Conflict analysis using Bayesian neural networks and generalized linear models

The study of conflict analysis has recently become more important due to current world events. Despite numerous quantitative analyses on the study of international conflict, the statistical results are often inconsistent with each other. The causes of conflict, however, are often stable and replicable when the prior probability of conflict is large. As there has been much conjecture about neural networks being able to cope with the complexity of such interconnected and interdependent data, we formulate a statistical version of a neural network model and compare the results to those of conventional statistical models. We then show how to apply Bayesian methods to the preferred model, with the aim of finding the posterior probabilities of conflict outbreak and hence being able to plan for conflict prevention.     

Bayesian inference in spherical linear models: robustness and conjugate analysis

The early work of Zellner on the multivariate Student-t linear model has been extended to Bayesian inference for linear models with dependent non-normal error terms, particularly through various papers by Osiewalski, Steel and coworkers. This article provides a full Bayesian analysis for a spherical linear model. The density generator of the spherical distribution is here allowed to depend both on the precision parameter φ and on the regression coefficients β. Another distinctive aspect of this paper is that proper priors for the precision parameter are discussed.The normal-chi-squared family of prior distributions is extended to a new class, which allows the posterior analysis to be carried out analytically. On the other hand, a direct joint modelling of the data vector and of the parameters leads to conjugate distributions for the regression and the precision parameters, both individually and jointly. It is shown that some model specifications lead to Bayes estimators that do not depend on the choice of the density generator, in agreement with previous results obtained in the literature under different assumptions. Finally, the distribution theory developed to tackle the main problem is useful on its own right.     

Marginal permutation invariant covariance matrices with applications to linear models

The goal of the present paper is to perform a comprehensive study of the covariance structures in balanced linear models containing random factors which are invariant with respect to marginal permutations of the random factors. We shall focus on model formulation and interpretation rather than the estimation of parameters. It is proven that permutation invariance implies a specific structure for the covariance matrices. Useful results are obtained for the spectra of permutation invariant covariance matrices. In particular, the reparameterization of random effects, i.e., imposing certain constraints, will be considered. There are many possibilities to choose reparameterization constraints in a linear model, however not every reparameterization keeps permutation invariance. The question is if there are natural restrictions on the random effects in a given model, i.e., such reparameterizations which are defined by the covariance structure of the corresponding factor. Examining relationships between the reparameterization conditions applied to the random factors of the models and the spectrum of the corresponding covariance matrices when permutation invariance is assumed, restrictions on the spectrum of the covariance matrix are obtained which lead to “sum-to-zero” reparameterization of the corresponding factor.     

Bayesian analysis of multivariate t linear mixed models using a combination of IBF and Gibbs samplers

The multivariate linear mixed model (MLMM) has become the most widely used tool for analyzing multi-outcome longitudinal data. Although it offers great flexibility for modeling the between- and within-subject correlation among multi-outcome repeated measures, the underlying normality assumption is vulnerable to potential atypical observations. We present a fully Bayesian approach to the multivariate t linear mixed model (MtLMM), which is a robust extension of MLMM with the random effects and errors jointly distributed as a multivariate t distribution. Owing to the introduction of too many hidden variables in the model, the conventional Markov chain Monte Carlo (MCMC) method may converge painfully slowly and thus fails to provide valid inference. To alleviate this problem, a computationally efficient inverse Bayes formulas (IBF) sampler coupled with the Gibbs scheme, called the IBF-Gibbs sampler, is developed and shown to be effective in drawing samples from the target distributions. The issues related to model determination and Bayesian predictive inference for future values are also investigated. The proposed methodologies are illustrated with a real example from an AIDS clinical trial and a careful simulation study.     

