Bayesian inference for Pr (Y

《Applied Mathematical Modelling》2014,38(5-6):1698-1709

Bayesian estimation for generalized exponential distribution based on progressive type-I interval censoring

In this study, we consider the Bayesian estimation of unknown parameters and reliability function of the generalized exponential distribution based on progressive type-I interval censoring. The Bayesian estimates of parameters and reliability function cannot be obtained as explicit forms by applying squared error loss and Linex loss functions, respectively; thus, we present the Lindley’s approximation to discuss these estimations. Then, the Bayesian estimates are compared with the maximum likelihood estimates by using the Monte Carlo simulations.     

Bayesian Monte Carlo estimation for profile hidden Markov models

Hidden Markov models are used as tools for pattern recognition in a number of areas, ranging from speech processing to biological sequence analysis. Profile hidden Markov models represent a class of so-called “left–right” models that have an architecture that is specifically relevant to classification of proteins into structural families based on their amino acid sequences. Standard learning methods for such models employ a variety of heuristics applied to the expectation-maximization implementation of the maximum likelihood estimation procedure in order to find the global maximum of the likelihood function. Here, we compare maximum likelihood estimation to fully Bayesian estimation of parameters for profile hidden Markov models with a small number of parameters. We find that, relative to maximum likelihood methods, Bayesian methods assign higher scores to data sequences that are distantly related to the pattern consensus, show better performance in classifying these sequences correctly, and continue to perform robustly with regard to misspecification of the number of model parameters. Though our study is limited in scope, we expect our results to remain relevant for models with a large number of parameters and other types of left–right hidden Markov models.     

基于贝叶斯统计方法的两总体基因表达数据分类

在疾病的诊断过程中,对疾病的精确分类是提高诊断准确率和疾病治愈率至 关重要的一个环节,DNA芯片技术的出现使得我们从微观的层次获得与疾病分类及诊断 密切相关的基因功能信息．但是DNA芯片技术得到的基因的表达模式数据具有多变量小 样本特点,使得分类过程极不稳定,因此我们首先筛选出表达模式发生显著性变化的基因 作为特征基因集合以减少变量个数,然后再根据此特征基因集合建立分类器对样本进行分 类．本文运用似然比检验筛选出特征基因,然后基于贝叶斯方法建立了统计分类模型,并 应用马尔科夫链蒙特卡罗(MCMC)抽样方法计算样本归类后验概率．最后我们将此模型 应用到两组真实的DNA芯片数据上,并将样本成功分类．     

A survey of sampling-based Bayesian analysis of financial data

The capability of implementing a complete Bayesian analysis of experimental data has emerged over recent years due to computational advances developed within the statistical community. The objective of this paper is to provide a practical exposition of these methods in the illustrative context of a financial event study. The customary assumption of Gaussian errors underlying development of the model is later supplemented by considering Student-t errors, thus permitting a Bayesian sensitivity analysis. The supplied data analysis illustrates the advantages of the sampling-based Bayesian approach in allowing investigation of quantities beyond the scope of classical methods.     

Comparison of the Bayesian and Randomised Decision Tree Ensembles within an Uncertainty Envelope Technique

Multiple Classifier Systems (MCSs) allow evaluation of the uncertainty of classification outcomes that is of crucial importance for safety critical applications. The uncertainty of classification is determined by a trade-off between the amount of data available for training, the classifier diversity and the required performance. The interpretability of MCSs can also give useful information for experts responsible for making reliable classifications. For this reason Decision Trees (DTs) seem to be attractive classification models for experts. The required diversity of MCSs exploiting such classification models can be achieved by using two techniques, the Bayesian model averaging and the randomised DT ensemble. Both techniques have revealed promising results when applied to real-world problems. In this paper we experimentally compare the classification uncertainty of the Bayesian model averaging with a restarting strategy and the randomised DT ensemble on a synthetic dataset and some domain problems commonly used in the machine learning community. To make the Bayesian DT averaging feasible, we use a Markov Chain Monte Carlo technique. The classification uncertainty is evaluated within an Uncertainty Envelope technique dealing with the class posterior distribution and a given confidence probability. Exploring a full posterior distribution, this technique produces realistic estimates which can be easily interpreted in statistical terms. In our experiments we found out that the Bayesian DTs are superior to the randomised DT ensembles within the Uncertainty Envelope technique.     

