Bayesian model comparison based on expected posterior priors for discrete decomposable graphical models

The implementation of the Bayesian paradigm to model comparison can be problematic. In particular, prior distributions on the parameter space of each candidate model require special care. While it is well known that improper priors cannot be routinely used for Bayesian model comparison, we claim that also the use of proper conventional priors under each model should be regarded as suspicious, especially when comparing models having different dimensions. The basic idea is that priors should not be assigned separately under each model; rather they should be related across models, in order to acquire some degree of compatibility, and thus allow fairer and more robust comparisons. In this connection, the intrinsic prior as well as the expected posterior prior (EPP) methodology represent a useful tool. In this paper we develop a procedure based on EPP to perform Bayesian model comparison for discrete undirected decomposable graphical models, although our method could be adapted to deal also with directed acyclic graph models. We present two possible approaches. One based on imaginary data, and one which makes use of a limited number of actual data. The methodology is illustrated through the analysis of a 2×3×4 contingency table.     

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.     

Bayesian penalized B-spline estimation approach for epidemic models

Ordinary differential equations (ODEs) are normally used to model dynamic processes in applied sciences such as biology, engineering, physics, and many other areas. In these models, the parameters are usually unknown, and thus they are often specified artificially or empirically. Alternatively, a feasible method is to estimate the parameters based on observed data. In this study, we propose a Bayesian penalized B-spline approach to estimate the parameters and initial values for ODEs used in epidemiology. We evaluated the efficiency of the proposed method based on simulations using the Markov chain Monte Carlo algorithm for the Kermack–McKendrick model. The proposed approach is also illustrated based on a real application to the transmission dynamics of hepatitis C virus in mainland China.     

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 bridge-randomized penalized quantile regression estimation for linear regression model with AP(q) perturbation

Bridge penalized regression has many desirable statistical properties such as unbiasedness, sparseness as well as ‘oracle’. In Bayesian framework, bridge regularized penalty can be implemented based on generalized Gaussian distribution (GGD) prior. In this paper, we incorporate Bayesian bridge-randomized penalty and its adaptive version into the quantile regression (QR) models with autoregressive perturbations to conduct Bayesian penalization estimation. Employing the working likelihood of the asymmetric Laplace distribution (ALD) perturbations, the Bayesian joint hierarchical models are established. Based on the mixture representations of the ALD and generalized Gaussian distribution (GGD) priors of coefficients, the hybrid algorithms based on Gibbs sampler and Metropolis-Hasting sampler are provided to conduct fully Bayesian posterior estimation. Finally, the proposed Bayesian procedures are illustrated by some simulation examples and applied to a real data application of the electricity consumption.     

Following a moving target—Monte Carlo inference for dynamic Bayesian models

Markov chain Monte Carlo (MCMC) sampling is a numerically intensive simulation technique which has greatly improved the practicality of Bayesian inference and prediction. However, MCMC sampling is too slow to be of practical use in problems involving a large number of posterior (target) distributions, as in dynamic modelling and predictive model selection. Alternative simulation techniques for tracking moving target distributions, known as particle filters, which combine importance sampling, importance resampling and MCMC sampling, tend to suffer from a progressive degeneration as the target sequence evolves. We propose a new technique, based on these same simulation methodologies, which does not suffer from this progressive degeneration.     

Reparameterization strategies for hidden Markov models and Bayesian approaches to maximum likelihood estimation

This paper synthesizes a global approach to both Bayesian and likelihood treatments of the estimation of the parameters of a hidden Markov model in the cases of normal and Poisson distributions. The first step of this global method is to construct a non-informative prior based on a reparameterization of the model; this prior is to be considered as a penalizing and bounding factor from a likelihood point of view. The second step takes advantage of the special structure of the posterior distribution to build up a simple Gibbs algorithm. The maximum likelihood estimator is then obtained by an iterative procedure replicating the original sample until the corresponding Bayes posterior expectation stabilizes on a local maximum of the original likelihood function.     

Model selection in finite mixture of regression models: a Bayesian approach with innovative weighted g priors and reversible jump Markov chain Monte Carlo implementation

Finite mixture of regression (FMR) models are aimed at characterizing subpopulation heterogeneity stemming from different sets of covariates that impact different groups in a population. We address the contemporary problem of simultaneously conducting covariate selection and determining the number of mixture components from a Bayesian perspective that can incorporate prior information. We propose a Gibbs sampling algorithm with reversible jump Markov chain Monte Carlo implementation to accomplish concurrent covariate selection and mixture component determination in FMR models. Our Bayesian approach contains innovative features compared to previously developed reversible jump algorithms. In addition, we introduce component-adaptive weighted g priors for regression coefficients, and illustrate their improved performance in covariate selection. Numerical studies show that the Gibbs sampler with reversible jump implementation performs well, and that the proposed weighted priors can be superior to non-adaptive unweighted priors.     

