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Showing papers on "Bayes' theorem published in 1990"


Journal ArticleDOI
TL;DR: The use of the Gibbs sampler as a method for calculating Bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models, including variance components, unordered and ordered means, hierarchical growth curves, and missing data in a crossover trial.
Abstract: The use of the Gibbs sampler as a method for calculating Bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models, including variance components, unordered and ordered means, hierarchical growth curves, and missing data in a crossover trial. In all cases the approach is straightforward to specify distributionally and to implement computationally, with output readily adapted for required inference summaries.

1,020 citations


Journal ArticleDOI
TL;DR: The multilayer perceptron, when trained as a classifier using backpropagation, is shown to approximate the Bayes optimal discriminant function.
Abstract: The multilayer perceptron, when trained as a classifier using backpropagation, is shown to approximate the Bayes optimal discriminant function. The result is demonstrated for both the two-class problem and multiple classes. It is shown that the outputs of the multilayer perceptron approximate the a posteriori probability functions of the classes being trained. The proof applies to any number of layers and any type of unit activation function, linear or nonlinear. >

866 citations


Journal ArticleDOI
TL;DR: A general approach to hierarchical Bayes changepoint models is presented, including an application to changing regressions, changing Poisson processes and changing Markov chains, which avoids sophisticated analytic and numerical high dimensional integration procedures.
Abstract: SUMMARY A general approach to hierarchical Bayes changepoint models is presented. In particular, desired marginal posterior densities are obtained utilizing the Gibbs sampler, an iterative Monte Carlo method. This approach avoids sophisticated analytic and numerical high dimensional integration procedures. We include an application to changing regressions, changing Poisson processes and changing Markov chains. Within these contexts we handle several previously inaccessible problems.

585 citations


Journal ArticleDOI
TL;DR: The authors examine the relative entropy distance D/sub n/ between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+c, where d is the dimension of the parameter vector.
Abstract: In the absence of knowledge of the true density function, Bayesian models take the joint density function for a sequence of n random variables to be an average of densities with respect to a prior. The authors examine the relative entropy distance D/sub n/ between the true density and the Bayesian density and show that the asymptotic distance is (d/2)(log n)+c, where d is the dimension of the parameter vector. Therefore, the relative entropy rate D/sub n//n converges to zero at rate (log n)/n. The constant c, which the authors explicitly identify, depends only on the prior density function and the Fisher information matrix evaluated at the true parameter value. Consequences are given for density estimation, universal data compression, composite hypothesis testing, and stock-market portfolio selection. >

517 citations


Journal ArticleDOI
TL;DR: In this article, the problem of finding Bayes estimators for cumulative hazard rates and related quantities, w.r.t. prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments was studied.
Abstract: Several authors have constructed nonparametric Bayes estimators for a cumulative distribution function based on (possibly right-censored) data. The prior distributions have, for example, been Dirichlet processes or, more generally, processes neutral to the right. The present article studies the related problem of finding Bayes estimators for cumulative hazard rates and related quantities, w.r.t. prior distributions that correspond to cumulative hazard rate processes with nonnegative independent increments. A particular class of prior processes, termed beta processes, is introduced and is shown to constitute a conjugate class. To arrive at these, a nonparametric time-discrete framework for survival data, which has some independent interest, is studied first. An important bonus of the approach based on cumulative hazards is that more complicated models for life history data than the simple life table situation can be treated, for example, time-inhomogeneous Markov chains. We find posterior distributions and derive Bayes estimators in such models and also present a semiparametric Bayesian analysis of the Cox regression model. The Bayes estimators are easy to interpret and easy to compute. In the limiting case of a vague prior the Bayes solution for a cumulative hazard is the Nelson-Aalen estimator and the Bayes solution for a survival probability is the Kaplan-Meier estimator.

515 citations


Journal ArticleDOI
TL;DR: It is concluded that probabilistic interpretations of conventional statistics are rarely justified, and that such interpretations may encourage misinterpretation of nonrandomized studies.
Abstract: This paper reviews the role of statistics in causal inference. Special attention is given to the need for randomization to justify causal inferences from conventional statistics, and the need for random sampling to justify descriptive inferences. In most epidemiologic studies, randomization and random sampling play little or no role in the assembly of study cohorts. I therefore conclude that probabilistic interpretations of conventional statistics are rarely justified, and that such interpretations may encourage misinterpretation of nonrandomized studies. Possible remedies for this problem include deemphasizing inferential statistics in favor of data descriptors, and adopting statistical techniques based on more realistic probability models than those in common use.

