Showing papers on "Bayes' theorem published in 1979"

Analysis of Probability as an Aid in the Clinical Diagnosis of Coronary-Artery Disease

Abstract: The diagnosis of coronary-artery disease has become increasingly complex. Many different results, obtained from tests with substantial imperfections, must be integrated into a diagnostic conclusion about the probability of disease in a given patient. To approach this problem in a practical manner, we reviewed the literature to estimate the pretest likelihood of disease (defined by age, sex and symptoms) and the sensitivity and specificity of four diagnostic tests: stress electrocardiography, cardiokymography, thallium scintigraphy and cardiac fluoroscopy. With this information, test results can be analyzed by use of Bayes' theorem of conditional probability. This approach has several advantages. It pools the diagnostic experience of many physicians and integrates fundamental pretest clinical descriptors with many varying test results to summarize reproducibly and meaningfully the probability of angiographic coronary-artery disease. This approach also aids, but does not replace, the physician's ju...

On the Reconciliation of Probability Assessments

Abstract: : This paper investigates the question of how to reconcile incoherent probability assessments, i.e., assessments that are inconsistent with the laws of probability. A general model for the analysis of probability assessments is introduced, and two approaches to the reconciliation problem are developed. In the internal approach, one estimates the subject's true probabilities on the basis of his assessments. In the external approach, an external observer updates his own coherent probabilities in the light of the assessments made by the subject. The two approaches are illustrated and discussed. Least-squares procedures for reconciliation are developed within the internal approach. (Author)

On Outlier Rejection Phenomena in Bayes Inference

Abstract: SUMMARY Inference is considered for a location parameter given a random sample. Outliers are not explicitly modelled, but rejection of extreme observations occurs naturally in any Bayesian analysis of data from distributions with suitably thick tails. For other distributions outlier rejection behaviour can never occur. These phenomena motivate new definitions of outlier-proneness and outlier-resistance. The definitions and methodology are Bayesian but the conclusions also have meaning for nonBayesians because they are proved for arbitrary prior distributions. Thus, for example, the t distribution is said to be outlier-prone because it is shown that any admissible inference procedure applied to a t sample will effectively ignore extreme outlying observations regardless of prior information. On the other hand, the normal distribution, for example, is said to be outlier-resistant because it never allows outlier rejection, regardless of prior information.

Why I am not an objective Bayesian; some reflections prompted by Rosenkrantz

Abstract: The objective Bayesian program has as its fundamental tenet (in addition to the three Bayesian postulates) the requirement that, from a given knowledge base a particular probability function is uniquely appropriate. This amounts to fixing initial probabilities, based on relatively little information, because Bayes' theorem (conditionalization) then determines the posterior probabilities when the belief state is altered by enlarging the knowledge base. Moreover, in order to reconstruct orthodox statistical procedures within a Bayesian framework, only privileged ‘ignorance’ probability functions will work.

Empirical Bayes Estimation of a Binomial Parameter Via Mixtures of Dirichlet Processes

Abstract: The theory of Dirichlet processes is applied to the empirical Bayes estimation problem in the binomial case. The approach is Bayesian rather than being empirical Bayesian. When the prior is a Dirichlet process the posterior is a mixture of Dirichlet processes. Explicit estimators are given for the case of 2 and 3 parameters and compared with other empirical Bayes estimators by way of examples. Since the number of calculations become enormous when the number of parameters gets larger than 2 or 3 we propose two approximations for estimators of a particular parameter and compare their performance using examples.

Nonparametric flow cytometry analysis.

Abstract: A nonparametric statistical test for the analysis of flow cytometry derived histograms is presented. The method involves smoothing and translocation of data, area normalization, channel by channel determination of the mean and S.D., and use of Bayes' theorem for unknown histogram classification. With this statistical method, different sets of histograms from numerous biological systems can be compared.

Application of Bayes's theorem to results of quantitative clinical chemical determinations.

Abstract: The diagnostic implication of a certain test result with regard to a certain condition can be expressed as a single number, L, the likelihood ratio of this result. This ratio allows Bayes's theorem to be written in a convenient form. We show that the practice of calculating predictive values for the results of quantitative tests by use of discrimination limits leads to incorrect predictive values. Including L values in laboratory reports seems a more logical approach to optimum interpretation of laboratory results than the use of discrimination values.

Limitations of electrocardiographic exercise testing.

