A Bayesian approach to the multivariate Behrens-Fisher problem under the assumption of proportional covariance matrices

Summary Two independent random samples of sizesN ₁ andN ₂ from multivariate normal populationsN _p (θ₁,∑₁) andN _p (θ₂,∑₂) are considered. Under the null hypothesisH ₀: θ₁=θ₂, a single θ is generated from aN _p(μ, Σ) prior distribution, while underH ₁: θ₁≠θ₂ two means are generated from the exchangeable priorN _p(μ,σ). In both cases Σ will be assumed to have a vague prior distribution. For a simple covariance structure, the Bayes factorB and minimum Bayes factor in favour of the null hypotheses is derived. The Bayes risk for each hypothesis is derived and a strategy is discussed for using the Bayes factor and Bayes risks to test the hypothesis.     

Non-parametric empirical Bayes procedure

In this paper we introduce an empirical Bayes procedure for estimating an unknown parameter, say θ. This procedure gives the empirical Bayes estimator for θ and its associated minimum posterior risk in closed forms without estimating the unknown prior density function of θ. In such procedure the posterior probability density function of θ is not required. A sufficient statistic for θ with conditional probability density function in the one parameter exponential family is required. Instead of estimating the unknown prior density function, the marginal density function of the sufficient statistic must be estimated. As special cases the empirical Bayes estimators and their respective minimum posterior risks of the failure rate for the exponential distribution, the unknown scale parameters of Weibull and gamma distributions are obtained in simple forms as special cases. Numerical results and a simulation study are introduced to (i) investigate how the number of available past experiments and the sample size of each influence the accuracy of the empirical Bayes estimator, (ii) make a comparison between the presented procedure and the Bayes procedure when the prior probability density function of the parameter θ is gamma.     

Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data

This paper describes the Bayesian inference and prediction of the inverse Weibull distribution for Type-II censored data. First we consider the Bayesian inference of the unknown parameter under a squared error loss function. Although we have discussed mainly the squared error loss function, any other loss function can easily be considered. A Gibbs sampling procedure is used to draw Markov Chain Monte Carlo (MCMC) samples, and they have in turn, been used to compute the Bayes estimates and also to construct the corresponding credible intervals with the help of an importance sampling technique. We have performed a simulation study in order to compare the proposed Bayes estimators with the maximum likelihood estimators. We further consider one-sample and two-sample Bayes prediction problems based on the observed sample and provide appropriate predictive intervals with a given coverage probability. A real life data set is used to illustrate the results derived. Some open problems are indicated for further research.     

基于后验预测分布的机器人焊接质量监控研究北大核心

针对KUKA机器人焊接质量监测工作量大、抽样样本小的特点,提出一种基于后验预测分布的贝叶斯动态监控方法。从历史数据中选择合适的数据,计算先验分布的超参数;再结合当前样本构建服从负二项分布的后验预测分布,实时计算控制限,实现对焊接质量的动态监测。结果表明:该方法优于传统似然估计法,有更强的异常检出力和稳健性。     

Specification uncertainty in modeling internet adoption: A developing city case analysis

Internet adoption fosters economic growth and development. Specifying policy control drivers is particularly relevant for developing countries. However, there is no consensus on the most relevant variables. We explored 33.6 million potential models to identify the most important determinants of household internet adoption using stochastic search variable selection and socioeconomic data from Medellín, Colombia. We found that monthly income, the head of household education and voting, and having a computer and cable television at home are the most relevant variables.     

Error rate estimation on the basis of posterior probabilities

The so-called posterior probability estimator, e, formed by averaging the minimum of the posterior probabilities over a set of initial or additional observations (which need not be classified) is considered in the context of estimating the overall actual error rate for the linear discriminant function appropriate for two multivariate normal populations with a common covariance matrix. The bias of e is examined by deriving asymptotic approximations under three different models, the normal, logistic, and mixture models. The properties of e are investigated further by a series of simulation experiments for the logistic and mixture models for which there are few other available estimators.     

用马尔可夫─贝叶斯方法预测孔隙度的分布

用地震资料预测储层参数可以为勘探和开发提供有价值的信息，可以提高钻井的成功率和布井的准确性。但地震所提供的信息存在着局限性，一般来说其精度较低。因此，要有效地预测储层参数，必须把地震、测井及地质学家的经验和推测结合起来。马尔可夫－贝叶斯方法就是这样一种方法。此方法不仅可以预测控制井点较少的区域，而且在没有井的区域，通过地质学家对点散图的解释，仍然可以得到区域储层参数的分布。本文用此方法仅做了孔隙度的估算，其它储层参数的估算道理相同。     

Change-point diagnostics in competing risks models: Two posterior predictive <Emphasis Type="Italic">p</Emphasis>-value approaches

This paper presents a Bayesian diagnostic procedure for examining change-point assumption in the competing risks model framework. It considers the family of distributions arising from the cause-specific model as reported by Chiang (Introduction to stochastic processes in biostatistics. Wiley, New York, 1968) upon which change-points are added to accommodate possible distributional heterogeneity. Model departure, due to misspecification of change-points associated with either the overall survival distribution or cause-specific probabilities, is quantified in terms of a sequence of cumulative-sum statistics between each pair of adjacent change-points assumed. When assessing the asymptotic behavior of each sequence of cumulative-sum statistics using its posterior predictive p-values, see Rubin (Ann Stat 12:1151–1172, 1984) and partial posterior predictive p-values as reported by Bayarri and Berger (J Am Stat Assoc 95:1127–1142, 2000), we show that both types of p-values attain their greatest departure from 0.5 at the change-point that is missed in the assumed model, from which a diagnostic procedure is formalized. Statistical power of these two types of p-values is discussed.      

Approximate Bayesian methods

Summary This paper develops asymptotic expansions for the ratios of integrals that occur in Bayesian analysis: for example, the posterior mean. The first term omitted isO(n ⁻²) and it is shown how the termO(n ⁻¹) can be of importance.     

Bayesian approach for measuring EEPROM process capability based on the one-sided indices CPU and CPL

The purpose of process capability analysis is to provide numerical measures on whether a process is capable of reproducing items meeting the manufacturing specifications. Capability analyses have received considerable recent research attention and increased usage in process assessments and purchasing decisions. Most existing research works on capability analysis focus on estimating and testing process capability based on the traditional distribution frequency approach. In this paper, we propose a Bayesian approach based on the indices C_PU and C_PL to measure EEPROM process capability, in which the specifications are one-sided rather than two-sided. We obtain the credible intervals of C_PU and C_PL and develop a Bayesian procedure for capability testing. The posterior probability p, for which the process under investigation is capable, is derived. The credible interval is a Bayesian analog of the classical lower confidence interval. A process satisfies the manufacturing capability requirements if all the points in the credible interval are greater than the pre-specified capability level w. To make this Bayesian procedure practical for in-plant applications, a real example of an EEPROM manufacturing process is investigated, demonstrating how the Bayesian procedure can be applied to actual data collected in the factories.