Showing papers on "Bayes' theorem published in 1984"
TL;DR: In this paper, three types of Bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician, and they are discussed in detail.
Abstract: A common reaction among applied statisticians is that the Bayesian statistician's energies in an applied problem must be directed at the a priori elicitation of one model specification from which an optimal design and all inferences follow automatically by applying Bayes's theorem to calculate conditional distributions of unknowns given knowns. I feel, however, that the applied Bayesian statistician's tool-kit should be more extensive and include tools that may be usefully labeled frequency calculations. Three types of Bayesianly justifiable and relevant frequency calculations are presented using examples to convey their use for the applied statistician.
1,284 citations
TL;DR: In this article, the authors propose new Bayes and empirical Bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters, with weight on the data being approximately the square root of that for the posterior expectation.
Abstract: In standard Bayes and empirical Bayes component decision problems, estimating inidividual parameters is the primary goal. In multiple comparison problems and in comparisons of histograms of estimates, however, the primary goal is to produce parameter estimates that can be considered as an ensemble. For example, the histogram of estimates should be a good estimate of the histogram of parameters. Standard methods of estimating by the posterior expectation do minimize symmetric, componentwise losses such as squared error, but they produce ensembles of estimates with a sample variance smaller than the posterior expected sample variance for parameters. In this article we propose new Bayes and empirical Bayes estimates that minimize a distance function between the empirical cdf of the estimates and the true parameters. These estimators are weighted averages of the prior mean and the data, with weight on the data being approximately the square root of that for the posterior expectation. We give theoreti...
207 citations
TL;DR: The Bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel (discrete) or continuous random variables, and it gives the optimum Bayes estimate for the scene value at each pixel.
Abstract: A new image segmentation algorithm is presented, based on recursive Bayes smoothing of images modeled by Markov random fields and corrupted by independent additive noise. The Bayes smoothing algorithm yields the a posteriori distribution of the scene value at each pixel, given the total noisy image, in a recursive way. The a posteriori distribution together with a criterion of optimality then determine a Bayes estimate of the scene. The algorithm presented is an extension of a 1-D Bayes smoothing algorithm to 2-D and it gives the optimum Bayes estimate for the scene value at each pixel. Computational concerns in 2-D, however, necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm. In particular, the scene (noiseless image) is modeled as a Markov mesh random field, a special class of Markov random fields, and the Bayes smoothing algorithm is applied on overlapping strips (horizontal/vertical) of the image consisting of several rows (columns). It is assumed that the signal (scene values) vector sequence along the strip is a vector Markov chain. Since signal correlation in one of the dimensions is not fully used along the edges of the strip, estimates are generated only along the middle sections of the strips. The overlapping strips are chosen such that the union of the middle sections of the strips gives the whole image. The Bayes smoothing algorithm presented here is valid for scene random fields consisting of multilevel (discrete) or continuous random variables.
199 citations
Book•
01 Jan 1984
TL;DR: Thomas Bayes and statistical Inference Classical Statistical Inference Bayes' Theorem Bayesian Methods for a Proportion Bayesian methods for Other Parameters Prior Distributions Bayesian Difficulties Bayesian Strengths
Abstract: Thomas Bayes and Statistical Inference Classical Statistical Inference Bayes' Theorem Bayesian Methods for a Proportion Bayesian Methods for Other Parameters Prior Distributions Bayesian Difficulties Bayesian Strengths
164 citations
TL;DR: In this article, the problem of estimating, in a Bayesian framework and in the presence of additive Gaussian noise, a signal which is a step function is considered, and best linear estimates and Bayes estimates are derived, evaluated and compared.
Abstract: Consider the problem of estimating, in a Bayesian framework and in the presence of additive Gaussian noise, a signal which is a step function. Best linear estimates and Bayes estimates are derived, evaluated and compared. A characterization of the Bayes estimates is presented. This characterization has an intuitive interpretation and also provides a way to compute the Bayes estimates with a number of operations of the order of $T^3$ where $T$ is the fixed time span. An approximation to the Bayes estimates is proposed which reduces the total number of operations to the order of $T$. The results are applied to the case where the Bayesian model fails to be satisfied using an empirical Bayes approach.
155 citations
TL;DR: The theory of Bayesian inference at a rather sophisticated mathematical level is discussed in this paper, which is based on lectures given to students who already have had a course in measure-theoretic probability and has the rather clipped style of notes.
