Bayesian inference for generalized extreme value distributions via Hamiltonian Monte Carlo

In this article, we propose to evaluate and compare Markov chain Monte Carlo (MCMC) methods to estimate the parameters in a generalized extreme value model. We employed the Bayesian approach using traditional Metropolis-Hastings methods, Hamiltonian Monte Carlo (HMC), and Riemann manifold HMC (RMHMC) methods to obtain the approximations to the posterior marginal distributions of interest. Applications to real datasets and simulation studies provide evidence that the extra analytical work involved in Hamiltonian Monte Carlo algorithms is compensated by a more efficient exploration of the parameter space.     

Markov chain Monte Carlo with the Integrated Nested Laplace Approximation

The Integrated Nested Laplace Approximation (INLA) has established itself as a widely used method for approximate inference on Bayesian hierarchical models which can be represented as a latent Gaussian model (LGM). INLA is based on producing an accurate approximation to the posterior marginal distributions of the parameters in the model and some other quantities of interest by using repeated approximations to intermediate distributions and integrals that appear in the computation of the posterior marginals. INLA focuses on models whose latent effects are a Gaussian Markov random field. For this reason, we have explored alternative ways of expanding the number of possible models that can be fitted using the INLA methodology. In this paper, we present a novel approach that combines INLA and Markov chain Monte Carlo (MCMC). The aim is to consider a wider range of models that can be fitted with INLA only when some of the parameters of the model have been fixed. We show how new values of these parameters can be drawn from their posterior by using conditional models fitted with INLA and standard MCMC algorithms, such as Metropolis–Hastings. Hence, this will extend the use of INLA to fit models that can be expressed as a conditional LGM. Also, this new approach can be used to build simpler MCMC samplers for complex models as it allows sampling only on a limited number of parameters in the model. We will demonstrate how our approach can extend the class of models that could benefit from INLA, and how the R-INLA package will ease its implementation. We will go through simple examples of this new approach before we discuss more advanced applications with datasets taken from the relevant literature. In particular, INLA within MCMC will be used to fit models with Laplace priors in a Bayesian Lasso model, imputation of missing covariates in linear models, fitting spatial econometrics models with complex nonlinear terms in the linear predictor and classification of data with mixture models. Furthermore, in some of the examples we could exploit INLA within MCMC to make joint inference on an ensemble of model parameters.     

Exact Bayesian modeling for bivariate Poisson data and extensions

Bivariate count data arise in several different disciplines (epidemiology, marketing, sports statistics just to name a few) and the bivariate Poisson distribution being a generalization of the Poisson distribution plays an important role in modelling such data. In the present paper we present a Bayesian estimation approach for the parameters of the bivariate Poisson model and provide the posterior distributions in closed forms. It is shown that the joint posterior distributions are finite mixtures of conditionally independent gamma distributions for which their full form can be easily deduced by a recursively updating scheme. Thus, the need of applying computationally demanding MCMC schemes for Bayesian inference in such models will be removed, since direct sampling from the posterior will become available, even in cases where the posterior distribution of functions of the parameters is not available in closed form. In addition, we define a class of prior distributions that possess an interesting conjugacy property which extends the typical notion of conjugacy, in the sense that both prior and posteriors belong to the same family of finite mixture models but with different number of components. Extension to certain other models including multivariate models or models with other marginal distributions are discussed.     

Bayesian Estimation of the Parameters of Bivariate Exponential Distributions

In this article, we develop an empirical Bayesian approach for the Bayesian estimation of parameters in four bivariate exponential (BVE) distributions. We have opted for gamma distribution as a prior for the parameters of the model in which the hyper parameters have been estimated based on the method of moments and maximum likelihood estimates (MLEs). A simulation study was conducted to compute empirical Bayesian estimates of the parameters and their standard errors. We use moment estimators or MLEs to estimate the hyper parameters of the prior distributions. Furthermore, we compare the posterior mode of parameters obtained by different prior distributions and the Bayesian estimates based on gamma priors are very close to the true values as compared to improper priors. We use MCMC method to obtain the posterior mean and compared the same using the improper priors and the classical estimates, MLEs.     

