Bayesian variable selection for proportional hazards models

The authors consider the problem of Bayesian variable selection for proportional hazards regression models with right censored data. They propose a semi-parametric approach in which a nonparametric prior is specified for the baseline hazard rate and a fully parametric prior is specified for the regression coefficients. For the baseline hazard, they use a discrete gamma process prior, and for the regression coefficients and the model space, they propose a semi-automatic parametric informative prior specification that focuses on the observables rather than the parameters. To implement the methodology, they propose a Markov chain Monte Carlo method to compute the posterior model probabilities. Examples using simulated and real data are given to demonstrate the methodology.     

Bayesian approach to estimation of intraclass correlation using reference prior

For the balanced variance component model when the intraclass correlation coefficient is of interest, Bayesian analysis is often appropriate. Berger and Bernardo’s (1992a) grouped ordering reference prior approach is used to analyze this model. The reference priors are developed and compared for the posterior inference with real and simulated data. We examine whether the reference priors satisfy the probability-matching criterion. Further, the reference prior is shown to be good in the sense of correct frequentist coverage probability of the posterior quantile.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

Bayesian analysis in multivariate regression models with conjugate priors

In this paper, we consider the full rank multivariate regression model with matrix elliptically contoured distributed errors. We formulate a conjugate prior distribution for matrix elliptical models and derive the posterior distributions of mean and scale matrices. In the sequel, some characteristics of regression matrix parameters are also proposed.     

Variational Inference of Linear Regression with Nonzero Prior Means

In this article, we employ the variational Bayesian method to study the parameter estimation problems of linear regression model, wherein some regressors are of Gaussian distribution with nonzero prior means. We obtain an analytical expression of the posterior parameter distribution, and then propose an iterative algorithm for the model. Simulations are carried out to test the performance of the proposed algorithm, and the simulation results confirm both the effectiveness and the reliability of the proposed algorithm.     

Variable selection in quantile regression via Gibbs sampling

Due to computational challenges and non-availability of conjugate prior distributions, Bayesian variable selection in quantile regression models is often a difficult task. In this paper, we address these two issues for quantile regression models. In particular, we develop an informative stochastic search variable selection (ISSVS) for quantile regression models that introduces an informative prior distribution. We adopt prior structures which incorporate historical data into the current data by quantifying them with a suitable prior distribution on the model parameters. This allows ISSVS to search more efficiently in the model space and choose the more likely models. In addition, a Gibbs sampler is derived to facilitate the computation of the posterior probabilities. A major advantage of ISSVS is that it avoids instability in the posterior estimates for the Gibbs sampler as well as convergence problems that may arise from choosing vague priors. Finally, the proposed methods are illustrated with both simulation and real data.     

A Direct Approach to Understanding Posterior Consistency of Bayesian Regression Problems

Previous approaches to establishing posterior consistency of Bayesian regression problems have used general theorems that involve verifying sufficient conditions for posterior consistency. In this article, we consider a direct approach by computing the posterior density explicitly and evaluating its asymptotic behavior. For this purpose, we deal with a sample size dependent prior based on a truncated regression function with increasing sample size, and evaluate the asymptotic properties of the resulting posterior. Based on a concept called posterior density consistency, we attempt to understand posterior consistency. As an application, we illustrate that the posterior density of an orthogonal semiparametric regression model is consistent.     

Bayesian variable selection for multioutcome models through shared shrinkage

Variable selection over a potentially large set of covariates in a linear model is quite popular. In the Bayesian context, common prior choices can lead to a posterior expectation of the regression coefficients that is a sparse (or nearly sparse) vector with a few nonzero components, those covariates that are most important. This article extends the “global‐local” shrinkage idea to a scenario where one wishes to model multiple response variables simultaneously. Here, we have developed a variable selection method for a K‐outcome model (multivariate regression) that identifies the most important covariates across all outcomes. The prior for all regression coefficients is a mean zero normal with coefficient‐specific variance term that consists of a predictor‐specific factor (shared local shrinkage parameter) and a model‐specific factor (global shrinkage term) that differs in each model. The performance of our modeling approach is evaluated through simulation studies and a data example.     

