Bayesian analysis of vector-autoregressive models with noninformative priors

In this paper, we investigate the properties of Bayes estimators of vector autoregression (VAR) coefficients and the covariance matrix under two commonly employed loss functions. We point out that the posterior mean of the variances of the VAR errors under the Jeffreys prior is likely to have an over-estimation bias. Our Bayesian computation results indicate that estimates using the constant prior on the VAR regression coefficients and the reference prior of Yang and Berger (Ann. Statist. 22 (1994) 1195) on the covariance matrix dominate the constant-Jeffreys prior estimates commonly used in applications of VAR models in macroeconomics. We also estimate a VAR model of consumption growth using both constant-reference and constant-Jeffreys priors.     

Calibrated Bayes factors for model comparison

A Bayes factor between two models can be greatly affected by the prior distributions on the model parameters. When prior information is weak, very dispersed proper prior distributions are known to create a problem for the Bayes factor when competing models differ in dimension, and it is of even greater concern when one of the models is of infinite dimension. Therefore, we propose an innovative method which uses training samples to calibrate the prior distributions so that they achieve a reasonable level of ‘information’. Then the calibrated Bayes factor can be computed over the remaining data. This method makes no assumption on model forms (parametric or nonparametric) and can be used with both proper and improper priors. We illustrate, through simulation studies and a real data example, that the calibrated Bayes factor yields robust and reliable model preferences under various situations.     

On the estimation problem of periodic autoregressive time series: symmetric and asymmetric innovations

Periodic autoregressive (PAR) models with symmetric innovations are widely used on time series analysis, whereas its asymmetric counterpart inference remains a challenge, because of a number of problems related to the existing computational methods. In this paper, we use an interesting relationship between periodic autoregressive and vector autoregressive (VAR) models to study maximum likelihood and Bayesian approaches to the inference of a PAR model with normal and skew-normal innovations, where different kinds of estimation methods for the unknown parameters are examined. Several technical difficulties which are usually complicated to handle are reported. Results are compared with the existing classical solutions and the practical implementations of the proposed algorithms are illustrated via comprehensive simulation studies. The methods developed in the study are applied and illustrate a real-time series. The Bayes factor is also used to compare the multivariate normal model versus the multivariate skew-normal model.     

Model determination for the variance component model using reference priors

For the balanced variance component model when the inference concerning intraclass correlation coefficient is of interest, Bayesian analysis is often appropriate. However, the question remains is to choose the appropriate prior. In this paper, we consider testing of the intraclass correlation coefficient under a default prior specification. Berger and Bernardo's (1992) On the development of the reference prior method. In: Bernardo, J.M., Berger, J.O., Dawid, A.P., Smith, A.F.M. (Eds.), Bayesian Statist. Vol. 4. Oxford University Press, London, pp. 35–60 reference priors are developed and are used to obtain the intrinsic Bayes factor (Berger and Pericchi, 1996) The intrinsic Bayes factor for model selection and prediction. J. Amer. statist. Assoc. 91, 109–122 for the nested models. Influence diagnostics using intrinsic Bayes factors are also developed. Finally, one simulated data is provided which illustrates the proposed methodology with appropriate simulation based on computational formulas. Then in order to overcome the difficulty in Bayesian computation, MCMC method, such as Gibbs sampler and Metropolis–Hastings algorithm, is employed.     

Equality and inequality constrained multivariate linear models: Objective model selection using constrained posterior priors

In objective Bayesian model selection, a well-known problem is that standard non-informative prior distributions cannot be used to obtain a sensible outcome of the Bayes factor because these priors are improper. The use of a small part of the data, i.e., a training sample, to obtain a proper posterior prior distribution has become a popular method to resolve this issue and seems to result in reasonable outcomes of default Bayes factors, such as the intrinsic Bayes factor or a Bayes factor based on the empirical expected-posterior prior.     

Nonparametric empirical Bayes method for comparison of treatment effects with application to stress urinary incontinence data

Hierarchical models are widely used in medical research to structure complicated models and produce statistical inferences. In a hierarchical model, observations are sampled conditional on some parameters and these parameters are sampled from a common prior distribution. Bayes and empirical Bayes (EB) methods have been effectively applied in analyzing these models. Despite many successes, parametric Bayes and EB methods may be sensitive to misspecification of prior distributions. In this paper, without specific restriction on the form of the prior distribution, we propose a nonparametric EB method to estimate the treatment effect of each group and develop a testing procedure to compare between-group differences. Simulation studies demonstrate that the proposed EB method was more efficient than some standard procedures. An illustrative example is provided with data from a clinical trial evaluating a new treatment for patients with stress urinary incontinence.     