Bayesian estimation of reliability using time-to-failure distribution of parametric degradation models

This paper focusses on computing the Bayesian reliability of components whose performance characteristics (degradation – fatigue and cracks) are observed during a specified period of time. Depending upon the nature of degradation data collected, we fit a monotone increasing or decreasing function for the data. Since the components are supposed to have different lifetimes, the rate of degradation is assumed to be a random variable. At a critical level of degradation, the time to failure distribution is obtained. The exponential and power degradation models are studied and exponential density function is assumed for the random variable representing the rate of degradation. The maximum likelihood estimator and Bayesian estimator of the parameter of exponential density function, predictive distribution, hierarchical Bayes approach and robustness of the posterior mean are presented. The Gibbs sampling algorithm is used to obtain the Bayesian estimates of the parameter. Illustrations are provided for the train wheel degradation data.     

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.     

Robust Bayesian inference in STAR models with neighbourhood effects

In this paper we propose a robust Bayesian procedure of estimation, testing, validation and selection of spatio-temporal autoregressive models (STAR) with neighbourhood effects applied to the appraisal of dwelling prices. The methodology does not depend on asymptotic results and, unlike previously procedures proposed in the literature, takes into account the uncertainty associated to the estimation of the neighbourhood parameters of the model, giving more realism to the analysis. Moreover, a sequential algorithm to elaborate fast on-line forecast, is provided. The methodology is illustrated by means of a practical case of the real estate market of Zaragoza.     

Bayesian estimation with integrated nested Laplace approximation for binary logit mixed models

In multilevel models for binary responses, estimation is computationally challenging due to the need to evaluate intractable integrals. In this paper, we investigate the performance of integrated nested Laplace approximation (INLA), a fast deterministic method for Bayesian inference. In particular, we conduct an extensive simulation study to compare the results obtained with INLA to the results obtained with a traditional stochastic method for Bayesian inference (MCMC Gibbs sampling), and with maximum likelihood through adaptive quadrature. Particular attention is devoted to the case of small number of clusters. The specification of the prior distribution for the cluster variance plays a crucial role and it turns out to be more relevant than the choice of the estimation method. The simulations show that INLA has an excellent performance as it achieves good accuracy (similar to MCMC) with reduced computational times (similar to adaptive quadrature).     

Bayesian methods for generalized linear models with covariates missing at random

The authors propose methods for Bayesian inference for generalized linear models with missing covariate data. They specify a parametric distribution for the covariates that is written as a sequence of one‐dimensional conditional distributions. They propose an informative class of joint prior distributions for the regression coefficients and the parameters arising from the covariate distributions. They examine the properties of the proposed prior and resulting posterior distributions. They also present a Bayesian criterion for comparing various models, and a calibration is derived for it. A detailed simulation is conducted and two real data sets are examined to demonstrate the methodology.     

Particle filtering for partially observed Gaussian state space models

Summary. Solving Bayesian estimation problems where the posterior distribution evolves over time through the accumulation of data has many applications for dynamic models. A large number of algorithms based on particle filtering methods, also known as sequential Monte Carlo algorithms, have recently been proposed to solve these problems. We propose a special particle filtering method which uses random mixtures of normal distributions to represent the posterior distributions of partially observed Gaussian state space models. This algorithm is based on a marginalization idea for improving efficiency and can lead to substantial gains over standard algorithms. It differs from previous algorithms which were only applicable to conditionally linear Gaussian state space models. Computer simulations are carried out to evaluate the performance of the proposed algorithm for dynamic tobit and probit models.     

Multivariate time series prediction using a hybridization of VARMA models and Bayesian networks

In this paper, a new hybrid model of vector autoregressive moving average (VARMA) models and Bayesian networks is proposed to improve the forecasting performance of multivariate time series. In the proposed model, the VARMA model, which is a popular linear model in time series forecasting, is specified to capture the linear characteristics. Then the errors of the VARMA model are clustered into some trends by K-means algorithm with Krzanowski–Lai cluster validity index determining the number of trends, and a Bayesian network is built to learn the relationship between the data and the trend of its corresponding VARMA error. Finally, the estimated values of the VARMA model are compensated by the probabilities of their corresponding VARMA errors belonging to each trend, which are obtained from the Bayesian network. Compared with VARMA models, the experimental results with a simulation study and two multivariate real-world data sets indicate that the proposed model can effectively improve the prediction performance.     