Non-informative priors for the inverse Weibull distribution

In this paper, we develop the non-informative priors for the inverse Weibull model when the parameters of interest are the scale and the shape parameters. We develop the first-order and the second-order matching priors for both parameters. For the scale parameter, we reveal that the second-order matching prior is not a highest posterior density (HPD) matching prior, does not match the alternative coverage probabilities up to the second order and is not a cumulative distribution function (CDF) matching prior. Also for the shape parameter, we reveal that the second-order matching prior is an HPD matching prior and a CDF matching prior and also matches the alternative coverage probabilities up to the second order. For both parameters, we reveal that the one-at-a-time reference prior is the second-order matching prior, but Jeffreys’ prior is not the first-order and the second-order matching prior. A simulation study is performed to compare the target coverage probabilities and a real example is given.     

Flexible objective Bayesian linear regression with applications in survival analysis

We study objective Bayesian inference for linear regression models with residual errors distributed according to the class of two-piece scale mixtures of normal distributions. These models allow for capturing departures from the usual assumption of normality of the errors in terms of heavy tails, asymmetry, and certain types of heteroscedasticity. We propose a general non-informative, scale-invariant, prior structure and provide sufficient conditions for the propriety of the posterior distribution of the model parameters, which cover cases when the response variables are censored. These results allow us to apply the proposed models in the context of survival analysis. This paper represents an extension to the Bayesian framework of the models proposed in [16]. We present a simulation study that shows good frequentist properties of the posterior credible intervals as well as point estimators associated to the proposed priors. We illustrate the performance of these models with real data in the context of survival analysis of cancer patients.     

Calibrating the prior distribution for a normal model with conjugate prior

For a normal model with a conjugate prior, we provide an in-depth examination of the effects of the hyperparameters on the long-run frequentist properties of posterior point and interval estimates. Under an assumed sampling model for the data-generating mechanism, we examine how hyperparameter values affect the mean-squared error (MSE) of posterior means and the true coverage of credible intervals. We develop two types of hyperparameter optimality. MSE optimal hyperparameters minimize the MSE of posterior point estimates. Credible interval optimal hyperparameters result in credible intervals that have a minimum length while still retaining nominal coverage. A poor choice of hyperparameters has a worse consequence on the credible interval coverage than on the MSE of posterior point estimates. We give an example to demonstrate how our results can be used to evaluate the potential consequences of hyperparameter choices.     

Bayesian elastic net single index quantile regression

Single index model conditional quantile regression is proposed in order to overcome the dimensionality problem in nonparametric quantile regression. In the proposed method, the Bayesian elastic net is suggested for single index quantile regression for estimation and variables selection. The Gaussian process prior is considered for unknown link function and a Gibbs sampler algorithm is adopted for posterior inference. The results of the simulation studies and numerical example indicate that our propose method, BENSIQReg, offers substantial improvements over two existing methods, SIQReg and BSIQReg. The BENSIQReg has consistently show a good convergent property, has the least value of median of mean absolute deviations and smallest standard deviations, compared to the other two methods.     

Adaptive Bayesian Procedures Using Random Series Priors

We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on Tecator data using functional regression models.     

Objective Bayesian multiple comparisons for normal variances

This paper considers the multiple comparisons problem for normal variances. We propose a solution based on a Bayesian model selection procedure to this problem in which no subjective input is considered. We construct the intrinsic and fractional priors for which the Bayes factors and model selection probabilities are well defined. The posterior probability of each model is used as a model selection tool. The behaviour of these Bayes factors is compared with the Bayesian information criterion of Schwarz and some frequentist tests.     