Abstract: This is a book about the theory of Bayesian inference at a rather sophisticated mathematical level. It is based on lectures given to students who already have had a course in measure-theoretic probability, and has the rather clipped style of notes. This led me to some difficulties of comprehension, especially when typographical errors occur, as in the definition of a random variable. Against this there is no unnecessary material and space for a few human touches. The development takes as fundamental the notion of expectation, though that word is scarcely used it does not appear in the inadequate index but has a brief mention on page 17. The book begins therefore with linear, non-negative, continuous operators and the treatment has the novelty that it does not require that the total probability be one: indeed, infinity is admitted, this having the advantage that improper distributions of the Jeffreys type can be included. There is an original and interesting account of marginal and conditional distributions with impropriety. For example, in discussing a uniform distribution over pairs (i,D of integers, the sets]=l and ]------2 both have infinite probability and cannot therefore be compared; so that conditional probabilities p( i=l / ]=l) , p(i=lff------2) require separate discussion. My own view is that this feature is not needed, for although improper distributions have some interest in low dimensions (and mainly in achieving an unnecessary match between Bayesian and Fisherian ideas) they fail in high dimensions, as Hartigan shows in chapter 9, where there is an admirable account of many normal means. A lesser objection is the complexity introduced by admitting impropriety: Bayes theorem takes 14 lines to state and 20 to prove. Chapter 5 is interestingly called \"Making Probabilities\" and discusses Jaynes' maximum entropy principle, Jeffreys' invariance, and similarity as ways of constructing distributions; those produced by the first two methods are typically improper. This attitude is continued into chapter 8 where exponential families are introduced as those minimizing information subject to constraints. There is a discussion of decision theory, as distinct from inference, but there is no attempt to consider utility: all is with respect to an undefined loss function. The consideration of the different types of admissibility is very brief and the opportunity to discuss the mathematically sensitive but practically meaningful aspects of this topic is lost. Other chapters are concerned with convergence, unbiasedness and confidence, multinomials, asymptotic normality, robustness and non-parametric procedures; the last being mainly devoted to a good account of the Dirichlet process. Before all this mathematics, the book begins with a brief account of the various theories of probability: logical, empirical and subjective. At the end of the account is a fascinating discussion of why the author thinks \"there is a probability 0.05 that there will be a large scale nuclear war between the U.S. and the U.S.S.R before 2000\". This connection between mathematics and reality is most warmly to be welcomed. The merit of this book lies in the novelty of the perspective presented. It is like looking at a courtyard from some unfamiliar window in an upper turret. Things look different from up there. Some corners of the courtyard are completely obscured. (It is suprising that there is no mention at all of the likelihood principle; and only an aside reference to likelihood.) Other matters are better appreciated because of the unfamiliar aspect normal means, for example. The book does not therefore present a balanced view of Bayesian theory but does provide an interesting and valuable account of many aspects of it and should command the attention of any statistical theorist.
85 citations
TL;DR: It is concluded that posttest probabilities calculated from Bayes' theorem more accurately classified patients with and without disease than did pretest probabilities, thus demonstrating the utility of the theorem in this application.
Abstract: One hundred fifty-four patients referred for coronary arteriography were prospectively studied with stress electrocardiography, stress thallium scintigraphy, cine fluoroscopy (for coronary calcifications), and coronary angiography. Pretest probabilities of coronary disease were determined based on age, sex, and type of chest pain. These and pooled literature values for the conditional probabilities of test results based on disease state were used in Bayes' theorem to calculate posttest probabilities of disease. The results of the three noninvasive tests were compared for statistical independence, a necessary condition for their simultaneous use in Bayes' theorem. The test results were found to demonstrate pairwise independence in patients with and those without disease. Some dependencies that were observed between the test results and the clinical variables of age and sex were not sufficient to invalidate application of the theorem. Sixty-eight of the study patients had at least one major coronary artery obstruction of greater than 50%. When these patients were divided into low-, intermediate-, and high-probability subgroups according to their pretest probabilities, noninvasive test results analyzed by Bayesian probability analysis appropriately advanced 17 of them by at least one probability subgroup while only seven were moved backward. Of the 76 patients without disease, 34 were appropriately moved into a lower probability subgroup while 10 were incorrectly moved up. We conclude that posttest probabilities calculated from Bayes' theorem more accurately classified patients with and without disease than did pretest probabilities, thus demonstrating the utility of the theorem in this application.
80 citations
Journal Article•
TL;DR: In this paper, the Bayesian maximum likelihood parametric classifier has been tested against the data-based formulation designated "linear discrimination analysis", using the "GLIKE" decision and "CLASSIFY" classification algorithms in the Landsat Mapping System.
