Imperfect debugging in software reliability: A Bayesian approach

The objective of studying software reliability is to assist software engineers in understanding more of the probabilistic nature of software failures during the debugging stages and to construct reliability models. In this paper, we consider modeling of a multiplicative failure rate whose components are evolving stochastically over testing stages and discuss its Bayesian estimation. In doing so, we focus on the modeling of parameters such as the fault detection rate per fault and the number of faults. We discuss how the proposed model can account for “imperfect debugging” under certain conditions. We use actual inter-failure data to carry out inference on model parameters via Markov chain Monte Carlo methods and present additional insights from Bayesian analysis.     

Bayesian inference for clustered extremes

We consider Bayesian inference for the extremes of dependent stationary series. We discuss the virtues of the Bayesian approach to inference for the extremal index, and for related characteristics of clustering behaviour. We develop an inference procedure based on an automatic declustering scheme, and using simulated data we implement and assess this procedure, making inferences for the extremal index, and for two cluster functionals. We then apply our procedure to a set of real data, specifically a time series of wind-speed measurements, where the clusters correspond to storms. Here the two cluster functionals selected previously correspond to the mean storm length and the mean inter-storm interval. We also consider inference for long-period return levels, advocating the posterior predictive distribution as being most representative of the information required by engineers interested in design level specifications.      

Call center service process analysis: Bayesian parametric and semi-parametric mixture modeling

The main focus of the call center research has been on models that assume all input distributions are known in queuing theory which gives birth to staffing and the estimation of operating characteristics. Studies investigating uncertainty of the input distributions and its implications on call center management are scarce. This study attempts to fill this gap by analyzing the call center service distribution behavior by using Bayesian parametric and semi-parametric mixture models that are capable of exhibiting non-standard behavior such as multi-modality, skewness and excess kurtosis motivated by real call center data. The study is motivated by the observation that different customer profiles might require different agent skill sets which can create additional sources of uncertainty in the behavior of service distributions. In estimating model parameters, Markov chain Monte Carlo methods such as the Gibbs sampler and the reversible jump algorithms are presented and the implications of using such models on system performance and staffing are discussed.     

Generalized Dynamic Factor Models for Mixed-Measurement Time Series

In this article, we propose generalized Bayesian dynamic factor models for jointly modeling mixed-measurement time series. The framework allows mixed-scale measurements associated with each time series, with different measurements having different distributions in the exponential family conditionally on time-varying latent factor(s). Efficient Bayesian computational algorithms are developed for posterior inference on both the latent factors and model parameters, based on a Metropolis–Hastings algorithm with adaptive proposals. The algorithm relies on a Greedy Density Kernel Approximation and parameter expansion with latent factor normalization. We tested the framework and algorithms in simulated studies and applied them to the analysis of intertwined credit and recovery risk for Moody’s rated firms from 1982 to 2008, illustrating the importance of jointly modeling mixed-measurement time series. The article has supplementary materials available online.     

A selective view of stochastic inference and modeling problems in nanoscale biophysics

Advances in nanotechnology enable scientists for the first time to study biological processes on a nanoscale molecule-by-molecule basis. They also raise challenges and opportunities for statisticians and applied probabilists. To exemplify the stochastic inference and modeling problems in the field, this paper discusses a few selected cases, ranging from likelihood inference, Bayesian data augmentation, and semi- and non-parametric inference of nanometric biochemical systems to the utilization of stochastic integro-differential equations and stochastic networks to model single-molecule biophysical processes. We discuss the statistical and probabilistic issues as well as the biophysical motivation and physical meaning behind the problems, emphasizing the analysis and modeling of real experimental data. This work was supported by the United States National Science Fundation Career Award (Grant No. DMS-0449204)     

A General Class of Change Point and Change Curve Modeling for Life Time Data

Change point hazard rate models arise in many life time data analysis, for example, in studying times until the undesirable side effects occur in clinical trials. In this paper we propose a general class of change point hazard model for survival data. This class includes and extends different types of change point models for survival data, e.g. cure rate model and lag model. Most classical approach develops estimates of model parameters, with particular interest in change point parameter and often the whole hazard function, but exclusively in terms of asymptotic properties. We propose a Bayesian approach, avoiding asymptotics and provide inference conditional upon the observed data. The proposed Bayesian models are fitted using Markov chain Monte Carlo method. We illustrate our proposed methodology with an application to modeling life times of the printed circuit board.     

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Step-stress accelerated degradation test (SSADT) is a useful tool for assessing the lifetime distribution of highly reliable products when the available test items are very few. In this paper, we discuss multiple-steps step-stress accelerated degradation models based on Wiener process, and we apply the objective Bayesian method for such analytically intractable models to obtain the noninformative priors (Jefferys prior and two Reference priors). Moreover, we show that their posterior distributions are proper, and we propose Gibbs sampling algorithms for the Bayesian inference based on the Jefferys prior and two Reference priors. Finally, we present some simulation studies to compare the objective Bayesian estimates with the other Bayesian estimate and the maximum likelihood estimates (MLEs). Simulation results demonstrate the superiority of objective Bayesian analysis method.     