Bayesian hierarchical linear mixed models for additive smoothing splines

Bayesian hierarchical models have been used for smoothing splines, thin-plate splines, and L-splines. In analyzing high dimensional data sets, additive models and backfitting methods are often used. A full Bayesian analysis for such models may include a large number of random effects, many of which are not intuitive, so researchers typically use noninformative improper or nearly improper priors. We investigate propriety of the posterior for these cases. Our findings extend known results for normal linear mixed models to certain cases with Bayesian additive smoothing spline models. Supported by National Science Foundation grant SES-0351523 and by National Institutes of Health grants R01-CA100760 and R01-MH071418.     

Bayesian analysis of conditional autoregressive models

Conditional autoregressive (CAR) models have been extensively used for the analysis of spatial data in diverse areas, such as demography, economy, epidemiology and geography, as models for both latent and observed variables. In the latter case, the most common inferential method has been maximum likelihood, and the Bayesian approach has not been used much. This work proposes default (automatic) Bayesian analyses of CAR models. Two versions of Jeffreys prior, the independence Jeffreys and Jeffreys-rule priors, are derived for the parameters of CAR models and properties of the priors and resulting posterior distributions are obtained. The two priors and their respective posteriors are compared based on simulated data. Also, frequentist properties of inferences based on maximum likelihood are compared with those based on the Jeffreys priors and the uniform prior. Finally, the proposed Bayesian analysis is illustrated by fitting a CAR model to a phosphate dataset from an archaeological region.     

Bayesian analysis of fixed-interval preventive-maintenance models

This article develops methods for making accurate decisionswhen scheduling preventive maintenance in systems where inter-eventtimes can be modelled by a delayed renewal process or delayedalternating renewal process. A practical application, relatingto the reliability and maintenance of a relatively low-levelcomponent (valve) in a continuous-process industry over a periodof six years is presented to demonstrate and compare the differentapproaches. Our analyses indicate a cost-effective recommendationfor maintenance practice in this context. Our main thrust relates to the use of Bayesian methodology inorder to obtain rational, admissible decisions. Particular advancesover previous research allow for informative prior distributions,better approximations which lead to improved accuracy, non-negligibledowntimes, and general lifetime distributions. General analyticsolutions are sought for the simpler models, in order to achieveaccuracy and insight. The resulting integrals can only be solvedto give an infinite series and one approximation to the requiredsolution is obtained by truncating this series. Two other approximationsare developed, based on expansions of the prior predictive andlog-posterior distributions. A simulation approach is also developed to include prior informationand hence provide alternative approximations to these optimaldecisions. With exponential lifetime distributions, the relevantposterior lifetime distributions are non-central Pareto. Thissimulation is simple to program, compared to the approximations,but requires more computing time. It is accurate and extendseasily to situations involving greater complexity. We considertwo such extensions, the inclusion of downtimes and Weibulllifetime distributions.     

Objective Bayesian analysis for CAR models

Objective priors, especially reference priors, have been studied extensively for spatial data in the last decade. In this paper, we study objective priors for a CAR model. In particular, the properties of the reference prior and the corresponding posterior are studied. Furthermore, we show that the frequentist coverage probabilities of posterior credible intervals depend only on the spatial dependence parameter $\rho $ , and not on the regression coefficient or the error variance. Based on the simulation study for comparing the reference and Jeffreys priors, the performance of two reference priors is similar and better than the Jeffreys priors. One spatial dataset is used for illustration.     

线性模型参数向量的近似贝叶斯估计

用线性贝叶斯方法去同时估计线性模型中回归系数和误差方差,并在不知道先验分布具体形式的情况下,得到了线性贝叶斯估计的表达式.在均方误差矩阵准则下,证明了其优于最小二乘估计和极大似然估计.与利用MCMC算法得到的贝叶斯估计相比,线性贝叶斯估计具有显式表达式并且更方便使用.对于几种不同的先验分布,数值模拟结果表明线性贝叶斯估...     

Consistent and asymptotically normal estimators for cyclically time-dependent linear models

We consider a general class of time series linear models where parameters switch according to a known fixed calendar. These parameters are estimated by means of quasi-generalized least squares estimators. conditions for strong consistency and asymptotic normality are given. Applications to cyclical ARMA models with non constant periods are considered.     

