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Showing papers on "Bayesian probability published in 1972"



Journal ArticleDOI
TL;DR: In this paper, a Bayesian selection rule based on non-informative priors for the parameters, and a multiple decision solution, that maximizes the weighted sum of the probabilities of correct selection, are presented; both of these yield a decision rule of the same form.
Abstract: In our approach to adaptive inference, selection of the true underlying family of distributions is needed. A Bayesian selection rule, which is based on non-informative priors for the parameters, and a multiple decision solution, that maximizes the weighted sum of the probabilities of correct selection, are presented; both of these yield a decision rule of the same form. A modified maximum likelihood solution is also given and is used to construct an adaptive estimate of the location of a symmetric distribution. By a Monte Carlo study, it is found that this adaptive estimator compares most favorably to more standard estimators.

37 citations



Journal ArticleDOI
TL;DR: A multiparameter Bayesian analysis which requires multiple integration of the (multivariate) posterior of the parameters of the time-to-failure distribution to obtain the expected loss resulting from a particular choice of burn-in time and item replacement age is described.
Abstract: An important problem facing a manufacturer is the determination of the amount of time to burn-in items (in order to eliminate early failures) and the age at which to replace items (to avoid failures due to wearout) This problem becomes difficult to solve if the time-to-failure distribution of an item is unknown and must be estimated from test and operational data This paper describes a method of statistical data analysis which is readily applied to the solution of this decision problem under a realistic but general loss (or gain) function The method is a multiparameter Bayesian analysis which requires multiple integration of the (multivariate) posterior of the parameters of the time-to-failure distribution to obtain the expected loss (or gain) resulting from a particular choice of burn-in time and item replacement age This integration is performed by a Monte Carlo Procedure using importance sampling An example demonstrates the flexibility of this method of analysis The data are a mixture of ``point'' and truncated data, which often create difficulties when using conventional methods of decision analysis In addition, since the method permits up to ten parameters for the family of time-to-failure distributions, a ``bathtub'' hazard rate function is used to generate the data for the example The results are presented in the form of Bayesian confidence intervals for the true hazard rate function and a presentation of the expected loss as a function of burn-in time and age at replacement

32 citations


Journal ArticleDOI
TL;DR: In this paper, four Ph.D.s in Biology made admissions judgments on 528 hypothetical applicants to their graduate program and compared Bayes' theorem and multiple regression analysis as descriptive models of the judges.

27 citations


Journal ArticleDOI
TL;DR: Variance-response models as discussed by the authors are a generalization of the traditional model of statistics that distinguishes response values only by their likelihood functions, and recognize the restricted identification of response values effectively precludes the need for any theory of sufficient statistics.
Abstract: The traditional model of statistics is examined in Section 1. The model, as such, distinguishes response values only by their likelihood functions. Recognition of this restricted identification effectively precludes the need for any theory of sufficient statistics. A probability space that has an attached observer-processor-reporter (OPR) mechanism is examined in Section 2 as a means of assessing the nature of reported information; such information may or may not be observational in character depending on properties of the OPR mechanism. A variation-response model is a probability space and a class of random variables: the probability space describes the sources of variation in the system under investigation and the class of random variables describes the possible presentations of this variation in the response of the system. Section 3 examines how realized values on the probability space are distinguished or identified by the model; Section 4 considers how distributions on the probability space are identified by response variable data. In Sections 5, 6, 7 the essentials of three contemporary approaches to inference are presented and each is accompanied by criticisms that proponents of the other methods might make. The key to these criticisms lies primarily in whether hypothetical information is added to or substantiated information is omitted from the assembled information concerning the system under investigation. In certain contexts the bayesian approach makes an arbitrary but typically consistent choice of input to its analyses; it is not the input suggested by standard invariance analysis. In those cases where a variation-response model is appropriate, the use of this more embracing model presents theoretical support for the bayesian choice (Section 8); but of course with the more embracing model the bayesian premises are not needed to obtain the usual bayesian result. An example in Section 9 illustrates the theoretical simplicity of classical probabilities for certain unknowns other than realized values on a probability space.