327 citations


Journal ArticleDOI
Eric A. Wan1
TL;DR: The relationship between minimizing a mean squared error and finding the optimal Bayesian classifier is reviewed and a number of confidence measures are proposed to evaluate the performance of the neural network classifier within a statistical framework.
Abstract: The relationship between minimizing a mean squared error and finding the optimal Bayesian classifier is reviewed. This provides a theoretical interpretation for the process by which neural networks are used in classification. A number of confidence measures are proposed to evaluate the performance of the neural network classifier within a statistical framework. >

301 citations


Journal ArticleDOI
TL;DR: Certain fundamental results and attractive features of the proposed approach in the context of the random field theory are discussed, and a systematic spatial estimation scheme is presented that satisfies a variety of useful properties beyond those implied by the traditional stochastic estimation methods.
Abstract: The purpose of this paper is to stress the importance of a Bayesian/maximum-entropy view toward the spatial estimation problem. According to this view, the estimation equations emerge through a process that balances two requirements: High prior information about the spatial variability and high posterior probability about the estimated map. The first requirement uses a variety of sources of prior information and involves the maximization of an entropy function. The second requirement leads to the maximization of a so-called Bayes function. Certain fundamental results and attractive features of the proposed approach in the context of the random field theory are discussed, and a systematic spatial estimation scheme is presented. The latter satisfies a variety of useful properties beyond those implied by the traditional stochastic estimation methods.

291 citations


Journal ArticleDOI
TL;DR: A new inference method, Highest Confidence First (HCF) estimation, is used to infer a unique labeling from the a posteriori distribution that is consistent with both prior knowledge and evidence.
Abstract: Integrating disparate sources of information has been recognized as one of the keys to the success of general purpose vision systems. Image clues such as shading, texture, stereo disparities and image flows provide uncertain, local and incomplete information about the three-dimensional scene. Spatial a priori knowledge plays the role of filling in missing information and smoothing out noise. This thesis proposes a solution to the longstanding open problem of visual integration. It reports a framework, based on Bayesian probability theory, for computing an intermediate representation of the scene from disparate sources of information. The computation is formulated as a labeling problem. Local visual observations for each image entity are reported as label likelihoods. They are combined consistently and coherently on hierarchically structured label trees with a new, computationally simple procedure. The pooled label likelihoods are fused with the a priori spatial knowledge encoded as Markov Random Fields (MRF's). The a posteriori distribution of the labelings are thus derived in a Bayesian formalism. A new inference method, Highest Confidence First (HCF) estimation, is used to infer a unique labeling from the a posteriori distribution. Unlike previous inference methods based on the MRF formalism, HCF is computationally efficient and predictable while meeting the principles of graceful degradation and least commitment. The results of the inference process are consistent with both observable evidence and a priori knowledge. The effectiveness of the approach is demonstrated with experiments on two image analysis problems: intensity edge detection and surface reconstruction. For edge detection, likelihood outputs from a set of local edge operators are integrated with a priori knowledge represented as an MRF probability distribution. For surface reconstruction, intensity information is integrated with sparse depth measurements and a priori knowledge. Coupled MRF's provide a unified treatment of surface reconstruction and segmentation, and an extension of HCF implements a solution method. Experiments using real image and depth data yield robust results. The framework can also be generalized to higher-level vision problems, as well as to other domains.

285 citations



Journal ArticleDOI
TL;DR: Several models of information integration are developed and analyzed within the context of a proto-typical pattern-recognition task to provide a measure of identifiability or the extent to which the models can be distinguished from one another.
Abstract: Several models of information integration are developed and analyzed within the context of a proto-typical pattern-recognition task. The central concerns are whether the models prescribe maximally efficient (optimal) integration and to what extent the models are psychologically valid. Evaluation, integration, and decision processes are specified for each model. Important features are whether evaluation is noisy, whether integration follows Bayes's theorem, and whether decision consists of a criterion rule or a relative goodness rule. Simulations of the models and predictions of the results by the same models are carried out to provide a measure of identifiability or the extent to which the models can be distinguished from one another. The models are also contrasted against empirical results from tasks with 2 and 4 response alternatives and with graded responses.