A two-state regime switching autoregressive model with an application to river flow analysis

We propose a regime switching autoregressive model and apply it to analyze daily water discharge series of River Tisza in Hungary. The dynamics is governed by two regimes, along which both the autoregressive coefficients and the innovation distributions are altering, moreover, the hidden regime indicator process is allowed to be non-Markovian. After examining stationarity and basic properties of the model, we turn to its estimation by Markov Chain Monte Carlo (MCMC) methods and propose two algorithms. The values of the latent process serve as auxiliary parameters in the first one, while the change points of the regimes do the same in the second one in a reversible jump MCMC setting. After comparing the mixing performance of the two methods, the model is fitted to the water discharge data. Simulations show that it reproduces the important features of the water discharge series such as the highly skewed marginal distribution and the asymmetric shape of the hydrograph.     

Data augmentation in multi-way contingency tables with fixed marginal totals

We describe and illustrate approaches to data augmentation in multi-way contingency tables for which partial information, in the form of subsets of marginal totals, is available. In such problems, interest lies in questions of inference about the parameters of models underlying the table together with imputation for the individual cell entries. We discuss questions of structure related to the implications for inference on cell counts arising from assumptions about log-linear model forms, and a class of simple and useful prior distributions on the parameters of log-linear models. We then discuss “local move” and “global move” Metropolis–Hastings simulation methods for exploring the posterior distributions for parameters and cell counts, focusing particularly on higher-dimensional problems. As a by-product, we note potential uses of the “global move” approach for inference about numbers of tables consistent with a prescribed subset of marginal counts. Illustration and comparison of MCMC approaches is given, and we conclude with discussion of areas for further developments and current open issues.     

Relabelling in Bayesian mixture models by pivotal units

Label switching is a well-known and fundamental problem in Bayesian estimation of finite mixture models. It arises when exploring complex posterior distributions by Markov Chain Monte Carlo (MCMC) algorithms, because the likelihood of the model is invariant to the relabelling of mixture components. If the MCMC sampler randomly switches labels, then it is unsuitable for exploring the posterior distributions for component-related parameters. In this paper, a new procedure based on the post-MCMC relabelling of the chains is proposed. The main idea of the method is to perform a clustering technique on the similarity matrix, obtained through the MCMC sample, whose elements are the probabilities that any two units in the observed sample are drawn from the same component. Although it cannot be generalized to any situation, it may be handy in many applications because of its simplicity and very low computational burden.     

基于非线性ECM模型的贝叶斯门限协整研究

 现有门限协整检验方法由于模型似然函数具有多峰、不连续特征,导致冗余参数识别存在困难,最优化计算相对复杂。本文提出基于非线性误差修正模型的贝叶斯门限协整分析,结合参数的后验条件分布设计MCMC抽样方案,进行贝叶斯门限协整检验;并利用Monte Carlo仿真研究了贝叶斯门限协整检验的有限样本性质,发现贝叶斯门限协整检验方法具有良好的有限样本性质。同时,利用不同期限的美国利率序列进行了实证研究,结果发现1个月与3个月利率之间、3个月与6个月利率之间以及3个月与1年利率之间均存在门限协整关系。研究结果表明：贝叶斯门限协整检验方法解决了冗余参数识别的难题,使计算变得相对简单,并提高了估计的精确度和检验的准确性。     

Data reconciliation of nonnormal observations with nonlinear constraints

This paper presents a new method for the reconciliation of data described by arbitrary continuous probability distributions, with the focus on nonlinear constraints. The main idea, already applied to linear constraints in a previous paper, is to restrict the joint prior probability distribution of the observed variables with model constraints to get a joint posterior probability distribution. Because in general the posterior probability density function cannot be calculated analytically, it is shown that it has decisive advantages to sample from the posterior distribution by a Markov chain Monte Carlo (MCMC) method. From the resulting sample of observed and unobserved variables various characteristics of the posterior distribution can be estimated, such as the mean, the full covariance matrix, marginal posterior densities, as well as marginal moments, quantiles, and HPD intervals. The procedure is illustrated by examples from material flow analysis and chemical engineering.     