19 citations


Journal ArticleDOI
TL;DR: In this paper, a Bayesian approach to the problem of comparing two multinomial distributions with k categories and a natural ordering of the categories is developed, and a convenient algorithm for computing the posterior probability that one distribution is stochastically larger than the other.
Abstract: A Bayesian approach to the problem of comparing two multinomial distributions with k categories and a natural ordering of the categories is developed. Assuming independent Dirichlet priors, we obtain a convenient algorithm for computing the posterior probability that one distribution is stochastically larger than the other. The important special case k = 2 reduces to computing the posterior probability that one binomial proportion exceeds another. We consider the more general problem of finding the posterior distribution of the ratio of two such proportions.

18 citations


Journal ArticleDOI
TL;DR: In this paper, five mathematical models whose components were subjective probability judgments were used to predict the job choices of persons seeking professional employment in the public schools, including heuristics and Bayesian models.

16 citations


Journal ArticleDOI
TL;DR: In classifying 861 M MPI profiles as either neurotic or psychotic, two variants of Bayesian analysis techniques had cross-validated hit rates as high as comparable analyses which correctly handled the interrelationships among the MMPI scales.

15 citations


Journal ArticleDOI
TL;DR: The weighted-average method was more frequent and its use increased in the presence of high variance among experts' judgments, while the cognitive abilities, tolerances for ambiguity, and risk-taking propensities of the subjects were not related to their choices of a consensus distribution.

15 citations


Journal ArticleDOI
Gary L. Crellin1
TL;DR: The application of the Bayesian approach depends on how the reliability decision is conceptualized as mentioned in this paper, and the difficulties of interpreting the probability function and assigning prior distributions restrict the presentation of a unified philosophy.
Abstract: The rudiments of Bayesian philosophy are introduced, and the mathematics of its application are surveyed; there is no uniformity of thought concerning either. The more extreme Bayesian philosophy, which allows subjective probabilities, is a means of plausible reasoning, or of making inferences, through inductive logic. Because inferences concerning reliability concepts are important in the decision process, this philosophy has a place in the reliability field. The difficulties of interpreting the probability function and of assigning prior distributions restrict the presentation of a unified philosophy. Thus, only techniques for describing prior probabilities under various circumstances can be given. Application of the Bayesian approach depends on how the reliability decision is conceptualized.


Journal ArticleDOI
TL;DR: The objective is to show that there is some substance to the classical statistician's opposition to Bayesian inference and that the issues are pertinent and meaningful to the reliability engineer.
Abstract: There appear to be two important developing trends in reliability. One is away from the use of statistics by reliability engineers; the other is toward increased use of Bayesian techniques. One source of the latter may be disillusionment with what is regarded as classical statistics; however, one source of the former may be dissatisfaction with what has been called Bayesian statistics. The purpose of this paper is to discuss these trends and to present a personal view of the issues between classical and Bayesian statistics. The objective is to show that there is some substance to the classical statistician's opposition to Bayesian inference and that the issues are pertinent and meaningful to the reliability engineer.

Journal ArticleDOI
TL;DR: In this article, a possible sequential life test incorporating prior information, applied to the simplest situation of components with exponentially distributed lifetimes, tested individually, is presented, and the results for conventional sequential lifetests are used as a rough yardstick against which to measure the properties of the Bayesian lifetest discussed in the paper.
Abstract: Detailed examination of the lifetimes of components and assemblies by means of the usual sequential probability ratio tests is often not feasible because of the prohibitive time or cost involved in such an examination. Situations commonly arise, however, where extraneous information is available (perhaps in the form of experience of similar situations or concerning the reputation of the supplier of the components or assemblies) which reflects on the current situation. Such information might be incorporated in a Bayesian analysis, but little work seems to have been done in this area. This paper presents a possible sequential life test incorporating prior information, applied to the simplest situation of components with exponentially distributed lifetimes, tested individually. The results for conventional sequential lifetests are used as a rough yardstick against which to measure the properties of the Bayesian lifetest discussed in the paper.