Abstract: The Coronary Artery Surgery Study on exercise stress testing, discussed this week in the Journal,1 is an important addition to a recent series of articles exploring the reliability and limitations of exercise testing in the detection of underlying coronary-artery disease.2 3 4 5 6 In particular, the results of the study demonstrate that clinicians must be just as cognizant of the important diagnostic implications of Bayes' theorem of conditional probability as are statisticians. Bayes' theorem states that, although the reliability of a diagnostic test is defined by the test's sensitivity* and specificity,* a test cannot be adequately interpreted without reference to the prevalence . . .

New error bounds with the nearest neighbor rule (Corresp.)

Abstract: A distribution-free lower bound on the Bayes error rate is formulated in terms of the asymptotic error rate of the nearest neighbor rule with a reject option. Next, a closed form expression for an upper bound of the k th nearest neighbor error rate in terms of the Bayes rate is established. These results are discussed in the framework of recent works on nonparametric estimation of the Bayes error rate.

Forecasts, decisions and uncertain probabilities

Abstract: In the traditional decision theories the role of forecasts is to a large extent swept under the carpet. I believe that a recognition of the connections between forecasts and decisions will be of benefit both for decision theory and for the art of forecasting. In this paper I have tried to analyse which factors, apart from the utilities of the outcomes of the decision alternatives, determine the value of a decision. I have outlined two answers to the question why a decision which is made on the basis of a forecast is better than a decision which is based on a guess. Neither of these answers is universally valid. An assumption which is necessary for the first answer, i.e. Good's result, is that Bayes' rule is accepted as a correct and generally applicable decision principle. The second answer, which was given with the aid of probability intervals, departed from a more general decision principle, the maximin criterion for expected utilities, which was formulated in order to evade some of the criticism against Bayes' rule. However, the argument leading to the ansser is based on the assumption that the probability intervals associated with the states of nature represent certain knowledge. For this reason this answer is only approximatively valid. As a number of quotations in the section on “the weight of evidence” show, it is not sufficient to describe the knowledge about the states of nature by a single number, representing the (subjective) probability of the state, but something else has to be invoked which measures the amount of information on which a decision is based. Several authors have tried to characterize this mysterious quantity, which here was called the weight of evidence. However, there seems to be little agreement as to how this quantity should be measured.

A Stepwise Bayesian Procedure

Abstract: Ordinarily a Bayesian estimation procedure uses one prior distribution to obtain a unique estimation rule (its Bayes rule). From the decision theoretical point of view, this procedure can be regarded as a convenient way to obtain admissible decision rules. However, many intuitively appealing, admissible estimation rules cannot be obtained directly in this way. We propose a new mechanism, called the Stepwise Bayesian Procedure (SBP). When the parameter space contains only finitely-many points and the loss function is strictly convex, this SBP can be used to obtain every admissible estimation rule. A relationship between SBP and the limiting Bayes rules is given.

