Likelihood and the Bayes procedure

Summary In this paper the likelihood function is considered to be the primary source of the objectivity of a Bayesian method. The necessity of using the expected behavior of the likelihood function for the choice of the prior distribution is emphasized. Numerical examples, including seasonal adjustment of time series, are given to illustrate the practical utility of the common-sense approach to Bayesian statistics proposed in this paper.     

Mixing Bayes and empirical Bayes inference to anticipate the realization of engineering concerns about variant system designs

Mixing Bayes and Empirical Bayes inference provides reliability estimates for variant system designs by using relevant failure data - observed and anticipated - about engineering changes arising due to modification and innovation. A coherent inference framework is proposed to predict the realization of engineering concerns during product development so that informed decisions can be made about the system design and the analysis conducted to prove reliability. The proposed method involves combining subjective prior distributions for the number of engineering concerns with empirical priors for the non-parametric distribution of time to realize these concerns in such a way that we can cross-tabulate classes of concerns to failure events within time partitions at an appropriate level of granularity. To support efficient implementation, a computationally convenient hypergeometric approximation is developed for the counting distributions appropriate to our underlying stochastic model. The accuracy of our approximation over first-order alternatives is examined, and demonstrated, through an evaluation experiment. An industrial application illustrates model implementation and shows how estimates can be updated using information arising during development test and analysis.     

The Bayes procedure in exponential reliability family models using conjugate convex tent prior family

A Bayes procedure for estimating the unknown parameter indexed to some of one parameter exponential family distributions is presented. In such a procedure, we shall use a new conjugate prior family that we shall call a conjugate convex tent family. A member of this family could be constructed by assuming a little information about the unknown parameter. Some of the needed parameters for constructing prior density function are the values r, p and q. Bayes estimators for using a priori symmetrical convex tent density can be obtained as special cases of the present work. Numerical simulation study is introduced by using the Monte Carlo method. In this study we have investigated the influence of the prior parameters r, p and q on the accuracy of the Bayes estimators.     

Comparison of empirical Bayes and full Bayes approaches for before-after road safety evaluations

The empirical Bayes (EB) approach has now gained wide acceptance among researchers as the much preferred one for the before-after evaluation of road safety treatments. In this approach, the before period crash experience at treated sites is used in conjunction with a crash prediction model for untreated reference sites to estimate the expected number of crashes that would have occurred without treatment. This estimate is compared to the count of crashes observed after treatment to evaluate the effect of the treatment. This procedure accounts for regression-to-the-mean effects that result from the natural tendency to select for treatment those sites with high observed crash frequencies. Of late, a fully Bayesian (FB) approach has been suggested as a useful, though complex alternative to the empirical Bayes approach in that it is believed to require less data for untreated reference sites, it better accounts for uncertainty in data used, and it provides more detailed causal inferences and more flexibility in selecting crash count distributions. This paper adds to the literature on comparing the two Bayesian approaches through empirical applications. The main application is an evaluation of the conversion of road segments from a four-lane to a three-lane cross-section with two-way left-turn lanes (also known as road diets). For completeness, the paper also summarizes the results of an earlier application pertaining to the evaluation of conversion of rural intersections from unsignalized to signalized control. For both applications, the estimated safety effects from the two approaches are comparable.     

Bayes approach in RDT using accelerated and long-term life data

A common problem of reliability demonstration testing (RDT) is the magnitude of total time on test required to demonstrate reliability to the consumer’s satisfaction, particularly in the case of high reliability components. One solution is the use of accelerated life testing (ALT) techniques. Another is to incorporate prior beliefs, engineering experience, or previous data into the testing framework. This may have the effect of reducing the amount of testing required in the RDT in order to reach a decision regarding conformance to the reliability specification. It is in this spirit that the use of a Bayesian approach can, in many cases, significantly reduce the amount of testing required.We demonstrate the use of this approach to estimate the acceleration factor in the Arrhenius reliability model based on long-term data given by a manufacturer of electronic components (EC). Using the Bayes approach we consider failure rate and acceleration factor to vary randomly according to some prior distributions. Bayes approach enables for a given type of technology the optimal choice of test plan for RDT under accelerated conditions when exacting reliability requirements must be met. These requirements are given by a hypothetical consumer by two different ways. The calculation of posterior consumer’s risk is demonstrated in both cases.The test plans are optimum in that they take into account Var{λ|data}, posterior risk, E{λ|data}, Median λ or other percentiles of λ at data observed at the accelerated conditions. The test setup assumes testing of units with time censoring.     