A ROBUST BAYES FACTOR FOR LINEAR MODELS

This paper proposes a new robust Bayes factor for comparing two linear models. The factor is based on a pseudo‐model for outliers and is more robust to outliers than the Bayes factor based on the variance‐inflation model for outliers. If an observation is considered an outlier for both models this new robust Bayes factor equals the Bayes factor calculated after removing the outlier. If an observation is considered an outlier for one model but not the other then this new robust Bayes factor equals the Bayes factor calculated without the observation, but a penalty is applied to the model considering the observation as an outlier. For moderate outliers where the variance‐inflation model is suitable, the two Bayes factors are similar. The new Bayes factor uses a single robustness parameter to describe a priori belief in the likelihood of outliers. Real and synthetic data illustrate the properties of the new robust Bayes factor and highlight the inferior properties of Bayes factors based on the variance‐inflation model for outliers.     

On selecting the best treatment in a generalized linear model

The problem of selecting the best treatment is studied under generalized linear models. For certain balanced designs, it is shown that simple rules are Bayes with respect to any non-informative prior on the treatment effects under any monotone invariant loss. When the nuisance parameters such as block effects are assumed to follow a uniform (improper) prior or a normal prior, Bayes rules are obtained for the normal linear model under more suitable balanced designs, keeping the generality of the loss and the generality of the non-informativeness on the prior of the treatment effects. These results are extended to certain types of informative priors on the treatment effects. When the designs are unbalanced, algorithms based on the Gibbs sampler and the Laplace method are provided to compute the Bayes rules.     

Sensitivity of the fractional Bayes factor to prior distributions

The authors derive a measure of the sensitivity of the fractional Bayes factor, an index which is used to compare models when the priors for their respective parameters are improper, or when there is concern about robustness of the prior specification. They prove that in a large class of problems, this measure is a decreasing function of the fraction of the sample used to update the prior distribution before the models are compared.     

A Generalized Bayes Rule for Prediction

In the case of prior knowledge about the unknown parameter, the Bayesian predictive density coincides with the Bayes estimator for the true density in the sense of the Kullback-Leibler divergence, but this is no longer true if we consider another loss function. In this paper we present a generalized Bayes rule to obtain Bayes density estimators with respect to any α-divergence, including the Kullback-Leibler divergence and the Hellinger distance. For curved exponential models, we study the asymptotic behaviour of these predictive densities. We show that, whatever prior we use, the generalized Bayes rule improves (in a non-Bayesian sense) the estimative density corresponding to a bias modification of the maximum likelihood estimator. It gives rise to a correspondence between choosing a prior density for the generalized Bayes rule and fixing a bias for the maximum likelihood estimator in the classical setting. A criterion for comparing and selecting prior densities is also given.     

Alternative Bayes factors for model selection

Several alternative Bayes factors have been recently proposed in order to solve the problem of the extreme sensitivity of the Bayes factor to the priors of models under comparison. Specifically, the impossibility of using the Bayes factor with standard noninformative priors for model comparison has led to the introduction of new automatic criteria, such as the posterior Bayes factor (Aitkin 1991), the intrinsic Bayes factors (Berger and Pericchi 1996b) and the fractional Bayes factor (O'Hagan 1995). We derive some interesting properties of the fractional Bayes factor that provide justifications for its use additional to the ones given by O'Hagan. We further argue that the use of the fractional Bayes factor, originally introduced to cope with improper priors, is also useful in a robust analysis. Finally, using usual classes of priors, we compare several alternative Bayes factors for the problem of testing the point null hypothesis in the univariate normal model.     

Bayes estimates in multivariate semiparametric linear models

Bayes estimates are derived in multivariate linear models with unknown distribution. The prior distribution is defined using a Dirichlet prior for the unknown error distribution and a normal-Wishart distribution for the parameters. The posterior distribution is determined and explicit expressions are given in the special cases of location-scale and two-sample models. The calculation of self-informative limits of Bayes estimates yields standard estimates.     

Objective Bayes Factors for Gaussian Directed Acyclic Graphical Models

Abstract. We propose an objective Bayesian method for the comparison of all Gaussian directed acyclic graphical models defined on a given set of variables. The method, which is based on the notion of fractional Bayes factor (BF), requires a single default (typically improper) prior on the space of unconstrained covariance matrices, together with a prior sample size hyper‐parameter, which can be set to its minimal value. We show that our approach produces genuine BFs. The implied prior on the concentration matrix of any complete graph is a data‐dependent Wishart distribution, and this in turn guarantees that Markov equivalent graphs are scored with the same marginal likelihood. We specialize our results to the smaller class of Gaussian decomposable undirected graphical models and show that in this case they coincide with those recently obtained using limiting versions of hyper‐inverse Wishart distributions as priors on the graph‐constrained covariance matrices.     