Abstract: The Bayesian maximum likelihood parametric classifier has been tested against the data-based formulation designated 'linear discrimination analysis', using the 'GLIKE' decision and "CLASSIFY' classification algorithms in the Landsat Mapping System. Identical supervised training sets, USGS land use/land cover classes, and various combinations of Landsat image and ancilliary geodata variables, were used to compare the algorithms' thematic mapping accuracy on a single-date summer subscene, with a cellularized USGS land use map of the same time frame furnishing the ground truth reference. CLASSIFY, which accepts a priori class probabilities, is found to be more accurate than GLIKE, which assumes equal class occurrences, for all three mapping variable sets and both levels of detail. These results may be generalized to direct accuracy, time, cost, and flexibility advantages of linear discriminant analysis over Bayesian methods.
70 citations
TL;DR: The utility of Bayes' theorem in the noninvasive diagnosis of coronary artery disease was analyzed in 147 patients who underwent electrocardiographic stress testing, thallium-201 perfusion imaging and coronary angiography, finding that it is useful clinically despite some evidence of test dependence.
Abstract: The utility of Bayes' theorem in the noninvasive diagnosis of coronary artery disease (CAD) was analyzed in 147 patients who underwent electrocardiographic stress testing, thallium-201 perfusion imaging and coronary angiography. Eighty-nine patients had typical anginal chest discomfort and 58 had atypical chest pain. Sensitivity and specificity of the tests and prevalence of CAD at each level of testing were tabulated and compared with the results generated from Bayes' theorem. The sensitivity of electrocardiographic stress was higher in patients with multivessel CAD than in patients with 1-vessel CAD. Sensitivity, but not specificity, of each test was dependent, in part, on the result of the other test. However, the probabilities calculated from Bayes' theorem when used for sequential testing are remarkably close to the tabulated data. Thus, Bayes' theorem is useful clinically despite some evidence of test dependence. Sequential test analysis by Bayes' theorem is most useful in establishing or ruling out a diagnosis when the pretest prevalence is approximately 50% and when the 2 tests are concordant.
66 citations
TL;DR: In this paper, the authors compare Bayesian policies with a comparable non-Bayesian policy for the n-period non-depletive inventory model and show that the quantity ordered under the non-balancing policy would be greater than or equal to that under a Bayesian policy.
Abstract: Although it is often the case that the parameters of the distribution of demand are not known with certainty and that a Bayesian formulation would be appropriate, such an approach is generally not used in inventory calculations for computational reasons. Since one often resorts to a non-Bayesian formulation, it is of interest to compare Bayesian policies with a comparable non-Bayesian policy. Using the concept of flexibility it was anticipated that the quantity ordered under the non-Bayesian policy would be greater than or equal to that under a Bayesian policy. This result is established for the n-period nondepletive inventory model. However, a two-period counterexample is given for the standard depletive inventory model.
55 citations
01 Jan 1984
TL;DR: The problem of using expert estimates is investigated and two models are proposed based on normal and lognormal likelihood functions, which represent the decision maker’s model of the credibility of the expert estimates.
Abstract: The problem of using expert estimates is investigated. These estimates are treated as evidence that must be evaluated by a decision maker and incorporated into his body of knowledge and beliefs. This is done coherently using Bayes’ theorem. Two models are proposed based on normal and lognormal likelihood functions, which represent the decision maker’s model of the credibility of the expert estimates. Several commonly used methods, e. g., taking the geometric average of the expert estimates, are investigated in the context of the general methods of this paper.
TL;DR: Formal probability analysis based on Bayes' theorem greatly enhances the diagnostic capabilities of exercise testing in the detection of coronary artery disease.
TL;DR: It is shown that, through the use of a new class of empirical Bayes methods, it is possible to obtain useful and reliable estimates of the joint validity of several predictors of academic performance at the departmental level.
Abstract: Graduate education in the United States is characterized by an enormous diversity of disciplines and the predominance of relatively small enrollments in individual departments. In this setting, a validity study based on a single department's data and employing classical statistical methods can only be of limited utility and applicability. In order to participate in the Graduate Record Examinations Validity Study Service, a department must have at least 25 students in its entering class. Only validities for single predictors are provided; estimates of the validity of two or more predictors, used jointly, are considered too unreliable because the corresponding prediction equations often possess implausible characteristic, such as negative coefficients. These constraints were introduced by the Validity Study Service to reduce the chance that the results in the report to a department would be overly influenced by statistical artifacts in the data and hence serve more to mislead than to inform.
In this study we investigated two statistical methods, empirical Bayes and cluster analysis, to determine whether their application to the problems faced by the Validity Study Service could result in useful improvements. Considerable effort was expended in developing and studying a new and more general class of empirical Bayes models that can accommodate the complex structure of the Validity Study Service data base.