Merging expert and empirical data for rare event frequency estimation: Pool homogenisation for empirical Bayes models

Empirical Bayes provides one approach to estimating the frequency of rare events as a weighted average of the frequencies of an event and a pool of events. The pool will draw upon, for example, events with similar precursors. The higher the degree of homogeneity of the pool, then the Empirical Bayes estimator will be more accurate. We propose and evaluate a new method using homogenisation factors under the assumption that events are generated from a Homogeneous Poisson Process. The homogenisation factors are scaling constants, which can be elicited through structured expert judgement and used to align the frequencies of different events, hence homogenising the pool. The estimation error relative to the homogeneity of the pool is examined theoretically indicating that reduced error is associated with larger pool homogeneity. The effects of misspecified expert assessments of the homogenisation factors are examined theoretically and through simulation experiments. Our results show that the proposed Empirical Bayes method using homogenisation factors is robust under different degrees of misspecification.     

The predictive validity of empirical Bayes estimates of road safety

This paper examines the predictive validity of empirical Bayes (EB) estimates of road safety. The predictive performance of EB-estimates was tested by applying five versions of EB-estimates of road safety:

• (1) 

  A simple estimate derived from the empirical distribution of accidents in a population of sites, by which the number of accidents predicted for period 2 for sites that recorded k accidents in period 1 equals the number of accidents for sites that recorded k + 1 accidents in period 1.

• (2) 

  Estimates derived from the parameters of a negative binomial distribution fitted to an empirical distribution of accidents in a population of sites by means of the method of moments.

• (3) 

  Estimates derived from the parameters of a negative binomial distribution fitted to an empirical distribution of accidents in a population of sites by means of the maximum likelihood technique.

• (4) 

  Estimates derived by combining the predictions of an accident prediction model and the recorded number of accidents for a site.

• (5) 

  Estimates derived by combining the predictions of a different version of an accident prediction model and the recorded number of accidents for a site.

All versions of EB-estimates are compared to the traditional, naïve assumption of treating the recorded number of accidents as an unbiased estimate of the expected number of accidents.To test the predictive performance of EB-estimates, data for two periods was used. EB-estimates based on data for the first period were treated as predictions of the number of accidents in the second period for road sections that had 0, 1, 2, etc., accidents in the first period, the idea being that the more accurate the prediction, the more accurate the result of a before-and-after study.All versions of EB-estimates were found to give considerably more correct predictions of the number of accidents in the second period than relying on the count of accidents in the first period as a prediction of the count in the second period. Smaller prediction errors were associated with predictions based on accident prediction models than predictions not based on such models.     

Safety effects of section control - An empirical Bayes evaluation

The safety effects of section control were investigated at 14 sites in Norway. A before–after study was conducted with the empirical Bayes method in order to control for regression to the mean (RTM). Effects of trend, volumes, speed limit changes and speed cameras at some of the sites in the before period are controlled for as well. For injury crashes a non-significant reduction by 12% was found. The number of killed or severely injured was found to be significantly reduced by 49% at the section control sites. The results indicate that the crash reductions in tunnels (most of which are undersea tunnels with section control on steep downhill segments) are at least of the same magnitude as on open roads. The results are consistent with findings from speed measurements, although the crash reductions are larger than one would expect as a result from the speed reductions. Downstream of the section control sites (up to 3 km in each direction) injury crashes were found to be significantly reduced by 46%. The number of KSI downstream of the section control sites is too small for drawing any conclusions. It is concluded that section control is effective in reducing both speed and crashes, especially serious crashes, and that spillover effects (crash reductions at non-enforcement sites) are more likely to occur than crash migration. The size of the effects that were found should be interpreted with caution because of the relatively short after periods for some of the sites and the sensitivity of the results to the outcomes of individual crashes.     