Bayesian analysis of constant elasticity of variance models

This paper focuses on the estimation of some models in finance and in particular, in interest rates. We analyse discretized versions of the constant elasticity of variance (CEV) models where the normal law showing up in the usual discretization of the diffusion part is replaced by a range of heavy‐tailed distributions. A further extension of the model is to allow the elasticity of variance to be a parameter itself. This generalized model allows great flexibility in modelling and simplifies the model implementation considerably using the scale mixtures representation. The mixing parameters provide a means to identify possible outliers and protect inference by down‐weighting the distorting effects of these outliers. For parameter estimation, Bayesian approach is adopted and implemented using the software WinBUGS (Bayesian inference using Gibbs sampler). Results from a real data analysis show that an exponential power distribution with a random shape parameter, which is highly leptokurtic compared with the normal distribution, forms the best CEV model for the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Bayesian estimation of linear functionals in the case of multidimensional normal distribution

In this paper we describe an approach for the construction of the substitutional method of obtaining Bayesian estimators H_B(f) for H(f). This method is used to obtain Bayesian estimators for the characteristic function, for moments of fixed orders and for the probability of linear inequalities in the case of a multivariate normal distribution. Reported at the XVI Seminar on Stability Problems for Stochastic Models, Eger, Hungary, 29 August — 3 September 1994. Received by the Editorial Board 15 April, 1995. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russian, 1995, Part II.     

Detecting stage-wise outliers in hierarchical Bayesian linear models of repeated measures data

We propose numerical and graphical methods for outlier detection in hierarchical Bayes modeling and analyses of repeated measures regression data from multiple subjects; data from a single subject are generically called a “curve”. The first-stage of our model has curve-specific regression coefficients with possibly autoregressive errors of a prespecified order. The first-stage regression vectors for different curves are linked in a second-stage modeling step, possibly involving additional regression variables. Detection of thestage at which the curve appears to be an outlier and themagnitude and specific component of the violation at that stage is accomplished by embedding the null model into a larger parametric model that can accommodate such unusual observations. We give two examples to illustrate the diagnostics, develop a BUGS program to compute them using MCMC techniques, and examine the sensitivity of the conclusions to the prior modeling assumptions.     

Invariant cross-sections and invariant linear subspaces

Existence theorems for linear subspaces invariant under a continuous mapping and contained in a given set are obtained from a general theorem on existence of invariant crosss-sections. Lecture delivered at a symposium on Series and Geometry in Linear Spaces, held at the Hebrew University of Jerusalem from March 16 till March 24, 1964. This work was supported in part by the National Science Foundation, Grant G-24865.     

Statistical Bayesian classifiers of normal populations and their comparative analysis

In the present paper, direct and substitutional Bayesian classification rules for normal populations are described and their properties are compared. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part III     

Independent sampling for Bayesian normal conditional autoregressive models with OpenCL acceleration

A new computational strategy produces independent samples from the joint posterior distribution for a broad class of Bayesian spatial and spatiotemporal conditional autoregressive models. The method is based on reparameterization and marginalization of the posterior distribution and massive parallelization of rejection sampling using graphical processing units (GPUs) or other accelerators. It enables very fast sampling for small to moderate-sized datasets (up to approximately 10,000 observations) and feasible sampling for much larger datasets. Even using a mid-range GPU and a high-end CPU, the GPU-based implementation is up to 30 times faster than the same algorithm run serially on a single CPU, and the numbers of effective samples per second are orders of magnitude higher than those obtained with popular Markov chain Monte Carlo software. The method has been implemented in the R package CARrampsOcl. This work provides both a practical computing strategy for fitting a popular class of Bayesian models and a proof of concept that GPU acceleration can make independent sampling from Bayesian joint posterior densities feasible.