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Showing papers on "Bayesian probability published in 1969"


Journal ArticleDOI
TL;DR: In this paper, the Weibull process with an unknown scale parameter was examined as a model for Bayesian decision making, and the analysis was extended by treating both the shape and scale parameters as unknown.
Abstract: Previously, the Weibull process with an unknown scale parameter was examined as a model for Bayesian decision making. The analysis is extended by treating both the shape and scale parameters as unknown. It is not possible to find a family of continuous joint prior distributions on the two parameters that is closed under sampling, so a family of prior distributions is used that places continuous distributions on the scale parameter and discrete distributions on the shape parameter. Prior and posterior analyses are examined and seen to be no more difficult than for the case in which only the scale parameter is treated as unknown, but preposterior analysis and determination of optimal sampling plans are considerably more complicated in this case. To illustrate the use of the present model, an example is presented in which it is necessary to make probability statements about the mean life and reliability of a long-life component both before and after life testing.

177 citations


Journal ArticleDOI

161 citations




Journal ArticleDOI
01 May 1969

55 citations



Journal ArticleDOI
TL;DR: In this article, the authors discuss the application of Bayes's theorem to those cases where the true state of the world is not known with certainty, and propose an algorithm that relaxes the requirement that the true data state be known with absolute certainty by postulating a true but unobservable elementary event, which gives rise to posterior probabilities which reflect the uncertainty of the data.

32 citations



Journal ArticleDOI
01 Oct 1969-Synthese
TL;DR: A comparison of Neyman's theory of interval estimation with the corresponding subjective Bayesian theory of credible intervals as mentioned in this paper shows that the Bayesian approach to the estimation of statistical parameters allows experimental procedures which, from the orthodox objective viewpoint, are clearly biased and clearly inadmissible.
Abstract: A comparison of Neyman's theory of interval estimation with the corresponding subjective Bayesian theory of ‘credible intervals’ shows that the Bayesian approach to the estimation of statistical parameters allows experimental procedures which, from the orthodox objective viewpoint, are clearly biased and clearly inadmissible. This demonstrated methodological difference focuses attention on the key difference in the two general theories, namely, that the orthodox theory is supposed to provide a known average frequency of successful estimates, whereas the Bayesian account provides only a coherent ordering of degrees of belief and a subsequent maximization of subjective expected utilities. To rebut the charge of allowing biased procedures, the Bayesian must attack the foundations of orthodox, objectivist methods. Two apparently popular avenues of attack are briefly considered and found wanting. The first is that orthodox methods fail to apply to the single case. The second is that orthodox methods are subject to a typical Humean regress. The conclusion is that orthodox objectivist methods remain viable in the face of the subjective Bayesian alternative — at least with respect to the problem of statistical estimation.

16 citations


Journal ArticleDOI
TL;DR: An empiric Bayes estimation procedure of the optimal Bayesian stock level is presented, and a dynamic programming method for obtaining the general Bayes sequential procedure is outlined.
Abstract: Bayesian determination of optimal stock levels is studied for the case of Poisson distribution of the demand variable, and prior gamma distribution of the expected demand. Bayes sequential procedure is derived, assuming that stock level can be adjusted at the beginning of each period so that a shortage can be immediately replenished and an overstock can be corrected. The Bayes sequential procedure is more difficult to obtain if this assumption is removed. A dynamic programming method for obtaining the general Bayes sequential procedure is outlined. Finally, an empiric Bayes estimation procedure of the optimal Bayesian stock level is presented.

15 citations


Journal ArticleDOI
TL;DR: In this paper, a Bayesian method of estimating ability is described that generalizes Kelley's regression estimate of true score based on a weighted average of observed score and the presumed known population mean true score.
Abstract: A Bayesian method of estimating ability is described that generalizes Kelley's regression estimate of true score based on a weighted average of observed score and the presumed known population mean true score. This Bayesian method uses all available data to estimate the population mean true score and incorporates this information into the estimate of each individual ability parameter. This method is superior to methods that do not include estimates of population mean values. A normal law error model is used to illustrate the method. Modal estimates of an intuitively attractive form from the joint posterior distribution of the ability parameters are identified as a solution to a set of nonlinear equations. The method is then illustrated in the context of a Poisson process model having both person-ability and item-difficulty parameters. The posterior marginal distributions for individual sets of ability and difficulty parameters are given and modal estimates are described. Modal estimates from the joint conditional distribution of the ability parameters given the item difficulty parameters are also given. The desirability of incorporating prior information is discussed and a method of accomplishing this is described. The application of these Bayesian methods to central prediction and sequential testing are discussed. It is suggested that the Bayesian method is uniquely appropriate to each of these problems. Finally, these ideas are also shown to be relevant to the problems of selecting predictor variables and multiple comparisons.

Journal ArticleDOI
TL;DR: Some Bayes estimates are obtained of an index of performance of a system that alternates between two states, up or down, in accordance with a Markov process, which measures the probability that the system will be up when needed.
Abstract: Some Bayes estimates are obtained of an index of performance of a system that alternates between two states, up or down, in accordance with a Markov process. The index considered is long-run availability, which measures the probability that the system will be up when needed. For the purpose of obtaining these estimates, two types of observations are considered: those that reveal only the state of system at isolated time points and those that continuously record the duration of the up and down times of the system.


Journal ArticleDOI
TL;DR: In this paper, it is suggested that estimates of regression parameters be regressed toward a mean value and that the resulting attenuated estimate of the multiple correlation be adopted, derived from a general Bayesian normal law analysis.
Abstract: Some work of Lindley on a Bayesian structural model is shown to be relevant to the problem of the selection of predictor variables. It is suggested that estimates of regression parameters be regressed toward a mean value and that the resulting attenuated estimate of the multiple correlation be adopted. These regressed estimates are derived from a general Bayesian normal law analysis.



Journal ArticleDOI
TL;DR: In this article, a Bayesian analysis of a model for misreadings is undertaken, which is essentially the Poisson model with multiplicative parameter structure first proposed and experimentally justified by Rasch.
Abstract: A Bayesian analysis of a model for misreadings is undertaken. The model is essentially the Poisson model with multiplicative parameter structure first proposed and experimentally justified by Rasch. Conditional and unconditional posterior densities, predictive densities and reliabilities are presented. Some results for point and interval estimation and examination design are derived from the basic results.