删失下的指数分布的贝叶斯估计

本文分别在Ⅱ型删失和随机删失下,表明了共轭先验下的指数分布的刻度参数的贝叶斯估计为具有如下形式的收缩估计(?)_(BE)=a■ bEθ,此处■为依赖样本θ的一个无偏估计且Eθ表示先验分布的期望。当采用平方损失函数时,a b=1;如果用加权平方损失函数,则a b＜1。     

刻度参数的贝叶斯经验贝叶斯估计

我们采用贝叶斯经验贝叶斯方法估计了一个特殊的指数分布族中的刻度参数,证明了刻度参数的贝叶斯经验贝叶斯估计几乎处处收敛到它的贝叶斯估计且获得了它的渐近最优性。最后,提出了一个模拟试验去验证贝叶斯经验贝叶斯估计的渐近最优性。     

Nonparametric Bayes Decision Theory

Summary A summary of the seminar with the same title is presented. Ferguson’s fundamental work on the theory of Dirichlet processes is elucidated and their shortcomings are discussed. Some modifications are also proposed and illustrated. Some of the intricate mathematical issues related to the definitions and the proofs are not discussed for the sake of clarity and brevity. The development related to unimodal processes, briefly mentioned in the last section, will appear as a joint work with Professor W.J. Hall elsewhere. Presently at Bell Labs, Murray Hill, N.J. 07974, U.S.A.     

Bayes Linear Estimators

A class of linear estimators, called Bayes linear estimators, is developed by finding, among all linear estimators, ones which have least average total mean squared error, averaged over parameter points. Ridge, generalized ridge, restricted least squares, subset least squares, least squares, best, and generalized inverse linear estimators are all either Bayes linear estimators or limits of Bayes linear estimators. Results on Bayes linear estimators are extended to affine estimators. “Bootstrapping” procedures, in which the data are recycled in the guise of prior information, are discussed.     

一类连续型单参数指数族参数的经验Bayes检验函数的收敛速度

本文研究了一类连续型单参数指数族参数的经验Bayes检验问题．利用独立同分布样本下概率密度函数的递归核估计和Bayes检验函数的单调性,重新构造一类连续型单参数指数族参数的EB检验函数．通过修改EB检验函数的构造方法,构造了单调的Bayes检验函数．在一定的条件下,获得了EB检验函数的收敛速度,改进了现有文献中的收敛速度阶的结果．最后给出满足定理条件的数值算例．     

Investigating the effects of the fixed and varying dispersion parameters of Poisson-gamma models on empirical Bayes estimates

Traditionally, transportation safety analysts have used the empirical Bayes (EB) method to improve the estimate of the long-term mean of individual sites; to correct for the regression-to-the-mean (RTM) bias in before-after studies; and to identify hotspots or high risk locations. The EB method combines two different sources of information: (1) the expected number of crashes estimated via crash prediction models, and (2) the observed number of crashes at individual sites. Crash prediction models have traditionally been estimated using a negative binomial (NB) (or Poisson-gamma) modeling framework due to the over-dispersion commonly found in crash data. A weight factor is used to assign the relative influence of each source of information on the EB estimate. This factor is estimated using the mean and variance functions of the NB model. With recent trends that illustrated the dispersion parameter to be dependent upon the covariates of NB models, especially for traffic flow-only models, as well as varying as a function of different time-periods, there is a need to determine how these models may affect EB estimates. The objectives of this study are to examine how commonly used functional forms as well as fixed and time-varying dispersion parameters affect the EB estimates. To accomplish the study objectives, several traffic flow-only crash prediction models were estimated using a sample of rural three-legged intersections located in California. Two types of aggregated and time-specific models were produced: (1) the traditional NB model with a fixed dispersion parameter and (2) the generalized NB model (GNB) with a time-varying dispersion parameter, which is also dependent upon the covariates of the model. Several statistical methods were used to compare the fitting performance of the various functional forms. The results of the study show that the selection of the functional form of NB models has an important effect on EB estimates both in terms of estimated values, weight factors, and dispersion parameters. Time-specific models with a varying dispersion parameter provide better statistical performance in terms of goodness-of-fit (GOF) than aggregated multi-year models. Furthermore, the identification of hazardous sites, using the EB method, can be significantly affected when a GNB model with a time-varying dispersion parameter is used. Thus, erroneously selecting a functional form may lead to select the wrong sites for treatment. The study concludes that transportation safety analysts should not automatically use an existing functional form for modeling motor vehicle crashes without conducting rigorous analyses to estimate the most appropriate functional form linking crashes with traffic flow.     