Computation of reference Bayesian inference for variance components in longitudinal studies

Generalized linear mixed models (GLMMs) have been applied widely in the analysis of longitudinal data. This model confers two important advantages, namely, the flexibility to include random effects and the ability to make inference about complex covariances. In practice, however, the inference of variance components can be a difficult task due to the complexity of the model itself and the dimensionality of the covariance matrix of random effects. Here we first discuss for GLMMs the relation between Bayesian posterior estimates and penalized quasi-likelihood (PQL) estimates, based on the generalization of Harville’s result for general linear models. Next, we perform fully Bayesian analyses for the random covariance matrix using three different reference priors, two with Jeffreys’ priors derived from approximate likelihoods and one with the approximate uniform shrinkage prior. Computations are carried out via the combination of asymptotic approximations and Markov chain Monte Carlo methods. Under the criterion of the squared Euclidean norm, we compare the performances of Bayesian estimates of variance components with that of PQL estimates when the responses are non-normal, and with that of the restricted maximum likelihood (REML) estimates when data are assumed normal. Three applications and simulations of binary, normal, and count responses with multiple random effects and of small sample sizes are illustrated. The analyses examine the differences in estimation performance when the covariance structure is complex, and demonstrate the equivalence between PQL and the posterior modes when the former can be derived. The results also show that the Bayesian approach, particularly under the approximate Jeffreys’ priors, outperforms other procedures.     

Bayesian Variable Selection for Median Regression

When the data has heavy tail feature or contains outliers, conventional variable selection methods based on penalized least squares or likelihood functions perform poorly. Based on Bayesian inference method, we study the Bayesian variable selection problem for median linear models. The Bayesian estimation method is proposed by using Bayesian model selection theory and Bayesian estimation method through selecting the Spike and Slab prior for regression coefficients, and the effective posterior Gibbs sampling procedure is also given. Extensive numerical simulations and Boston house price data analysis are used to illustrate the effectiveness of the proposed method.     

Nonparametric Bayesian Modeling for Stochastic Order

In comparing two populations, sometimes a model incorporating stochastic order is desired. Customarily, such modeling is done parametrically. The objective of this paper is to formulate nonparametric (possibly semiparametric) stochastic order specifications providing richer, more flexible modeling. We adopt a fully Bayesian approach using Dirichlet process mixing. An attractive feature of the Bayesian approach is that full inference is available regarding the population distributions. Prior information can conveniently be incorporated. Also, prior stochastic order is preserved to the posterior analysis. Apart from the two sample setting, the approach handles the matched pairs problem, the k-sample slippage problem, ordered ANOVA and ordered regression models. We illustrate by comparing two rather small samples, one of diabetic men, the other of diabetic women. Measurements are of androstenedione levels. Males are anticipated to produce levels which will tend to be higher than those of females.     

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??When the data has heavy tail feature or contains outliers, conventional variable selection methods based on penalized least squares or likelihood functions perform poorly. Based on Bayesian inference method, we study the Bayesian variable selection problem for median linear models. The Bayesian estimation method is proposed by using Bayesian model selection theory and Bayesian estimation method through selecting the Spike and Slab prior for regression coefficients, and the effective posterior Gibbs sampling procedure is also given. Extensive numerical simulations and Boston house price data analysis are used to illustrate the effectiveness of the proposed method.     

Falling and explosive,dormant, and rising markets via multiple‐regime financial time series models

A multiple‐regime threshold nonlinear financial time series model, with a fat‐tailed error distribution, is discussed and Bayesian estimation and inference are considered. Furthermore, approximate Bayesian posterior model comparison among competing models with different numbers of regimes is considered which is effectively a test for the number of required regimes. An adaptive Markov chain Monte Carlo (MCMC) sampling scheme is designed, while importance sampling is employed to estimate Bayesian residuals for model diagnostic testing. Our modeling framework provides a parsimonious representation of well‐known stylized features of financial time series and facilitates statistical inference in the presence of high or explosive persistence and dynamic conditional volatility. We focus on the three‐regime case where the main feature of the model is to capturing of mean and volatility asymmetries in financial markets, while allowing an explosive volatility regime. A simulation study highlights the properties of our MCMC estimators and the accuracy and favourable performance as a model selection tool, compared with a deviance criterion, of the posterior model probability approximation method. An empirical study of eight international oil and gas markets provides strong support for the three‐regime model over its competitors, in most markets, in terms of model posterior probability and in showing three distinct regime behaviours: falling/explosive, dormant and rising markets. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust and data-driven approaches to call centers

We propose both robust and data-driven approaches to a fluid model of call centers that incorporates random arrival rates with abandonment to determine staff levels and dynamic routing policies. We test the resulting models with real data obtained from the call center of a US bank. Computational results show that the robust fluid model is significantly more tractable as compared to the data-driven one and produces overall better solutions to call centers in most experiments.     