A Hybrid Approximation Bayesian Test of Variance Components for Longitudinal Data

The test of variance components of possibly correlated random effects in generalized linear mixed models (GLMMs) can be used to examine if there exists heterogeneous effects. The Bayesian test with Bayes factors offers a flexible method. In this article, we focus on the performance of Bayesian tests under three reference priors and a conjugate prior: an approximate uniform shrinkage prior, modified approximate Jeffreys' prior, half-normal unit information prior and Wishart prior. To compute Bayes factors, we propose a hybrid approximation approach combining a simulated version of Laplace's method and importance sampling techniques to test the variance components in GLMMs.     

Bayesian model selection for join point regression with application to age-adjusted cancer rates

Summary.  The method of Bayesian model selection for join point regression models is developed. Given a set of K +1 join point models M ₀,  M ₁, …,  M _K with 0, 1, …,  K join points respec-tively, the posterior distributions of the parameters and competing models M _k are computed by Markov chain Monte Carlo simulations. The Bayes information criterion BIC is used to select the model M _k with the smallest value of BIC as the best model. Another approach based on the Bayes factor selects the model M _k with the largest posterior probability as the best model when the prior distribution of M _k is discrete uniform. Both methods are applied to analyse the observed US cancer incidence rates for some selected cancer sites. The graphs of the join point models fitted to the data are produced by using the methods proposed and compared with the method of Kim and co-workers that is based on a series of permutation tests. The analyses show that the Bayes factor is sensitive to the prior specification of the variance σ ², and that the model which is selected by BIC fits the data as well as the model that is selected by the permutation test and has the advantage of producing the posterior distribution for the join points. The Bayesian join point model and model selection method that are presented here will be integrated in the National Cancer Institute's join point software ( http://www.srab.cancer.gov/joinpoint/ ) and will be available to the public.     

Adaptive Shrinkage in Bayesian Vector Autoregressive Models

Vector autoregressive (VAR) models are frequently used for forecasting and impulse response analysis. For both applications, shrinkage priors can help improving inference. In this article, we apply the Normal-Gamma shrinkage prior to the VAR with stochastic volatility case and derive its relevant conditional posterior distributions. This framework imposes a set of normally distributed priors on the autoregressive coefficients and the covariance parameters of the VAR along with Gamma priors on a set of local and global prior scaling parameters. In a second step, we modify this prior setup by introducing another layer of shrinkage with scaling parameters that push certain regions of the parameter space to zero. Two simulation exercises show that the proposed framework yields more precise estimates of model parameters and impulse response functions. In addition, a forecasting exercise applied to U.S. data shows that this prior performs well relative to other commonly used specifications in terms of point and density predictions. Finally, performing structural inference suggests that responses to monetary policy shocks appear to be reasonable.     

Bain: a program for Bayesian testing of order constrained hypotheses in structural equation models

This paper presents a new statistical method and accompanying software for the evaluation of order constrained hypotheses in structural equation models (SEM). The method is based on a large sample approximation of the Bayes factor using a prior with a data-based correlational structure. An efficient algorithm is written into an R package to ensure fast computation. The package, referred to as Bain, is easy to use for applied researchers. Two classical examples from the SEM literature are used to illustrate the methodology and software.     

A bayesian analysis of the multinomial model for a dichotomous response with nonrespondents

This paper studies the mu1tinomial model 2x2 contingency table data with some cell counts missing .Various hypotheses of interest including row-column independence are tested by using Bayes factors which represent the ratio of the posterior odds to the prior odds for the null hypothesis. The Dirichlet-Beta family of prior distributions is considered for the multinomial parameters cond itional on the complement of the null hypothesis. The Bayes factor for the incomplete data is a mixture of the Bayes factors for different possibilities for the full data.     

Testing equality of regression coefficients in heteroscedastic normal regression models

This article addresses the problem of testing whether the vectors of regression coefficients are equal for two independent normal regression models when the error variances are unknown. This problem poses severe difficulties both to the frequentist and Bayesian approaches to statistical inference. In the former approach, normal hypothesis testing theory does not apply because of the unrelated variances. In the latter, the prior distributions typically used for the parameters are improper and hence the Bayes factor-based solution cannot be used.We propose a Bayesian solution to this problem in which no subjective input is considered. We first generate “objective” proper prior distributions (intrinsic priors) for which the Bayes factor and model posterior probabilities are well defined. The posterior probability of each model is used as a model selection tool. This consistent procedure of testing hypotheses is compared with some of the frequentist approximate tests proposed in the literature.     

Compatible prior distributions for directed acyclic graph models

Summary.  The application of certain Bayesian techniques, such as the Bayes factor and model averaging, requires the specification of prior distributions on the parameters of alternative models. We propose a new method for constructing compatible priors on the parameters of models nested in a given directed acyclic graph model, using a conditioning approach. We define a class of parameterizations that is consistent with the modular structure of the directed acyclic graph and derive a procedure, that is invariant within this class, which we name reference conditioning.