The Bayes linear approach to inference and decision-making for a reliability programme

In reliability modelling it is conventional to build sophisticated models of the probabilistic behaviour of the component lifetimes in a system in order to deduce information about the probabilistic behaviour of the system lifetime. Decision modelling of the reliability programme requires a priori, therefore, an even more sophisticated set of models in order to capture the evidence the decision maker believes may be obtained from different types of data acquisition.Bayes linear analysis is a methodology that uses expectation rather than probability as the fundamental expression of uncertainty. By working only with expected values, a simpler level of modelling is needed as compared to full probability models.In this paper we shall consider the Bayes linear approach to the estimation of a mean time to failure MTTF of a component. The model built will take account of the variance in our estimate of the MTTF, based on a variety of sources of information.     

Alternative Bayes factors: Sample size determination and discriminatory power assessment

Alternative Bayes factors are families of methods used for hypothesis testing and model selection when sensitivity to priors is a concern and also when prior information is weak or lacking. This paper deals with two related problems that arise in the practical use of these model choice criteria: sample size determination and evaluation of discriminatory power. We propose a pre-experimental approach to cope with both these issues. Specifically, extending the evidential approach of Royall (J Am Stat Assoc 95(451):760–780, 2000) and following De Santis (J Stat Plan Inference 124(1):121–144, 2004), we propose a criterion for sample size choice based on the predictive probability of observing decisive and correct evidence. The basic idea is to select the minimal sample size that guarantees a sufficiently high pre-experimental probability that an alternative Bayes factor provides strong evidence in favor of the true hypothesis. It is also argued that a predictive analysis is a natural approach to the measurement of discriminatory power of alternative Bayes factors. The necessity of measuring discrimination ability depends on the fact that alternative Bayes factors are, in general, less sensitive to prior specifications than ordinary Bayes factors and that this gain in robustness corresponds to a reduced discriminative power. Finally, implementation of the predictive approach with improper priors is discussed and possible strategies are proposed.      

刻度指数族参数的经验Bayes估计收敛速度的改进

本文对刻度指数族在加权平方损失下获得了参数的Bayes估计,利用基于Bessel函数的核估计方法构造了相应的经验Bayes(EB)估计,证明了所提出的EB估计具有收敛速度O((n-1In10)λδ-2 δ),此处δ≥2,1/2<λ<1.最后,给出了一个例子,说明适合定理条件的先验分布是存在的.     

Invariancia de las reglas admisibles y Bayes respecto de transformaciones monotonas

From an optimality point of view the solution of a decision problem is related to classes of optimal strategies: admissible, Bayes, etc. which are closely related to boundaries of the risk set S such as lower-boundary, Bayes boundary, positive Bayes boundary. In this paper we present some results concerning invariance properties of such boundaries when the set is transformed by means of a continuous monotonic increasing functionW.

     

On the overhaul effect for repairable mechanical units: a Bayes approach

This paper deals with the statistical analysis, from a Bayes viewpoint, of the failure data of repairable mechanical units subjected to minimal repairs and periodic overhauls. A proportional age reduction model is assumed to model the effect of overhauls on the reliability of the unit, whereas the failure process between two successive overhaul epochs is modeled by the power law process. Point and interval estimation of model parameters, as well as of quantities of large interest, are provided on the basis of a number of suitable prior densities, which reflect different degrees of belief on the failure/repair process. Hypothesis tests on the effectiveness of performed overhauls are developed on the basis of Bayes factor, and some guidelines to perform sensitivity analysis versus the prior information are provided. Finally, a numerical application illustrates the proposed inference and testing procedures.     

Bayes estimation of the general hazard rate model

In reliability theory and life testing models, the life time distributions are often specified by choosing a relevant hazard rate function. Here a general hazard rate function h(t)=a+bt^c−1, where c, a, b are constants greater than zero, is considered. The parameter c is assumed to be known. The Bayes estimators of (a,b) based on the data of type II/item-censored testing without replacement are obtained. A large simulation study using Monte Carlo Method is done to compare the performance of Bayes with regression estimators of (a,b). The criterion for comparison is made based on the Bayes risk associated with the respective estimator. Also, the influence of the number of failed items on the accuracy of the estimators (Bayes and regression) is investigated. Estimations for the parameters (a,b) of the linearly increasing hazard rate model h(t)=a+bt, where a, b are greater than zero, can be obtained as the special case, letting c=2.