The principal methodological conclusions of this study are that, through the use of a new class of empirical Bayes methods, it is possible to obtain, at the departmental level, useful and reliable estimates of the joint validity of several predictors of academic performance and that these methods may be further refined to address the question of differential predictive validity, again at the departmental level. These results have important practical implications for the GRE Validity Study Service.
03 Dec 1984
TL;DR: This paper describes a method for extracting information from data to form the knowledge base for a probabilistic expert system that consists of joint probabilities that show significant Probabilistic connections between the associated attribute values.
Abstract: This paper describes a method for extracting information from data to form the knowledge base for a probabilistic expert system. The information the method finds consists of joint probabilities that show significant probabilistic connections between the associated attribute values. These joint probabilities can be combined with information about particular cases to compute particular conditional probabilities as described in [3]. The search procedure and significance test required are presented for different types of data. The significance test requires finding if the minimum message length (in the information theory sense) required to encode the data is reduced if the joint probability being tested is given explicitly. This significance test is derived from Bayes' theorem and is shown to find the hypothesis (i.e. set of significant joint probabilities) with the highest posterior probability given the data.
TL;DR: The relationship between Stein estimation of a multivariate normal mean and Bayesian analysis is considered in this article, where the necessity to involve prior information is discussed, and various methods of so doing are reviewed These include direct Bayesian analyses, ad hoc utilization of prior information, restricted class Bayesian and Γ-minimax analyses, and Type II maximum likelihood (empirical Bayes) methods.
TL;DR: A new family of life distributions, called the exponential-Weibull wear-out distributions, are developed, for systems whose failure rate function is a constant up to a change-point wear- out point and strictly increasing afterward.
Abstract: We develop a new family of life distributions, called the exponential-Weibull wear-out distributions, for systems whose failure rate function is a constant up to a change-point wear-out point and strictly increasing afterward. We derive properties of these wear-out distributions and develop a Bayes adaptive procedure for estimating the change point. Recursive formulas are given for determining the posterior probability that the change has occurred and its Bayes estimator. Results of numerical simulations are given to illustrate the properties of the adaptive procedure.
TL;DR: The result of applying the design methodology for designing suboptimal rules based on the Bayesian approach to an example shows that this approach is potentially a useful one.
Abstract: The formulation of the decision making process of a failure detection algorithm as a Bayes sequential decision problem provides a simple conceptualization of the decision rule design problem As the optimal Bayes rule is not computable, a methodology that is based on the Bayesian approach and aimed at a reduced computational requirement is developed for designing suboptimal rules A numerical algorithm is constructed to facilitate the design and performance evaluation of these suboptimal rules The result of applying this design methodology to an example shows that this approach is potentially a useful one
TL;DR: A new variable has been introduced, the predictive factor (c), which is calculated as follows: c=c1/(c1+1−c2).
Abstract: According to Bayes' rule the predictive value (PV) of a diagnostic test (= probability of disease if the test is positive) depends on the prevalence of the disease (= a priori probability), the sensitivity (c1) and the specificity (c2) of the test A new variable has been introduced, the predictive factor (c), which is calculated as follows:c=c1/(c1+1−c2) Since the PV only depends on this factor and on the prevalence, the calculation is much easier and a general graphical solution is possible This simplification renders several additional advantages and facilitates the understanding of the dependence of PV on prevalence
TL;DR: In this paper, a simple two-components of variation empirical Bayes model is proposed for use in estimating the between-expert variability curve in the presence of such biases, and compared with the existing method.
Abstract: Several recent nuclear power plant probabilistic risk assessments (PRAs) have utilized broadened Reactor Safety Study (RSS) component failure rate population variability curves to compensate for such things as expert overvaluation bias in the estimates upon which the curves are based. A simple two-components of variation empirical Bayes model is proposed for use in estimating the between-expert variability curve in the presence of such biases. Under certain conditions this curve is a population variability curve. Comparisons are made with the existing method. The popular procedure appears to be generally much more conservative than the empirical Bayes method in removing such biases. In one case the broadened curve based on the popular method is more than two orders of magnitude broader than the empirical Bayes curve. In another case it is found that the maximum justifiable degree of broadening of the RSS curve is to increase ..cap alpha.. from 5% to 12%, which is significantly less than 20% value recommended in the popular approach. 15 references, 1 figure, 5 tables.
TL;DR: In this paper, the optimal critical regions for pre-test estimators were developed in the context of the normal linear regression model with a conjugate prior, where the criterion is Bayes risk and where the pre-testing involves a single coefficient.