The use of Jeffreys priors for the Student-t distribution

The Jeffreys-rule prior and the marginal independence Jeffreys prior are recently proposed in Fonseca et al. [Objective Bayesian analysis for the Student-t regression model, Biometrika 95 (2008), pp. 325–333] as objective priors for the Student-t regression model. The authors showed that the priors provide proper posterior distributions and perform favourably in parameter estimation. Motivated by a practical financial risk management application, we compare the performance of the two Jeffreys priors with other priors proposed in the literature in a problem of estimating high quantiles for the Student-t model with unknown degrees of freedom. Through an asymptotic analysis and a simulation study, we show that both Jeffreys priors perform better in using a specific quantile of the Bayesian predictive distribution to approximate the true quantile.     

Numerical Bayesian inference with arbitrary prior

The purpose of the paper, is to explain how recent advances in Markov Chain Monte Carlo integration can facilitate the routine Bayesian analysis of the linear model when the prior distribution is completely user dependent. The method is based on a Metropolis-Hastings algorithm with a Student-t source distribution that can generate posterior moments as well as marginal posterior densities for model parameters. The method is illustrated with numerical examples where the combination of prior and likelihood information leads to multimodal posteriors due to prior-likelihood conflicts, and to cases where prior information can be summarized by symmetric stable Paretian distributions.     

Censored Exponential Data: Large Deviations for MLEs and Posterior Distributions

In this article, we consider several statistical models for censored exponential data. We prove a large deviation result for the maximum likelihood estimators (MLEs) of each model, and a unique result for the posterior distributions which works well for all the cases. Finally, comparing the large deviation rate functions for MLEs and posterior distributions, we show that a typical feature fails for one model; moreover, we illustrate the relation between this fact and a well-known result for curved exponential models.     

SELECTION OF WEIGHTS FOR WEIGHTED MODEL AVERAGING

We address the task of choosing prior weights for models that are to be used for weighted model averaging. Models that are very similar should usually be given smaller weights than models that are quite distinct. Otherwise, the importance of a model in the weighted average could be increased by augmenting the set of models with duplicates of the model or virtual duplicates of it. Similarly, the importance of a particular model feature (a certain covariate, say) could be exaggerated by including many models with that feature. Ways of forming a correlation matrix that reflects the similarity between models are suggested. Then, weighting schemes are proposed that assign prior weights to models on the basis of this matrix. The weighting schemes give smaller weights to models that are more highly correlated. Other desirable properties of a weighting scheme are identified, and we examine the extent to which these properties are held by the proposed methods. The weighting schemes are applied to real data, and prior weights, posterior weights and Bayesian model averages are determined. For these data, empirical Bayes methods were used to form the correlation matrices that yield the prior weights. Predictive variances are examined, as empirical Bayes methods can result in unrealistically small variances.     

Detecting structural change in linear models

Consider a sequence of independent observations which change their marginal distribution at most once somewhere in the sequence and one is not certain where the change has occurred. One would be interested in detecting the change and determining the two distributions which would describe the sequence. On the other hand if no change had occurred, one would want to know the common distribution of the observations. This study develops a Bayesian test for detecting a switch from one linear model to another. The test is based on the marginal posterior mass function of the switch point and the posterior probability of a stable model. This test and an informal sequential procedure of Smith are illustrated with data generated from an unstable linear regression model, which changes the linear relationship between the dependent and independent variables     

Objective Testing Procedures in Linear Models: Calibration of the p-values

Abstract.  An optimal Bayesian decision procedure for testing hypothesis in normal linear models based on intrinsic model posterior probabilities is considered. It is proven that these posterior probabilities are simple functions of the classical F -statistic, thus the evaluation of the procedure can be carried out analytically through the frequentist analysis of the posterior probability of the null. An asymptotic analysis proves that, under mild conditions on the design matrix, the procedure is consistent. For any testing hypothesis it is also seen that there is a one-to-one mapping – which we call calibration curve – between the posterior probability of the null hypothesis and the classical bi p -value. This curve adds substantial knowledge about the possible discrepancies between the Bayesian and the p -value measures of evidence for testing hypothesis. It permits a better understanding of the serious difficulties that are encountered in linear models for interpreting the p -values. A specific illustration of the variable selection problem is given.