown, another Bayesian estimator is given for all unknown parameters under a conjugate prior. Bayesian model selection problem is also being considered under the double-exponential priors. By the convergence of ρ-mixing sequence, the consistency and asymptotic normality of the Bayesian estimators of the regression coefficients are proved. Simulation results indicate that our Bayesian estimators are not strongly dependent on the priors, and are robust.     

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.