TL;DR: In this paper, a class of prior distributions is defined to reflect exchangeability of a set of binomial probabilities, indexed by the hyperparameter K, which indicates the precision of the user's prior belief about the similarity of the probabilities.
Abstract: A class of prior distributions is defined to reflect exchangeability of a set of binomial probabilities. The class is indexed by the hyperparameter K, which indicates the precision of the user's prior belief about the similarity of the probabilities. By estimating the unknown value of K from the marginal distribution, simple new point and interval estimators are proposed.
TL;DR: In this paper, prediction bounds for the order statistics are dealt with for a series of samples drawn from exponential, Pareto and power function populations, and the bounds assist in knowing the nature of predicted statistics even without actually having the sample observations.
Abstract: In this paper, prediction bounds for the order statistics are dealt with. For this purpose, predictive distributions derived by Bayesian approach, are utilized. In particular, bounds for the smallest and the largest order statistics are set when a series of samples are drawn from exponential, Pareto and power function populations. These bounds assist in knowing the nature of these predicted statistics even without actually having the sample observations.
01 Mar 1984
TL;DR: The Bayes smoothing algorithm presented is an extension of a 1-D algorithm to 2-D and it yields the a posteriori distribution and the optimum Bayes estimate of the scene value at each pixel, using the total noisy image data.
Abstract: A new image segmentation algorithm is presented, based on recursive Bayes smoothing of images modeled by Markov random fields and corrupted by independent additive noise The Bayes smoothing algorithm presented is an extension of a 1-D algorithm to 2-D and it yields the a posteriori distribution and the optimum Bayes estimate of the scene value at each pixel, using the total noisy image data Computational concerns in 2-D, however, necessitate certain simplifying assumptions on the model and approximations on the implementation of the algorithm In particular, the scene (noiseless image) is modeled as a Markov mesh random field and the algorithm is applied on (horizontal/vertical) strips of the image The Bayes smoothing algorithm is applied to segmentation of two level test images and remotely sensed SAR data obtained from SEASAT, yielding remarkably good segmentation results even for very low signal to noise ratios
TL;DR: In this article, the authors considered the Bayesian sequential estimation for the mean θ of a one parameter exponential family of distributions and showed that the A.P.O. procedure is asymptotically non-deficient.
Abstract: Bayesian sequential estimation for the mean θ of a one parameter exponential family fo distributions is considered. Squared error loss for estimation error and linear sampling cost are assumed. The asymptotically pointwise optimal (A.P.O.) procedure continues until Un< cn where U is the leading term in the expansion of the posterior loss. It is then shown that the A.P.O. procedure is asymptotically non-deficient: that is, the difference between the Bayes risk fo the A.P.O. procedure and the Bayes risk of the optimal procedure is of smaller order of magnitude than C, As C → 0, where c is the cost of a single observation.
TL;DR: In this article, the problem of finding admissible estimates for a fairly general class of parametric functions in the so-called "non-regular" type of densities has been considered.
Abstract: In this paper we have considered the problem of finding admissible estimates for a fairly general class of parametric functions in the so called “non-regular” type of densities Following Karlin s (1958) technique, we have established the ad-missibility of generalized Bayes estimates and Pitman estimates. Some examples are discussed.
TL;DR: A predictive test is used to demonstrate the superiority of the Bayesian approach to ordinary least squares and to compare the performance of several Bayesian estimators.
Abstract: A large regression model is constructed to convert data from an insurance company's annual statement into an industry-wide standard expected total expense figure. Bayesian methods are used to estimate parameters. A predictive test is used to demonstrate the superiority of the Bayesian approach to ordinary least squares and to compare the performance of several Bayesian estimators.
TL;DR: In this paper, it is shown how one may construct tests and confidence regions for the unknown structural parameters in empirical linear Bayes estimation problems, under the assumption that the design variables independently follow a common statistical law.
Abstract: It is shown how one may construct tests and confidence regions for the unknown structural parameters in empirical linear Bayes estimation problems. The case of the collateral units having varying “designs” (i.e. regressor and covariance matrices) may be treated under the assumption that the design variables independently follow a common statistical law. The results are of an asymptotic nature.
TL;DR: In this article, the reliability of a 2-unit hot-standby redundant system with an imperfect switch was assessed through Bayes theorem and the prior knowledge of experts was used in the assessment.
Abstract: This study assesses the reliability of a 2-unit hot-standby redundant system with an imperfect switch. The prior knowledge of experts is used in the assessment through Bayes theorem. Bayes interval limits for the system reliability are presented. The predictive distribution of the first failure time of the system is formulated.