Journal ArticleDOI
TL;DR: A highly specific drug-resistance assay was developed in which human tumor colonies were cultured in soft agar and drugs were tested at high concentrations for long exposure times, allowing the construction of a nomogram for determining assay-predicted probability of response.
Abstract: Bayes' theorem has been used to describe the relationship between the accuracy of a predictive test (posttest probability) and the overall incidence of what is being tested (pretest probability). Bayes' theorem indicates that laboratory assays will be accurate in the prediction of clinical drug resistance in tumors with high overall response rates (e.g., previously untreated breast cancer) only when the assays are extremely (greater than 98%) specific for drug resistance. We developed a highly specific drug-resistance assay in which human tumor colonies were cultured in soft agar and drugs were tested at high concentrations for long exposure times. Coefficients for concentration x time exceeded those reported in contemporaneous studies by about 100-fold. We reviewed 450 correlations between assay results and clinical response over an 8-year period. Results were analyzed by subsets, including different tumor histologies, single agents, and drug combinations. Extreme drug resistance (an assay result greater than or equal to SD below the median) was identified with greater than 99% specificity. Only one of 127 patients with tumors showing extreme drug resistance responded to chemotherapy. This negligible posttest probability of response was independent of pretest (expected) probability of response. Once this population of patients with tumors showing extreme drug resistance had been identified, posttest response probabilities for the remaining cohorts of patients varied according to both assay results and pretest response probabilities, precisely according to predictions based on Bayes' theorem. This finding allowed the construction of a nomogram for determining assay-predicted probability of response.

Journal ArticleDOI
TL;DR: In this paper, the authors compared the conventional method of measuring ability, which is based on items with assumed true parameter values obtained from a pretest, compared to a Bayesian method that deals with the uncertainties of such items.
Abstract: The conventional method of measuring ability, which is based on items with assumed true parameter values obtained from a pretest, is compared to a Bayesian method that deals with the uncertainties of such items. Computational expressions are presented for approximating the posterior mean and variance of ability under the three-parameter logistic (3PL) model. A 1987 American College Testing Program (ACT) math test is used to demonstrate that the standard practice of using maximum likelihood or empirical Bayes techniques may seriously underestimate the uncertainty in estimated ability when the pretest sample is only moderately large.

Journal ArticleDOI
TL;DR: Early forecasting and the adaptive capability are the two major payoffs from using Hierarchical Bayes procedures, which contrasts with other estimation approaches which either use a linear model or provide reasonable forecasts only after the inflection point of the time series of the sales.
Abstract: A method for obtaining early forecasts for the sales of new durable products based on Hierarchical Bayes procedures is presented. The Bass model is implemented within this framework by using a nonlinear regression approach. The linear regression model has been shown to have numerous shortcomings. Two stages of prior distributions use sales data from a variety of dissimilar new products. The first prior distribution describes the variation among the parameters of the products, and the second prior distribution expresses the uncertainty about the hyperparameters of the first prior. Before observing sales data for a new product launch, the forecasts are the expectation of the first stage prior distribution. As sales data become available, the forecasts adapt to the unique features of the product. Early forecasting and the adaptive capability are the two major payoffs from using Hierarchical Bayes procedures. This contrasts with other estimation approaches which either use a linear model or provide reasonable forecasts only after the inflection point of the time series of the sales. The paper also indicates how the Hierarchical Bayes procedure can be extended to include exogenous variables.