Dynamically Rescaled Hamiltonian Monte Carlo for Bayesian Hierarchical Models

Dynamically rescaled Hamiltonian Monte Carlo is introduced as a computationally fast and easily implemented method for performing full Bayesian analysis in hierarchical statistical models. The method relies on introducing a modified parameterization so that the reparameterized target distribution has close to constant scaling properties, and thus is easily sampled using standard (Euclidian metric) Hamiltonian Monte Carlo. Provided that the parameterizations of the conditional distributions specifying the hierarchical model are “constant information parameterizations” (CIPs), the relation between the modified- and original parameterization is bijective, explicitly computed, and admit exploitation of sparsity in the numerical linear algebra involved. CIPs for a large catalogue of statistical models are presented, and from the catalogue, it is clear that many CIPs are currently routinely used in statistical computing. A relation between the proposed methodology and a class of explicitly integrated Riemann manifold Hamiltonian Monte Carlo methods is discussed. The methodology is illustrated on several example models, including a model for inflation rates with multiple levels of nonlinearly dependent latent variables. Supplementary materials for this article are available online.     

A markov chain sampler for contingency table exact inference

Summary  In the inference of contingency table, when the cell counts are not large enough for asymptotic approximation, conditioning exact method is used and often computationally impractical for large tables. Instead, various sampling methods can be used. Based on permutation, the Monte Carlo sampling may become again impractical for large tables. For this, existing the Markov chain method is to sample a few elements of the table at each iteration and is inefficient. Here we consider a Markov chain, in which a sub-table of user specified size is updated at each iteration, and it achieves high sampling efficiency. Some theoretical properties of the chain and its applications to some commonly used tables are discussed. As an illustration, this method is applied to the exact test of the Hardy-Weinberg equilibrium in the population genetics context.     

Robust and accurate inference for generalized linear models

In the framework of generalized linear models, the nonrobustness of classical estimators and tests for the parameters is a well known problem, and alternative methods have been proposed in the literature. These methods are robust and can cope with deviations from the assumed distribution. However, they are based on first order asymptotic theory, and their accuracy in moderate to small samples is still an open question. In this paper, we propose a test statistic which combines robustness and good accuracy for moderate to small sample sizes. We combine results from Cantoni and Ronchetti [E. Cantoni, E. Ronchetti, Robust inference for generalized linear models, Journal of the American Statistical Association 96 (2001) 1022–1030] and Robinson, Ronchetti and Young [J. Robinson, E. Ronchetti, G.A. Young, Saddlepoint approximations and tests based on multivariate M-estimators, The Annals of Statistics 31 (2003) 1154–1169] to obtain a robust test statistic for hypothesis testing and variable selection, which is asymptotically χ²-distributed as the three classical tests but with a relative error of order O(n⁻¹). This leads to reliable inference in the presence of small deviations from the assumed model distribution, and to accurate testing and variable selection, even in moderate to small samples.     

Adaptive Bayesian Nonstationary Modeling for Large Spatial Datasets Using Covariance Approximations

Gaussian process models have been widely used in spatial statistics but face tremendous modeling and computational challenges for very large nonstationary spatial datasets. To address these challenges, we develop a Bayesian modeling approach using a nonstationary covariance function constructed based on adaptively selected partitions. The partitioned nonstationary class allows one to knit together local covariance parameters into a valid global nonstationary covariance for prediction, where the local covariance parameters are allowed to be estimated within each partition to reduce computational cost. To further facilitate the computations in local covariance estimation and global prediction, we use the full-scale covariance approximation (FSA) approach for the Bayesian inference of our model. One of our contributions is to model the partitions stochastically by embedding a modified treed partitioning process into the hierarchical models that leads to automated partitioning and substantial computational benefits. We illustrate the utility of our method with simulation studies and the global Total Ozone Matrix Spectrometer (TOMS) data. Supplementary materials for this article are available online.     

Pareto分布形状参数的E-Bayes估计和多层Bayes估计及其应用

给出了参数的E-Bayes估计的定义,对Pareto分布在尺度参数已知时,在平方损失下给出了形状参数的E-Bayes估计和多层Bayes估计,并且用Monte Carlo方法给出了模拟算例.最后,结合高尔夫球手收入数据的实际问题进行了计算,结果表明本文提出的方法可行且便于应用.     

Simulation of conditioned diffusion and application to parameter estimation

In this paper, we propose some algorithms for the simulation of the distribution of certain diffusions conditioned on a terminal point. We prove that the conditional distribution is absolutely continuous with respect to the distribution of another diffusion which is easy for simulation, and the formula for the density is given explicitly. An example of parameter estimation for a Duffing–Van der Pol oscillator is given as an application.     

Bayesian total loss estimation using shared random effects

The pricing of insurance policies requires estimates of the total loss. The traditional compound model imposes an independence assumption on the number of claims and their individual sizes. Bivariate models, which model both variables jointly, eliminate this assumption. A regression approach allows policy holder characteristics and product features to be included in the model. This article presents a bivariate model that uses joint random effects across both response variables to induce dependence effects. Bayesian posterior estimation is done using Markov Chain Monte Carlo (MCMC) methods. A real data example demonstrates that our proposed model exhibits better fitting and forecasting capabilities than existing models.     

