Showing papers on "Bayes' theorem published in 1976"
330 citations
01 Jan 1976
TL;DR: For many years, statistics textbooks have followed this canonical procedure: (1) the reader is warned not to use the discredited methods of Bayes and Laplace, (2) an orthodox method is extolled as superior and applied to a few simple problems, (3) the corresponding Bayesian solutions are not worked out or described in any way as discussed by the authors.
Abstract: For many years, statistics textbooks have followed this ‘canonical’ procedure: (1) the reader is warned not to use the discredited methods of Bayes and Laplace, (2) an orthodox method is extolled as superior and applied to a few simple problems, (3) the corresponding Bayesian solutions are not worked out or described in any way. The net result is that no evidence whatsoever is offered to substantiate the claim of superiority of the orthodox method.
281 citations
TL;DR: This article showed that the conjunctive probability of events A and B (PAB) and disjunctive probability (PA∪B) can be predicted from information bearing upon the likelihood of A alone and B alone (PA and PB) or from information about B given A (P B A ) and A given B given B (P A B ), respectively.
113 citations
TL;DR: A consistent estimator is discussed which is computationally more efficient than estimators based on Parzen's estimation and its relation between the distance of a sample from the decision boundary and its contribution to the error is derived.
Abstract: The L^{ \alpha} -distance between posterior density functions (PDF's) is proposed as a separability measure to replace the probability of error as a criterion for feature extraction in pattern recognition. Upper and lower bounds on Bayes error are derived for \alpha > 0 . If \alpha = 1 , the lower and upper bounds coincide; an increase (or decrease) in \alpha loosens these bounds. For \alpha = 2 , the upper bound equals the best commonly used bound and is equal to the asymptotic probability of error of the first nearest neighbor classifier. The case when \alpha = 1 is used for estimation of the probability of error in different problem situations, and a comparison is made with other methods. It is shown how unclassified samples may also be used to improve the variance of the estimated error. For the family of exponential probability density functions (pdf's), the relation between the distance of a sample from the decision boundary and its contribution to the error is derived. In the nonparametric case, a consistent estimator is discussed which is computationally more efficient than estimators based on Parzen's estimation. A set of computer simulation experiments are reported to demonstrate the statistical advantages of the separability measure with \alpha = 1 when used in an error estimation scheme.
98 citations
96 citations
45 citations
TL;DR: In this article, a sequence of empirical Bayes estimators is defined for estimating a distribution function and the sequence is shown to be asymptotically optimal relative to a Ferguson Dirichlet process prior.
Abstract: A sequence of empirical Bayes estimators is defined for estimating a distribution function. The sequence is shown to be asymptotically optimal relative to a Ferguson Dirichlet process prior. Exact risk expressions are derived and the rate, at which the overall expected loss approaches the minimum Bayes risk, is exhibited. The empirical Bayes approach, based on the Dirichlet process, is also applied to the problem of estimating the mean of a distribution.
34 citations
TL;DR: A quickly implementable table look-up rule based on Ashby’s representation of sets in his constraint analysis that is comparable to the optimum Bayes rule on simulated Gaussian Data.
Abstract: The table look-up rule problem can be described by the question: what is a good way for the table to represent the decision regions in the N-dimensional measurement space. This paper describes a quickly implementable table look-up rule based on Ashby’s representation of sets in his constraint analysis. A decision region for category c in the N-dimensional measurement space is considered to be the intersection of the inverse projections of the decision regions determined for category c by Bayes rules in smaller dimensional projection spaces. Error bounds for this composite decision rule are derived: any entry in the confusion matrix for the composite decision rule is bounded above by the minimum of that entry taken over all the confusion matrices of the Bayes decision rules in the smaller dimensional projection spaces. On simulated Gaussian Data, probability of error with the table look-up rule is comparable to the optimum Bayes rule.
33 citations
TL;DR: The posterior distribution function is developed in terms of an expansion in shifted Chebyshev polynomials of the second kind whose convergence properties and numerical evaluation are well suited for practical applications.