Journal ArticleDOI
TL;DR: The first five sections of the paper describe the Bayesian paradigm for statistics and its relationship with other attitudes towards inference and an attempt is made to appreciate how accurate formulae like the extension of the conversation, the product law and Bayes rule are in evaluating probabilities.
Abstract: The first five sections of the paper describe the Bayesian paradigm for statistics and its relationship with other attitudes towards inference. Section 1 outlines Wald's major contributions and explains how they omit the vital consideration of coherence. When this point is included the Bayesian view results, with the main difference that Waldean ideas require the concept of the sample space, whereas the Bayesian approach may dispense with it, using a probability distribution over parameter space instead. Section 2 relates statistical ideas to the problem of inference in science. Scientific inference is essentially the passage from observed, past data to unobserved, future data. The roles of models and theories in doing this are explored. The Bayesian view is that all this should be accomplished entirely within the calculus of probability and Section 3 justifies this choice by various axiom systems. The claim is made that this leads to a quite different paradigm from that of classical statistics and, in particular, prob- lems in the latter paradigm cease to have importance within the other. Point estimation provides an illustration. Some counter-examples to the Bayesian view are discussed. It is important that statistical conclusions should be usable in making decisions. Section 4 explains how the Bayesian view achieves this practi- cality by introducing utilities and the principle of maximizing expected utility. Practitioners are often unhappy with the ideas of basing inferences on one number, probability, or action on another, an expectation, so these points are considered and the methods justified. Section 5 discusses why the Bayesian viewpoint has not achieved the success that its logic suggests. Points discussed include the relationship between the inferences and the practical situation, for example with multiple comparisons; and the lack of the need to confine attention to normality or the exponential family. Its extensive use by nonstatisticians is documented. The most important objection to the Bayesian view is that which rightly says that probabilities are hard to assess. Consequently Section 6 considers how this might be done and an attempt is made to appreciate how accurate formulae like the extension of the conversation, the product law and Bayes rule are in evaluating probabilities.

Journal ArticleDOI
TL;DR: Here Bayesian analysis is extended to the problem of selecting the model which is most probable in view of the data and all the prior information, and in addition to the analytic calculation, two examples are given.


Book
10 Oct 1990
TL;DR: In this article, a posterior analysis based on distributions for robust maximum likelihood type estimates is proposed for reconstruction of digital images. But it is not suitable for the reconstruction of 3D images.
Abstract: Basic concepts.- Bayes' Theorem.- Prior density functions.- Point estimation.- Confidence regions.- Hypothesis testing.- Predictive analysis.- Numerical techniques.- Models and special applications.- Linear models.- Nonlinear models.- Mixed models.- Linear models with unknown variance and covariance components.- Classification.- Posterior analysis based on distributions for robust maximum likelihood type estimates.- Reconstruction of digital images.

Journal ArticleDOI
TL;DR: Bayesian methods simplify the analysis of data from sequential clinical trials and avoid certain paradoxes of frequentist inference, and offer a natural setting for the synthesis of expert opinion in deciding policy matters.
Abstract: Attitudes of biostatisticians toward implementation of the Bayesian paradigm have changed during the past decade due to the increased availability of computational tools for realistic problems. Empirical Bayes' methods, already widely used in the analysis of longitudinal data, promise to improve cancer incidence maps by accounting for overdispersion and spatial correlation. Hierarchical Bayes' methods offer a natural framework in which to demonstrate the bioequivalence of pharmacologic compounds. Their use for quantitative risk assessment and carcinogenesis bioassay is more controversial, however, due to uncertainty regarding specification of informative priors. Bayesian methods simplify the analysis of data from sequential clinical trials and avoid certain paradoxes of frequentist inference. They offer a natural setting for the synthesis of expert opinion in deciding policy matters. Both frequentist and Bayes' methods have a place in biostatistical practice.

Journal ArticleDOI
TL;DR: In this article, a conditional bias correction method is proposed to correct the shortness of the EM intervals, since they do not attain the desired coverage probability in the EB sense defined by Morris (1983a, b).
Abstract: Parametric empirical Bayes (EB) methods of point estimation date to the landmark paper by James and Stein (1961). Interval estimation through parametric empirical Bayes techniques has a somewhat shorter history, which was summarized by Laird and Louis (1987). In the exchangeable case, one obtains a “naive” EB confidence interval by simply taking appropriate percentiles of the estimated posterior distribution of the parameter, where the estimation of the prior parameters (“hyperparameters”) is accomplished through the marginal distribution of the data. Unfortunately, these “naive” intervals tend to be too short, since they fail to account for the variability in the estimation of the hyperparameters. That is, they do not attain the desired coverage probability in the EB sense defined by Morris (1983a, b). They also provide no statement of conditional calibration (Rubin 1984). In this article we propose a conditional bias correction method for developing EM intervals that corrects these deficiencies...