Bayesian semiparametric reproductive dispersion mixed models for non-normal longitudinal data: estimation and case influence analysis

Semiparametric reproductive dispersion mixed model (SPRDMM) is a natural extension of the reproductive dispersion model and the semiparametric mixed model. In this paper, we relax the normality assumption of random effects in SPRDMM and use a truncated and centred Dirichlet process prior to specify random effects, and present the Bayesian P-spline to approximate the smoothing unknown function. A hybrid algorithm combining the block Gibbs sampler and the Metropolis–Hastings algorithm is implemented to sample observations from the posterior distribution. Also, we develop Bayesian case deletion influence measure for SPRDMM based on the φ-divergence and present those computationally feasible formulas. Several simulation studies and a real example are presented to illustrate the proposed methodologies.     

Exact and Monte Carlo calculations of integrated likelihoods for the latent class model

The latent class model or multivariate multinomial mixture is a powerful approach for clustering categorical data. It uses a conditional independence assumption given the latent class to which a statistical unit is belonging. In this paper, we exploit the fact that a fully Bayesian analysis with Jeffreys non-informative prior distributions does not involve technical difficulty to propose an exact expression of the integrated complete-data likelihood, which is known as being a meaningful model selection criterion in a clustering perspective. Similarly, a Monte Carlo approximation of the integrated observed-data likelihood can be obtained in two steps: an exact integration over the parameters is followed by an approximation of the sum over all possible partitions through an importance sampling strategy. Then, the exact and the approximate criteria experimentally compete, respectively, with their standard asymptotic BIC approximations for choosing the number of mixture components. Numerical experiments on simulated data and a biological example highlight that asymptotic criteria are usually dramatically more conservative than the non-asymptotic presented criteria, not only for moderate sample sizes as expected but also for quite large sample sizes. This research highlights that asymptotic standard criteria could often fail to select some interesting structures present in the data.     

Bayesian inference for sinh-normal/independent nonlinear regression models

Sinh-normal/independent distributions are a class of symmetric heavy-tailed distributions that include the sinh-normal distribution as a special case, which has been used extensively in Birnbaum–Saunders regression models. Here, we explore the use of Markov Chain Monte Carlo methods to develop a Bayesian analysis in nonlinear regression models when Sinh-normal/independent distributions are assumed for the random errors term, and it provides a robust alternative to the sinh-normal nonlinear regression model. Bayesian mechanisms for parameter estimation, residual analysis and influence diagnostics are then developed, which extend the results of Farias and Lemonte [Bayesian inference for the Birnbaum-Saunders nonlinear regression model, Stat. Methods Appl. 20 (2011), pp. 423-438] who used the Sinh-normal/independent distributions with known scale parameter. Some special cases, based on the sinh-Student-t (sinh-St), sinh-slash (sinh-SL) and sinh-contaminated normal (sinh-CN) distributions are discussed in detail. Two real datasets are finally analyzed to illustrate the developed procedures.     

A Bayesian approach for parameter estimation in multi-stage models

Multi-stage time evolving models are common statistical models for biological systems, especially insect populations. In stage-duration distribution models, parameter estimation for the models use the Laplace transform method. This method involves assumptions such as known constant shapes, known constant rates or the same overall hazard rate for all stages. These assumptions are strong and restrictive. The main aim of this paper is to weaken these assumptions by using a Bayesian approach. In particular, a Metropolis-Hastings algorithm based on deterministic transformations is used to estimate parameters. We will use two models, one which has no hazard rates, and the other has stage-wise constant hazard rates. These methods are validated in simulation studies followed by a case study of cattle parasites. The results show that the proposed methods are able to estimate the parameters comparably well, as opposed to using the Laplace transform methods.     

A note on Bayesian estimation of traffic intensity in single-server Markovian queues

ABSTRACT

In queuing theory, a major interest of researchers is studying the behavior and formation process and analyzing the performance characteristics of queues, particularly the traffic intensity, which is defined as the ratio between the arrival rate and the service rate. How these parameters can be estimated using some statistical inferential method is the mathematical problem treated here. This article aims to obtain better Bayesian estimates for the traffic intensity of M/M/1 queues, which, in Kendall notation, stand for Markovian single-server infinity queues. The Jeffreys prior is proposed to obtain the posterior and predictive distributions of some parameters of interest. Samples are obtained through simulation and some performance characteristics are analyzed. It is observed from the Bayes factor that Jeffreys prior is competitive, among informative and non-informative prior distributions, and presents the best performance in many of the cases tested.