Mixture of Distributions in the Biparametric Exponential Family: A Bayesian Approach

In this article we propose mixture of distributions belonging to the biparametric exponential family, considering joint modeling of the mean and variance (or dispersion) parameters. As special cases we consider mixtures of normal and gamma distributions. A novel Bayesian methodology, using Markov Chain Monte Carlo (MCMC) methods, is proposed to obtain the posterior summaries of interest. We include simulations and real data examples to illustrate de performance of the proposal.     

Bayesian Time Series Analysis of Structural Changes in Level and Trend

In this article we consider the problem of detecting changes in level and trend in time series model in which the number of change-points is unknown. The approach of Bayesian stochastic search model selection is introduced to detect the configuration of changes in a time series. The number and positions of change-points are determined by a sequence of change-dependent parameters. The sequence is estimated by its posterior distribution via the maximum a posteriori (MAP) estimation. Markov chain Monte Carlo (MCMC) method is used to estimate posterior distributions of parameters. Some actual data examples including a time series of traffic accidents and two hydrological time series are analyzed.     

Bayesian analysis of circular distributions based on non-negative trigonometric sums

ABSTRACT

Fernández-Durán [Circular distributions based on nonnegative trigonometric sums. Biometrics. 2004;60:499–503] developed a new family of circular distributions based on non-negative trigonometric sums that is suitable for modelling data sets that present skewness and/or multimodality. In this paper, a Bayesian approach to deriving estimates of the unknown parameters of this family of distributions is presented. Because the parameter space is the surface of a hypersphere and the dimension of the hypersphere is an unknown parameter of the distribution, the Bayesian inference must be based on transdimensional Markov Chain Monte Carlo (MCMC) algorithms to obtain samples from the high-dimensional posterior distribution. The MCMC algorithm explores the parameter space by moving along great circles on the surface of the hypersphere. The methodology is illustrated with real and simulated data sets.     

Influential Observations in the Functional Measurement Error Model

In this work we propose Bayesian measures to quantify the influence of observations on the structural parameters of the simple measurement error model (MEM). Different influence measures, like those based on q-divergence between posterior distributions and Bayes risk, are studied to evaluate the influence. A strategy based on the perturbation function and MCMC samples is used to compute these measures. The samples from the posterior distributions are obtained by using the Metropolis-Hastings algorithm and assuming specific proper prior distributions. The results are illustrated with an application to a real example modeled with MEM in the literature.     

Bayesian analysis of the amended Davidson model for paired comparison using noninformative and informative priors

In recent years, numerous statisticians have focused their attention on the Bayesian analysis of different paired comparison models. While studying paired comparison techniques, the Davidson model is considered to be one of the famous paired comparison models in the available literature. In this article, we have introduced an amendment in the Davidson model which has been commenced to accommodate the option of not distinguishing the effects of two treatments when they are compared pairwise. Having made this amendment, the Bayesian analysis of the Amended Davidson model is performed using the noninformative (uniform and Jeffreys’) and informative (Dirichlet–gamma–gamma) priors. To study the model and to perform the Bayesian analysis with the help of an example, we have obtained the joint and marginal posterior distributions of the parameters, their posterior estimates, graphical presentations of the marginal densities, preference and predictive probabilities and the posterior probabilities to compare the treatment parameters.     