Journal ArticleDOI
TL;DR: In this article, Bayes estimators of the parameters of the Marshall-Olkin exponential distribution are obtained when random samples from series and parallel systems are available, with respect to the quadratic loss function, and the prior distribution allows for prior dependence among the components of the parameter vector.
Abstract: SUMMARY Bayes estimators of the parameters of the Marshall-Olkin exponential distribution are obtained when random samples from series and parallel systems are available. The estimators are with respect to the quadratic loss function, and the prior distribution allows for prior dependence among the components of the parameter vector. Exact and approximate highest posterior density credible ellipsoids for the parameters are also obtained. In contrast with series sampling, the Bayes estimators under parallel sampling are not in closed form, and numerical procedures are required to obtain estimates. Bayes estimators of the reliability functions are also given. The gain in asymptotic precision of parallel estimates over series estimates is also ascertained theoretically.

Journal ArticleDOI
TL;DR: TheBurn-in problem is reframed in a decision-making context and the role of the predictive life distribution in the burn-in decision problem is highlighted.
Abstract: The burn-in problem is reframed in a decision-making context. The role of the predictive life distribution in the burn-in decision problem is highlighted. The case in which the predictive distribution is a mixture of exponentials is discussed extensively.

Journal ArticleDOI
TL;DR: In this paper, lower bounds for the Bayes risk under scaled quadratic loss were derived from the information inequality, which is a direct extension of an earlier result of Fabian and Hannan.
Abstract: This paper presents lower bounds, derived from the information inequality, for the Bayes risk under scaled quadratic loss. Some numerical results are also presented which give some idea concerning the precision of these bounds. An appendix contains a proof of the information inequality without conditions on the estimator. This result is a direct extension of an earlier result of Fabian and Hannan.