Book ChapterDOI
TL;DR: In this paper, a Bayesian approach to reliability demonstration testing is described and differences between the Bayesian viewpoint and the commonly employed classical approach are highlighted, and a procedure for selecting a specific inverted gamma probability density to characterize the prior distribution of the MTBF of electronic hardware is developed.
Abstract: A Bayesian approach to reliability demonstration testing is described and differences between the Bayesian viewpoint and the commonly employed classical approach are highlighted. A procedure for selecting a specific inverted gamma probability density to characterize the prior distribution of the MTBF of electronic hardware is developed and a table of Bayesian demonstration plans for a practical range of input parameters is provided. In addition, procedures for implementation of the plans and two illustrative examples are given. Finally, two commonly employed classical plans are compared to a Bayesian plan illustrating the efficiency of the latter in terms of demonstration test time requirements.

Journal ArticleDOI
TL;DR: In this article, a Bayesian decision model is constructed around a conjugate probability density function for the Weibull hazard rate, and prior, posterior, and preposterior analysis of this decision model are discussed.
Abstract: A two-parameter Weibull distribution is assumed to be the appropriate statistical life-model of an engineering device. The hazard rate of this device is the relevant quantity in terms of which statistical decisions are to be made. A Bayesian decision model is constructed around a conjugate probability density function for the Weibull hazard rate. Prior, posterior, and preposterior analysis of this decision model are discussed. The results indicate the rational decision before and after sampling, and permit the optimization of sequential single-item sampling schemes. Such schemes are of particular importance in the reliability testing of high-cost equipment.

01 Jan 1972
TL;DR: A model for visual space perception is proposed that contains desirable features in the theories of Gibson and Brunswik and it is compared with signal detection theory models.
Abstract: A model for visual space perception is proposed that contains desirable features in the theories of Gibson and Brunswik. This model is a Bayesian processor of proximal stimuli which contains three important elements: an internal model of the Markov process describing the knowledge of the distal world, the a priori distribution of the state of the Markov process, and an internal model relating state to proximal stimuli. The universality of the model is discussed and it is compared with signal detection theory models. Experimental results of Kinchla are used as a special case.

Journal ArticleDOI
01 Nov 1972
TL;DR: It is shown that the maximization of the mean Bhattacharyya distance minimizes an upper bound on the error probability.
Abstract: A relationship between the probability of misrecognition and the expected Bhattacharyya distance is examined, and it is shown that the maximization of the mean Bhattacharyya distance minimizes an upper bound on the error probability.

01 Jan 1972
TL;DR: This paper is concerned with the problem of testing H: 0 .
Abstract: In 1959 Chernoff [7] initiated the study of the asymptotic theory of sequential Bayes tests as the cost of observation tends to zero. He dealt with the case of a finite parameter space. The definitive generalization of the line of attack initiated in that paper was given by Kiefer and Sacks in [13]. Their work as well as that of Chernoff, the intervening papers of Albert [1], Bessler [3], and Schwarz [19], and the subsequent work of the authors [4] used implicitly or expli'-itly the theory of large deviations and applied only to situations where hypothesis and alternative were separated or at least an indifference region was present. In the meantime in 1961 Chernoff [8] began to study the problem of testing H: 0 . 0 versus K: 0 > 0 on the basis of observation of a Wiener process with drift 0 per unit time as an approximation to the discrete time normal observations problem. Having made the striking observation that study of the asymptotic behavior of the Bayes procedures for any normal prior was in this case equivalent to the study of the Bayes procedure with Lebesgue measure as prior and unit cost of observation, he reduced this problem for suitable loss functions to the solution of a free boundary problem for the heat equation. In subsequent work ([2], [9J, [10] and [16]) the nature of this solution was investigated by Chernoff and others. In this paper we are concerned with the problem of testing H: 0 . 0 versus K: 0 > 0 by sampling sequentially from a member of one parameter exponential (Koopman-Darmois) family of distributions (see equation (3.1)) at cost c per observation. We will assume the simple zero-one loss structure in which an error in decision costs one unit while being right costs nothing.