A Comparison of Two Sequential Metropolis-Hastings Algorithms with Standard Simulation Techniques in Bayesian Inference in Reliability Models Involving a Generalized Gamma Distribution

In this paper we consider the generalized gamma distribution as introduced in Gåsemyr and Natvig (1998). This distribution enters naturally in Bayesian inference in exponential survival models with left censoring. In the paper mentioned above it is shown that the weighted sum of products of generalized gamma distributions is a conjugate prior for the parameters of component lifetimes, having autopsy data in a Marshall-Olkin shock model. A corresponding result is shown in Gåsemyr and Natvig (1999) for independent, exponentially distributed component lifetimes in a model with partial monitoring of components with applications to preventive system maintenance. A discussion in the present paper strongly indicates that expressing the posterior distribution in terms of the generalized gamma distribution is computationally efficient compared to using the ordinary gamma distribution in such models. Furthermore, we present two types of sequential Metropolis-Hastings algorithms that may be used in Bayesian inference in situations where exact methods are intractable. Finally these types of algorithms are compared with standard simulation techniques and analytical results in arriving at the posterior distribution of the parameters of component lifetimes in special cases of the mentioned models. It seems that one of these types of algorithms may be very favorable when prior assessments are updated by several data sets and when there are significant discrepancies between the prior assessments and the data.     

Variational Bayes With Intractable Likelihood

Variational Bayes (VB) is rapidly becoming a popular tool for Bayesian inference in statistical modeling. However, the existing VB algorithms are restricted to cases where the likelihood is tractable, which precludes their use in many interesting situations such as in state--space models and in approximate Bayesian computation (ABC), where application of VB methods was previously impossible. This article extends the scope of application of VB to cases where the likelihood is intractable, but can be estimated unbiasedly. The proposed VB method therefore makes it possible to carry out Bayesian inference in many statistical applications, including state--space models and ABC. The method is generic in the sense that it can be applied to almost all statistical models without requiring too much model-based derivation, which is a drawback of many existing VB algorithms. We also show how the proposed method can be used to obtain highly accurate VB approximations of marginal posterior distributions. Supplementary material for this article is available online.     

A directed topic model applied to call center improvement

We propose subject matter expert refined topic (SMERT) allocation, a generative probabilistic model applicable to clustering freestyle text. SMERT models are three‐level hierarchical Bayesian models in which each item is modeled as a finite mixture over a set of topics. In addition to discrete data inputs, we introduce binomial inputs. These ‘high‐level’ data inputs permit the ‘boosting’ or affirming of terms in the topic definitions and the ‘zapping’ of other terms. We also present a collapsed Gibbs sampler for efficient estimation. The methods are illustrated using real world data from a call center. Also, we compare SMERT with three alternative approaches and two criteria. Copyright © 2015 John Wiley & Sons, Ltd.     

Bayesian analysis of quantile regression for censored dynamic panel data

This paper develops a Bayesian approach to analyzing quantile regression models for censored dynamic panel data. We employ a likelihood-based approach using the asymmetric Laplace error distribution and introduce lagged observed responses into the conditional quantile function. We also deal with the initial conditions problem in dynamic panel data models by introducing correlated random effects into the model. For posterior inference, we propose a Gibbs sampling algorithm based on a location-scale mixture representation of the asymmetric Laplace distribution. It is shown that the mixture representation provides fully tractable conditional posterior densities and considerably simplifies existing estimation procedures for quantile regression models. In addition, we explain how the proposed Gibbs sampler can be utilized for the calculation of marginal likelihood and the modal estimation. Our approach is illustrated with real data on medical expenditures.     

Bayesian Registration of Functions With a Gaussian Process Prior

We present a Bayesian framework for registration of real-valued functional data. At the core of our approach is a series of transformations of the data and functional parameters, developed under a differential geometric framework. We aim to avoid discretization of functional objects for as long as possible, thus minimizing the potential pitfalls associated with high-dimensional Bayesian inference. Approximate draws from the posterior distribution are obtained using a novel Markov chain Monte Carlo (MCMC) algorithm, which is well suited for estimation of functions. We illustrate our approach via pairwise and multiple functional data registration, using both simulated and real datasets. Supplementary material for this article is available online.     

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Most regression modeling is based on traditional mean regression which results in non-robust estimation results for non-normal errors. Compared to conventional mean regression, composite quantile regression (CQR) may produce more robust parameters estimation. Based on a composite asymmetric Laplace distribution (CALD), we build a Bayesian hierarchical model for the weighted CQR (WCQR). The Gibbs sampler algorithm of Bayesian WCQR is developed to implement posterior inference. Finally, the proposed method are illustrated by some simulation studies and a real data analysis.