Abstract: We present a method for computing Bayes confidence limits for the reliability of an arbitrary series-parallel system consisting of failure independent modules. Each module has an internal structure of one or more failure independent devices in series, and each device consists of one operating component with identical nonoperating standby components. Each component is assumed to have exponential distribution of life with unknown failure rates that must be estimated from test data. We outline a systematic procedure to compute the posterior distribution of system reliability, from which exact confidence limits can be obtained. The posterior distribution function is developed in terms of an expansion in shifted Chebyshev polynomials of the second kind whose convergence properties and numerical evaluation are well suited for practical applications.
22 citations
TL;DR: In this paper, a Bayesian approach with vague prior distributions is taken for parameter estimation in normal regression when some of the observations are missing, and joint estimators of the parameters are obtained as the joint mode of the posterior distribution.
Abstract: We are concerned with the problem of parameter estimation in normal regression when some of the observations are missing. A Bayesian approach with vague prior distributions is taken. No assumption is made about the independent variables for which no observations are missing, but the missing components are assumed to be normally distributed with a mean that can depend on the other variables. Joint estimators of the parameters are obtained as the joint mode of the posterior distribution.
22 citations
TL;DR: A mixed-data model is presented and shown to represent a general class of models of which non-mixed-data models are special cases and can take advantage of a potential reduction in the number of parameters that must be estimated.
Journal Article•
TL;DR: A retrospective study to assess the feasibility of computer-assisted prognostication by discriminant analysis and the Bayesian classification procedure based on clinical information collected on patients with acute myocardial infarction found not all of the 44 variables used for analysis were necessary to reach the same level of predictive accuracy.
Abstract: A retrospective study was carried out to assess the feasibility of computer-assisted prognostication by discriminant analysis and the Bayesian classification procedure based on clinical information collected on patients with acute myocardial infarction. The overall accuracy was 94.2% in predicting hospital death but the prediction of late death after discharge was less accurate. It was found that not all of the 44 variables used for analysis were necessary to reach the same level of predictive accuracy--16 to 20 variables would result in almost the identical prediction. The Bayesian classification procedure was applied to estimate probabilities of individual patients belonging to the different prognostic categories.
TL;DR: In this article, a smoothed Bayes control procedure was developed for controlling the output of a production process when the quality characteristic is continuous with a linear shift in its basic level.
Abstract: A smoothed Bayes control procedure has been developed for controlling the output of a production process when the quality characteristic is continuous with a linear shift in its basic level. The procedure uses Bayesian estimation with exponential smooth..
TL;DR: In this article, a Bayes decision theoretic approach consists in seeking a decision that minimizes the Bayes risk function, and also evaluates the decision taken and compares the expected cost of delaying the construction to the worth of additional information resulting from such a delay.
Abstract: Economic and sociopolitical aspects of land reclamation in areas that necessitate drainage are combined with technical problems to yield a set of possible decisions, among which an optimum design is chosen. The loss or objective function is the sum of the expected damage caused by the submersion of given crops resulting from extreme rainfall events, and the initial cost of the reclamation. When the parameter uncertainty in the probability distribution function of extreme events is taken into account in the loss function, a Bayes risk function is obtained. The Bayes decision theoretic (BDT) approach consists in seeking a decision that minimizes the Bayes risk function. The BDT also evaluates the decision taken and compares the expected cost of delaying the construction to the worth of additional information resulting from such a delay. The practical example of an intensive agricultural system with different type soils and crops illustrates the methodology. Crop loss functions and probability distributions of events are assumed on the basis of empirical observations.
TL;DR: In this paper, two different kinds of evaluation criteria were used: the first refers to the goodness of individual assessments, defined as the score of a proper scoring rule, and the second criterion referred to the information inherent in the assessments.