Book ChapterDOI
TL;DR: In this paper, a model for conditioning a forecast during periods when a time series exhibits atypical behavior is proposed, which is based on a Bayesian approach, requiring the assessment of a probability distribution to the proportionate change believed to result from the impending abnormal situation.
Abstract: A model is offered for conditioning a forecast during periods when a time series exhibits atypical behavior. The model is based on a Bayesian approach, requiring the assessment of a probability distribution to the proportionate change believed to result from the impending atypical situation.

Journal ArticleDOI
TL;DR: In this paper, the authors apply Bayes' Equation to hypotheses concerning reliability and obtain posterior probabilities for the reliability hypotheses, which are consistent with the prior beliefs and the available test results.
Abstract: The rudiments of applying Bayes' Equation to hypotheses concerning reliability are introduced in a simple manner. The application is a means of obtaining posterior probabilities, for the reliability hypotheses, which are consistent with the prior beliefs and the available test results. The posterior distributions, from which decision theory could formally arrive at optimal estimates, are greatly dependent on the prior distributions. Thus, the discussion centers about the desired properties of a prior and its effects on the posterior for various data situations. Formulations for both continuous-conjugate and discrete representations of the prior beliefs are discussed and contrasted. The use of discrete priors offers many advantages over the use of continuous-conjugate priors.




Journal ArticleDOI
Myron A. Wilson1
TL;DR: In this article, two methods of Bayesian reliability measurement are described, one based on handbook data and the other based on a discrete probability distribution with nonuniform cell widths.
Abstract: Experience with two methods of Bayesian reliability measurement is described. An aerospace subsystem was evaluated assuming continuous gamma-distributed component failure rates. Priors were developed by conventional reliability prediction methods based on handbook data. The ``strength'' of the prior was expressed in terms of variance about a predicted mean. Comparative evaluation was also made by a classical technique during a test program extending over 10 months. The Bayesian method was preferred though problems inherent in the method were apparent. More recently, a complex marine system was evaluated over a one-year period using a Bayesian formulation in which the failure rate is described by a discrete probability distribution with nonuniform cell widths. This technique avoids some of the operational problems of continuous formulations. Experience with Bayesian methods leaves little doubt of their utility as evaluation tools. The philosophical problems, however, remain as intransigent as ever.

01 Dec 1972
TL;DR: In this paper, the authors examined the problem of finding confidence intervals for the reliability function R(t) of two independent exponential series systems via a sampling theory approach from the Bayesian point of view.
Abstract: : Leiberman and Ross have recently given a solution to the problem of finding confidence intervals for the reliability function R(t) of two independent exponential series systems via a sampling theory approach. In the paper the authors examine the problem from the Bayesian point of view. The authors deal with both the uncensored and the censored cases, and illustrate the work with examples. (Author)

ReportDOI
01 Sep 1972
TL;DR: Research on the decomposition of utility estimates showed that the model of a weighted linear average is relatively insensitive to nonadditive combination rules but highly sensitive to the nonlinearity of utility functions.
Abstract: : The report describes research intended to develop procedures for eliciting judgments of probability and utility that could be employed efficiently in a decision theoretic analysis. Experiments on probability estimation investigated the reinforcing effects of a proper scoring rule upon probability estimates; the use of Bayesian procedures to revise probability estimates; and, the relative merits of probabilities and odds as response modes. Research on the decomposition of utility estimates showed that the model of a weighted linear average is relatively insensitive to nonadditive combination rules but highly sensitive to the nonlinearity of utility functions; when utilities are estimated it makes relatively little difference whether or not the judgments are decomposed; and, the procedures of decomposing utility estimates were feasible for use on a real world problem where water-quality engineers rather than college students served as subjects.


Journal ArticleDOI
TL;DR: In this article, the role of sufficient statistics in decision making for an arbitrary optimality criterion and particularly for the Bayesian criterion is analyzed for a simple inventory problem of the slow-mover type with the special feature that the probability distribution of the demand interarrival times is unknown.
Abstract: Summary In this paper some tools will be developed for analysing a simple inventory problem of the slow-mover type with the special feature that the probability distribution of the demand interarrival times is unknown. The role of sufficient statistics in decision making is analysed for an arbitrary optimality criterion and particularly for the Bayesian criterion. A computational approach for the case of a Bayesian criterion is presented together with some numerical results.