Book
01 May 1990
TL;DR: This updated and expanded edition of Marvin H. Gruber's Regression Estimators presents, compares, and contrasts the development and properties of ridge-type estimators from these two philosophically different points of view.
Abstract: Preface Part I: Introduction and Mathematical Preliminaries 1. Introduction 1.1. The Purpose of This Book 1.2. Least Square Estimators and the Need for Alternatives 1.3. Historical Survey 1.4. The Structure of the Book 2. Mathematical and Statistical Preliminaries 2.0. Introduction 2.1. Matrix Theory Results 2.2. The Bayes Estimator (BE) 2.3. Admissible Estimators 2.4. The Minimax Estimator 2.5. Criterion for Comparing Estimators: Theobald's 1974 Result 2.6. Some Useful Inequalities: Some Miscellaneous Useful Matrix Results 2.7. Summary Part II: The Estimators, Their Derivations, and Their Relationships 3. The Estimators 3.0. The Least Square Estimator and Its Properties 3.1. The Generalized Ridge Regression Estimator 3.2. The Mixed Estimators 3.3. The Linear Minimax Estimator 3.4. The Bayes Estimator 3.6. Summary 4. How the Different Estimators Are Related 4.0. Introduction 4.1. Alternative Forms of the Bayes Estimator Full-Rank Case 4.2. Alternative Forms of the Bayes Estimator Non-Full-Rank Case Estimable Parametric Functions 4.3. Equivalence of the Generalized Ridge Estimator and the BayesEstimator 4.4. Equivalence of the Mixed Estimator and the Bayes Estimator 4.5. Ridge Estimators in the Literature as Special Cases of the BE, Minimax Estimators, or Mixed Estimators 4.6. An Extension of the Gauss-Markov Theorem 4.7. Generalities 4.8. Summary Part III: Comparing the Efficiency of the Estimators 5. Measures of Efficiency of the Estimators 5.0. Introduction 5.1. The Different Kinds of Mean Square Error 5.2. Zellner's Balanced Loss Function 5.3. The LINEX Loss Function 5.4. Linear Admissibility 5.5. Summary 6. The Average Mean Square Error 6.0. Introduction 6.1. The Forms of the MSE for the Minimax, Bayes, and Mixed Estimators 6.2. The Relationship between the Average Variance and the MSE 6.3. The Average MSE of the Bayes Estimator 6.4. Alternative Forms of the MSE of the Mixed Estimator 6.5. Comparison of the MSE of Different BEs 6.6. Comparison of the MSE of the Ridge and Contraction Estimators 6.7. Comparison of the Average MSE of the Two-Parameter Liu Estimator and the Ordinary Ridge Regression Estimator 6.8. Summary 7. The MSE Neglecting the Prior Assumptions 7.0. Introduction 7.1. The MSE of the BE 7.2. The MSE of the Mixed Estimators Neglecting PriorAssumptions 7.3. Comparison of the Conditional MSE of the Bayes and Least Square Estimators and Comparison of the Conditional and Average MSE 7.4. Comparison of the MSE of a Mixed Estimator with That of the LS Estimators 7.5. Comparison of the MSE of Two Bayes Estimators 7.6. Summary 8. The MSE for Incorrect Prior Assumptions 8.0. Introduction 8.1. The Bayes Estimator and Its MSE 8.2. The Minimax Estimator 8.3. The Mixed Estimator 8.4. Contaminated Priors 8.5. Contaminated (Mixed) Bayes Estimators 8.6. Summary Part IV: Applications 9. The Kalman Filter 9.0. Introduction 9.1. The Kalman Filter as a Bayes Estimator 9.2. The Kalman Filter as a Recursive Least Square Estimator,and the Connection with the Mixed Estimator 9.3. The Minimax Estimator 9.4. The Generalized Ridge Estimator 9.5. The Average Mean Square Error 9.6. The MSE for Incorrect Initial Prior Assumptions 9.7. Applications 9.8. Recursive Ridge Regression 9.9. Summary 10. Experimental Design Models 10.0. Introduction 10.1. The One-Way ANOVA Model 10.2. The Bayes and Empirical Bayes Estimators 10.3. The Two- Way Classification 10.4. The Bayes and Empirical Bayes Estimators 10.5. Summary Appendix to Section 10.2. Calculation of the MSE of Section 10.2 11. How Penalized Splines and Ridge- Type EstimatorsAre Related 11.0. Introduction 11.1. Splines as a Special Kind of Regression Model 11.2. Penalized Splines 11.3. The Best Linear Unbiased Predictor (BLUP) 11.4. Two Examples 11.5. Summary Part V: Alternative Measures of Efficiency 12. Estimation Using Zellner's Balanced Loss Function 12.0. Introduction 12.1. Zellner's Balanced Loss Function 12.2. The Estimators from Different Points of View 12.3. The Average Mean Square Error 12.4. The Risk without Averaging over a Prior Distribution 12.5. Some Optimal Ridge Estimators 12.6. Summary 13. The LINEX and Other Asymmetric Loss Functions 13.0. Introduction 13.1. The LINEX Loss Function 13.2. The Bayes Risk for a Regression Estimator 13.3. The Frequentist Risk 13.4. Summary 14. Distances between Ridge-Type Estimators, andInformation Geometry 14.0. Introduction 14.1. The Relevant Differential Geometry 14.2. The Distance between Two Linear Bayes Estimators, Based on the Prior Distributions 14.3. The Distance between Distributions of Ridge-Type Estimators from a Non-Bayesian Point of View 14.4. Distances between the Mixed Estimators 14.5. An Example Using the Kalman Filter 14.6. Summary References Author Index Subject Index

Journal ArticleDOI
TL;DR: In this article, the authors consider the interface between hierarchical Bayes and frequentist shrinkage estimation, and compare the hierarchical and minimax estimators for multivariate normal mean estimators.
Abstract: In shrinkage estimation of a multivariate normal mean, the two dominant approaches to construction of estimators have been the hierarchical or empirical Bayes approach and the minimax approach. The first has been most extensively used in practice, because of its greater flexibility in adapting to varying situations, while the second has seen the most extensive theoretical development. In this paper we consider several topics on the interface of these approaches, concentrating, in particular, on the interface between hierarchical Bayes and frequentist shrinkage estimation. The hierarchical Bayes setup considered is quite general, allowing (and encouraging) utilization of subjective second stage prior distributions to represent knowledge about the actual location of the normal means. (The first stage of the prior is used, as usual, to model suspected relationships among the means.) We begin by providing convenient representations for the hierarchical Bayes estimators to be considered, as well as formulas for their associated posterior covariance matrices and unbiased estimators of matrical mean square error; these are typically proposed by Bayesians and frequentists, respectively, as possible "error matrices" for use in evaluating the accuracy of the estimators. These two measures of accuracy are extensively compared in a special case, to highlight some general features of their differences. Risks and various estimated risks or losses (with respect to quadratic loss) of the hierarchical Bayes estimators are also considered. Some rather surprising minimax results are established (such as one in which minimaxity holds for any subjective second stage prior of the mean), and the various risks and estimated risks are extensively compared. Finally, a conceptually trivial (but often calculationally difficult) method of verifying minimaxity is illustrated, based on numerical maximization of the unbiased estimator of risk (using certain convenient calculational formulas for hierarchical Bayes estimators), and is applied to an illustrative example.