TL;DR: Representations for y in terms of its innovations and following a Girsanov-type measure transformation are derived to develop a measure form of Bayes' rule that provides a convenient tool for the study of estimation and decision problems arising in a variety of applications including communication and control.
Abstract: The observation process y considered is an additive composition of continuous and discontinuous components. The additive Gaussian, point, and jump process models, treated separately in the past, are all included here simultaneously. Representations for y in terms of its innovations and following a Girsanov-type measure transformation are derived. These are then used to develop a measure form of Bayes' rule that provides a convenient tool for the study of estimation and decision problems arising in a variety of applications including communication and control.
TL;DR: In this paper, a sequence of empirical Bayes estimators is defined for estimating, in a two-sample problem, the probability that X ≥ Y. The sequence is shown to be asymptotically optimal relative to a Ferguson Dirichlet process prior.
Abstract: A sequence of empirical Bayes estimators is defined for estimating, in a two-sample problem, the probability that X ≥ Y. The sequence is shown to be asymptotically optimal relative to a Ferguson Dirichlet process prior.
TL;DR: Given a situation where two persons know the evaluations they receive at each step of a series of performances, a process model is proposed to account for changes in their expectations as a function of the given evaluations.
Abstract: This paper focuses on the effect of performance evaluations on expectations about future performances: given a situation where two persons know the evaluations they receive at each step of a series of performances, a process model is proposed to account for changes in their expectations as a function of the given evaluations. The approach consists of looking at the problem as a special case of information processing, and of using Bayes' theorem for the construction of the model. Thus, instead of asking about the effect on a person's expectations of a certain evaluation, we ask how a piece of evidence affects the subjective probability attached to a given hypothesis. A few illustrative runs are presented and discussed.
01 Jan 1976
TL;DR: In this article, a robustness study of Bayes estimates of the failure intensity is carried out, and implications for the robustness of fitted priors are discussed, for the gamma, uniform, and inverted uniform priors which are fitted to actual failure data.
Abstract: : The number of times a piece of equipment fails in a fixed time T is assumed to have the Poisson sampling density. The parameter of interest is the failure intensity, the reciprocal of the mean time between failures. The loss function of interest is the squared error loss function. A robustness study of Bayes estimates of the failure intensity is carried out. The Bayes estimate of the failure intensity for a general prior is expressed in terms of the marginal density of the observed number of failures, and implications for the robustness of Bayes estimates for fitted priors is discussed. Numerical examples are given for the gamma, uniform, and inverted uniform priors which are fitted to actual failure data. In addition to comparisons among the Bayes estimates of failure intensity for fitted priors, the Bayes estimate under the fitted gamma prior is compared to the corresponding estimates under other, subjectively chosen, gamma priors.
TL;DR: In this paper, the optimality of the set S(x) = {θ0: decide θ = θ0] (usually an interval) is investigated for the case in which the testing rules are Bayes, and the main result is that S (x) is minimum Bayes risk with respect to a loss function which is the integral of the loss functions for testing.
Abstract: Given a family of rules for the three-decision testing problems of locating a scalar parameter θ relative to θ0, with decisions θ θ0, and given a sample x, the set S(x) = {θ0: decide θ = θ0) (usually an interval) is an intuitively appealing set estimate of θ. This paper considers the optimality of S(x) for the case in which the testing rules are Bayes. The main result is that when the losses for the testing problems have a particular structure, S(x) is minimum Bayes risk with respect to a loss function which is the integral (in a certain sense) of the loss functions for testing.
Journal Article•
01 Dec 1976
TL;DR: A quasi-Bayes approach is motivated, and discussed in detail for some versions of a two-class decision problem, which mimics closely the formal Bayes solution, whilst involving only a minimal amount of computation.
Abstract: Unsupervised Bayes sequential learning procedures for classification and estimation are often useless in practice because of computational constraints. In this paper, a quasi-Bayes approach is motivated, and discussed in detail for some versions of a two-class decision problem. The proposed procedure mimics closely the formal Bayes solution, whilst involving only a minimal amount of computation. Some numerical illustrations are provided, and the approach is compared with a number of other proposed learning procedures.