A Pivotal Allocation-Based Algorithm for Solving the Label-Switching Problem in Bayesian Mixture Models

In Bayesian analysis of mixture models, the label-switching problem occurs as a result of the posterior distribution being invariant to any permutation of cluster indices under symmetric priors. To solve this problem, we propose a novel relabeling algorithm and its variants by investigating an approximate posterior distribution of the latent allocation variables instead of dealing with the component parameters directly. We demonstrate that our relabeling algorithm can be formulated in a rigorous framework based on information theory. Under some circumstances, it is shown to resemble the classical Kullback-Leibler relabeling algorithm and include the recently proposed equivalence classes representatives relabeling algorithm as a special case. Using simulation studies and real data examples, we illustrate the efficiency of our algorithm in dealing with various label-switching phenomena. Supplemental materials for this article are available online.     

A note on the Bayesian inference for generalized multivariate gamma distribution

In this paper, we show that the derivation of Lemma 3 of Das and Dey (2010) needs to be corrected by using a logical transformation, instead of the ad-hoc transformation which is partially motivated by its univariate equivalent transformation. The correct derivation is presented by two approaches.     

Nonparametric Bayesian Modeling for Multivariate Ordinal Data

This article proposes a probability model for k-dimensional ordinal outcomes, that is, it considers inference for data recorded in k-dimensional contingency tables with ordinal factors. The proposed approach is based on full posterior inference, assuming a flexible underlying prior probability model for the contingency table cell probabilities. We use a variation of the traditional multivariate probit model, with latent scores that determine the observed data. In our model, a mixture of normals prior replaces the usual single multivariate normal model for the latent variables. By augmenting the prior model to a mixture of normals we generalize inference in two important ways. First, we allow for varying local dependence structure across the contingency table. Second, inference in ordinal multivariate probit models is plagued by problems related to the choice and resampling of cutoffs defined for these latent variables. We show how the proposed mixture model approach entirely removes these problems. We illustrate the methodology with two examples, one simulated dataset and one dataset of interrater agreement.     

Statistical inference for a bivariate exponential distribution based on grouped data

Consider the bivarate exponential distribution due to Marshall and Olkin [2], whose survival function is F(x,y)=exp[-λ1x-λ2y-λ12 max(x,y)] (x≥0,y≥0)with unknown pexameters λ1＞0,λ2＞0 andλ12≥0. Based on grouped data, a new estimstor for ,λ1, ,λ2 and ,λ12 is derlved and its asymptotic perties are discussed.Bealdes, some test procedures of equalmarginals and independence are gven. A simulation result is given, too.     

Bayesian Methods for Wavelet Series in Single-Index Models

Single-index models have found applications in econometrics and biometrics, where multidimensional regression models are often encountered. This article proposes a nonparametric estimation approach that combines wavelet methods for nonequispaced designs with Bayesian models. We consider a wavelet series expansion of the unknown regression function and set prior distributions for the wavelet coefficients and the other model parameters. To ensure model identifiability, the direction parameter is represented via its polar coordinates. We employ ad hoc hierarchical mixture priors that perform shrinkage on wavelet coefficients and use Markov chain Monte Carlo methods for a posteriori inference. We investigate an independence-type Metropolis-Hastings algorithm to produce samples for the direction parameter. Our method leads to simultaneous estimates of the link function and of the index parameters. We present results on both simulated and real data, where we look at comparisons with other methods.     

Statistical analysis of two-parameter generalized Birnbaum-Saunders Cauchy distribution

The image features of density function and failure rate function are studied in detail for two-parameter generalized Birnbaum-Saunders Cauchy fatigue life distribution. The logarithmic moment estimation and other two point estimations of parameters are proposed under full sample, and the precisions of point estimations are investigated by Monte-Carlo simulations. The approximate interval estimations of parameters are given by using Taylor expansion, and the precisions of approximate interval estimations are investigated by Monte-Carlo simulations. Finally, several examples show the feasibility of the methods.     

Fast and Accurate Approximation of the Full Conditional for Gamma Shape Parameters

The gamma distribution arises frequently in Bayesian models, but there is not an easy-to-use conjugate prior for the shape parameter of a gamma. This inconvenience is usually dealt with by using either Metropolis–Hastings moves, rejection sampling methods, or numerical integration. However, in models with a large number of shape parameters, these existing methods are slower or more complicated than one would like, making them burdensome in practice. It turns out that the full conditional distribution of the gamma shape parameter is well approximated by a gamma distribution, even for small sample sizes, when the prior on the shape parameter is also a gamma distribution. This article introduces a quick and easy algorithm for finding a gamma distribution that approximates the full conditional distribution of the shape parameter. We empirically demonstrate the speed and accuracy of the approximation across a wide range of conditions. If exactness is required, the approximation can be used as a proposal distribution for Metropolis–Hastings. Supplementary material for this article is available online.