Journal ArticleDOI
TL;DR: In this article, a hierarchical Bayesian linear model is used to predict outstanding claims on a porfolio of general insurance policies, which can be expressed in the form of a linear model.
Abstract: The subject of predicting outstanding claims on a porfolio of general insurance policies is approached via the theory of hierarchical Bayesian linear models. This is particularly appropriate since the chain ladder technique can be expressed in the form of a linear model. The statistical methods which are applied allow the practitioner to use different modelling assumptions from those implied by a classical formulation, and to arrive at forecasts which have a greater degree of inherent stability. The results can also be used for other linear models. By using a statistical structure, a sound approach to the chain ladder technique can be derived. The Bayesian results allow the input of collateral information in a formal manner. Empirical Bayes results are derived which can be interpreted as credibility estimates. The statistical assumptions which are made in the modelling procedure are clearly set out and can be tested by the practitioner. The results based on the statistical theory form one part of the reserving procedure, and should be followed by expert interpretation and analysis. An illustration of the use of Bayesian and empirical Bayes estimation methods is given.

Journal ArticleDOI
01 May 1990
TL;DR: A Bayesian detection model is formulated for a distributed system of sensors, wherein each sensor provides the central processor with a detection probability rather than an observation vector or a detection decision.
Abstract: A Bayesian detection model is formulated for a distributed system of sensors, wherein each sensor provides the central processor with a detection probability rather than an observation vector or a detection decision. Sufficiency relations are developed for comparing alternative sensor systems in terms of their likelihood functions. The sufficiency relations, characteristic Bayes risks, and receiver operating characteristics provide equivalent criteria for establishing a dominance order of sensor systems. Parametric likelihood functions drawn from the beta family of densities are presented, and analytic solutions for the decision model and dominance conditions are derived. The theory is illustrated with numerical examples highlighting the behavior of the model and benefits of fusing the detection probabilities. >

Book ChapterDOI
01 Jan 1990
TL;DR: An empirical evaluation of three inference methods for uncertain reasoning is presented in the context of Pathfinder, a large expert system for the diagnosis of lymph-node pathology.
Abstract: In this paper, an empirical evaluation of three inference methods for uncertain reasoning is presented in the context of Pathfinder, a large expert system for the diagnosis of lymph-node pathology. The inference procedures evaluated are (1) Bayes' theorem, assuming evidence is conditionally independent given each hypothesis; (2) odds–likelihood updating, assuming evidence is conditionally independent given each hypothesis and given the negation of each hypothesis; and (3) a inference method related to the Dempster–Shafer theory of belief. Both expert-rating and decision-theoretic metrics are used to compare the diagnostic accuracy of the inference methods.

Journal ArticleDOI
Attila Csenki1
TL;DR: The concepts of Bayes prediction analysis are used to obtain predictive distributions of the next time to failure of software when its past failure behavior is known and can show an improved predictive performance for some data sets even when compared with some more sophisticated software-reliability models.
Abstract: The concepts of Bayes prediction analysis are used to obtain predictive distributions of the next time to failure of software when its past failure behavior is known. The technique is applied to the Jelinski-Moranda software-reliability model, which in turn can show an improved predictive performance for some data sets even when compared with some more sophisticated software-reliability models. A Bayes software-reliability model is presented which can be applied to obtain the next time to failure PDF (probability distribution function) and CDF (cumulative distribution function) for all testing protocols. The number of initial faults and the per-fault failure rate are assumed to be s-independent and Poisson and gamma distributed respectively. For certain data sets, the technique yields better predictions than some alternative methods if the frequential likelihood and U-plot criteria are adopted. >

Journal ArticleDOI
TL;DR: In this article, the authors derived a constructive result about the approximation of a Bayes estimator by a mixture of these primitive estimators with respect to a uniform distribution on spheres.