Estimation,prediction and interpretation of NGG random effects models: an application to Kevlar fibre failure times

We propose a class of Bayesian semiparametric mixed-effects models; its distinctive feature is the randomness of the grouping of observations, which can be inferred from the data. The model can be viewed under a more natural perspective, as a Bayesian semiparametric regression model on the log-scale; hence, in the original scale, the error is a mixture of Weibull densities mixed on both parameters by a normalized generalized gamma random measure, encompassing the Dirichlet process. As an estimate of the posterior distribution of the clustering of the random-effects parameters, we consider the partition minimizing the posterior expectation of a suitable class of loss functions. As a merely illustrative application of our model we consider a Kevlar fibre lifetime dataset (with censoring). We implement an MCMC scheme, obtaining posterior credibility intervals for the predictive distributions and for the quantiles of the failure times under different stress levels. Compared to a previous parametric Bayesian analysis, we obtain narrower credibility intervals and a better fit to the data. We found that there are three main clusters among the random-effects parameters, in accordance with previous frequentist analysis.     

Using the Reversible Jump MCMC Procedure for Identifying and Estimating Univariate TAR Models

One way that has been used for identifying and estimating threshold autoregressive (TAR) models for nonlinear time series follows the Markov chain Monte Carlo (MCMC) approach via the Gibbs sampler. This route has major computational difficulties, specifically, in getting convergence to the parameter distributions. In this article, a new procedure for identifying a TAR model and for estimating its parameters is developed by following the reversible jump MCMC procedure. It is found that the proposed procedure conveys a Markov chain with convergence properties.     

Particle learning for Bayesian semi-parametric stochastic volatility model

This article designs a Sequential Monte Carlo (SMC) algorithm for estimation of Bayesian semi-parametric Stochastic Volatility model for financial data. In particular, it makes use of one of the most recent particle filters called Particle Learning (PL). SMC methods are especially well suited for state-space models and can be seen as a cost-efficient alternative to Markov Chain Monte Carlo (MCMC), since they allow for online type inference. The posterior distributions are updated as new data is observed, which is exceedingly costly using MCMC. Also, PL allows for consistent online model comparison using sequential predictive log Bayes factors. A simulated data is used in order to compare the posterior outputs for the PL and MCMC schemes, which are shown to be almost identical. Finally, a short real data application is included.     

Assessing convergence diagnostic tests for Bayesian Cox regression

The Markov chain Monte Carlo (MCMC) method generates samples from the posterior distribution and uses these samples to approximate expectations of quantities of interest. For the process, researchers have to decide whether the Markov chain has reached the desired posterior distribution. Using convergence diagnostic tests are very important to decide whether the Markov chain has reached the target distribution. Our interest in this study was to compare the performances of convergence diagnostic tests for all parameters of Bayesian Cox regression model with different number of iterations by using a simulation and a real lung cancer dataset.     

Double Generalized Spatial Econometric Models

The authors offer a unified method extending traditional spatial dependence with normally distributed error terms to a new class of spatial models based on the biparametric exponential family of distributions. Joint modeling of the mean and variance (or precision) parameters is proposed in this family of distributions, including spatial correlation. The proposed models are applied for analyzing Colombian land concentration, assuming that the variable of interest follows normal, gamma, and beta distributions. In all cases, the models were fitted using Bayesian methodology with the Markov Chain Monte Carlo (MCMC) algorithm for sampling from joint posterior distribution of the model parameters.     

Collapsing of Non‐centred Parameterized MCMC Algorithms with Applications to Epidemic Models

Data augmentation is required for the implementation of many Markov chain Monte Carlo (MCMC) algorithms. The inclusion of augmented data can often lead to conditional distributions from well‐known probability distributions for some of the parameters in the model. In such cases, collapsing (integrating out parameters) has been shown to improve the performance of MCMC algorithms. We show how integrating out the infection rate parameter in epidemic models leads to efficient MCMC algorithms for two very different epidemic scenarios, final outcome data from a multitype SIR epidemic and longitudinal data from a spatial SI epidemic. The resulting MCMC algorithms give fresh insight into real‐life epidemic data sets.