TL;DR: In this article, a probabilistic item response model and a latent trait estimation procedure were developed for adaptive testing. But this procedure relies on the item selection during the test administration, making use of information about the subject as he provides it and updating of ability estimates is handled naturally by repeated application of Bayes theorem.
Abstract: The use of response‐contingent testing to estimate mental abilities has relied hitherto on pre‐programming of items. An obvious extension is to perform the item selection during the test administration, making use of information about the subject as he provides it The updating of ability estimates is handled naturally by repeated application of Bayes theorem. Using a probablistic item response model, a latent trait estimation procedure is developed. Results of an actual implementation of adaptive testing are reported and applications of the procedure in a number of measurement settings are suggested.
TL;DR: In this paper, the Dirichiet process priors p of Ferguson have been used as the prior distributions on the space of distribution functions on the real line, and when p is unknown partially empirical Bayes procedures are asymptotically optimal with rates 0{jT1/2} and OCCmCn+l.
Abstract: This paper considers the generalised empirical Bayes two-action (testing) and multiple action problems concerning a distribution function The Dirichiet process priors p of Ferguson have been used as the prior distributions on the space of distribution functions on the real line. The two-action component problem is considered in detail and when p is unknown partially empirical Bayes procedures {6 } which are asymptotically optimal with rates 0{jT1/2) and OCCmCn+l))
01 Jan 1976
TL;DR: In this paper, the authors investigate the behavior of a two-parameter failure model subject to both behaving as random variables, and show that using the coefficient of variation as the prior in a Bayesian setting makes sense.
Abstract: : In the last five or six years there has been a considerable amount of rising interest in the Bayesian approach to reliability. To a practicing reliability scientist, such an approach would seem quite appealing because it provides a way for the formulation of a distributional form of the unknown parameters inherent within the failure model, based on prior convictions or information available to him. Evans and Drake have given an excellent account for the use of Bayesian theory in reliability. Furthermore, from a practical point of view, Feduccia states that employing the reliability prediction and its associated measure of uncertainty, the coefficient of variation as the 'prior' in a Bayesian setting makes sense. The aims of the present study is to investigate the behavior of a two parameter failure model subject to both behaving as random variables.
01 Jan 1976
TL;DR: This paper will present a procedure for Bayes estimation on the mean vector, covariance matrix, and a priori probability of the single-class classifier using labeled samples from the class of interest and unlabeled samples drawn from the mixture density function.
Abstract: Normal procedures used for designing a Bayes classifier to classify wheat as the major crop of interest require not only training samples of wheat but also those of nonwheat. Therefore, ground truth must be available for the class of interest plus all confusion classes. The single-class Bayes classifier classifies data into the class of interest or the class 'other' but requires training samples only from the class of interest. This paper will present a procedure for Bayes estimation on the mean vector, covariance matrix, and a priori probability of the single-class classifier using labeled samples from the class of interest and unlabeled samples drawn from the mixture density function.
TL;DR: In this paper, the authors exploit the results of Lloyd (1952) to obtain optimal linear estimators based on order statistics of location or/and scale parameter (s) of a continuous univariate data distribution.
Abstract: Summary An estimator which is a linear function of the observations and which minimises the expected square error within the class of linear estimators is called an “optimal linear” estimator. Such an estimator may also be regarded as a “linear Bayes” estimator in the spirit of Hartigan (1969). Optimal linear estimators of the unknown mean of a given data distribution have been described by various authors; corresponding “linear empirical Bayes” estimators have also been developed. The present paper exploits the results of Lloyd (1952) to obtain optimal linear estimators based on order statistics of location or/and scale parameter (s) of a continuous univariate data distribution. Related “linear empirical Bayes” estimators which can be applied in the absence of the exact knowledge of the optimal estimators are also developed. This approach allows one to extend the results to the case of censored samples.