A Bayesian Probability View of Phylogenetic Systematics

Abstract: Harper, Charles W., Jr., (School of Geology and Geophysics, University of Oklahoma, 830 Van Vleet Oval, Rm. 109, Norman, Oklahoma 73019) 1979. A bayesian probability view of phylogenetic systematics. Syst. Zool. 28:547-553.-The development of hypotheses in phylogenetic systematics involves two activities: (1) the construction of phylogenetic hypotheses, and (2) subjecting these hypotheses to critical tests. Both may be analyzed within the framework of Bayes' theorem. Both are, in my view, inherently probabilistic and, as such, do not fit well into Popper's rigidly deductive models of scientific methodology. Regarding item (1), phylogenetic hypotheses are constructed, wholly or in part, by applying an ordering principle of shared derived character states to M taxa out of a group of N taxa. In order to establish a rational basis for when to apply the ordering principle (when to believe it), the principle is analyzed in terms of a relative frequency model. Application of Bayes' theorem to this model, first for M = 2, and then for M > 2, establishes a necessary condition for applying the ordering principle. As it turns out, the conditions under which one might rationally apply the principle vary markedly from one value of N (and M) to the next; this conclusion provides a good incentive for following a probabilistic approach when applying the ordering principle. The testing of phylogenetic hypotheses (item 2 above) is also viewed in terms of Bayes' theorem. These probabilistic approaches contrast markedly with the strictly deductive view of scientific methodology developed by Popper. [Phylogenetic systematics; phyletics; phylogeny construction; Bayesian probability.] "Popper regards Carnap as going beyond the pale of true Humean skepticism in saying that hypotheses can ever be probable. Carnap frowns upon Popper for nearly the same reason: it would on his view be altogether unwarranted ever to accept a hypothesis, tentatively or in any other way. All we can hope for is to be able to assign ,some degree of probability to the hypothesis" (Kyburg, 1970:175-176). "No one knows when the truth goes by" (Linda Ronstadt, "Simple dreams," 1977). Phylogenetic systematics, in its original sense as practiced by Willi Hennig (1966), involves 1) the development of phylogenetic hypotheses, and 2) the design of a classification based on them. Although I would differ with Hennig regarding details as to how we should go about accomplishing these two objectives, I am convinced that this basic twopart approach of Hennig is sound-the one we should follow. I shall concern myself with the first of these two steps-the development of phylogenetic hypotheses. It, in turn, involves two activities: 1) the construction of phylogenetic hypotheses, and 2) subjecting these hypotheses to critical tests. Both activities may be analyzed within the framework of Bayes' theorem. Both are, in my view, inherently probabilistic and, as such, do not fit well into Karl Popper's rigidly deductive models of scientific methodology. A common attitude concerning the explication of the scientific method is that we should start with the hypothesis and disregard any considerations as to how we arrived at the hypothesis in the first place. Thus, Popper (1968:31) states: "The initial stage, the act of conceiving or inventing a theory, seems to me neither to call for logical analysis nor be susceptible to it." Also, Popperians at least, commonly argue that we should propose bold, daring, and improbable hypotheses (Harsany, 1960:332-33). Both of these suggestions simply cannot apply in the case of phylogenetic inference, owing to the enormous number of possible phylogenetic hypotheses. Felsenstein (1978:31, table 2) has listed the total number of possible phylogenetic trees for n taxa; for n = 10, for example, there are in excess of 102 billion possible trees.

Bayesian subset selection for additive and linear loss function

Abstract: Given k independent samples of common size n from k populations πj,…,πk with distribution the problem is to select a non-empty subset form {πj,…,πk}, which is associated with "good" (large) θ-values. We consider this problem from a Bayesian approach. By choosing additive and especially linear loss functions we try to fill a gap lying in between the results of Deely and Gupta (1968) and more recent papers due to Goel and Rubin (1977), Gupta and Hsu (1978) and other authors. It is shown that under acertain "normal model" Seal's procedure turns out to be Bayes w.r.t. an unrealistic loss function where as Gupta's maximunl means procedure turns out to be ( for large n) asymptotically Bayes w.r. t. more realistic additive loss functions. Finally, in the appendix sonie bounds for are derived (where are fixed known and to approximate the Bayes rules w.r.t. linear loss functions in cases where n is finite.

Bayes' theorem, the exercise ECG, and coronary artery disease.

Abstract: Bayes' theorem was published posthumously by The Reverend Thomas Bayes in 1763. 1 This formula, which allows one to calculate predictability "in the doctrine of chance," has been of interest to mathematicians and statisticians ever since that time, although its value has only recently been emphasized in medicine, particularly in laboratory science. Galen and Gambino 2 have stressed the importance of the Bayes' equation and have prepared tables for determining the predictive value of any laboratory test. The predictability of a measurement can be estimated with precision if its sensitivity (true positives/all diseased patients), specificity (true negatives/all subjects without disease), and the prevalence of the disease under question are known. The predictive value of a positive test is the percent of all positive tests that are true positives: it can be shown to equal the product of prevalence and sensitivity divided by a denominator of that same product plus the

A Pattern Deformational Model and Bayes Error-Correcting Recognition System

Abstract: A pattern deformational model is proposed in this paper. Pattern deformations are categorized into two types: local deformation and structural deformation. A structure-preserving local deformation can be decomposed into a syntactic deformation followed by a semantic deformation, the former being induced on primitive structures and the latter on primitive properties. Bayes error-correcting parsing algorithms are proposed accordingly which not only can perform normal syntax analysis but also can make statistical decisions. An optimum Bayes error-correcting recognition system is then formulated for pattern classification. The system can be considered as a hybrid pattern classifier which uses both syntactic and statistical pattern recognition techniques.

A bayesian nonparametric estimator of survival probability assuming increasing failure rate

Abstract: Bayesian nonparametric estimators of the survival function, the failure rate function, and the density function are obtainedusing jump processes as prior distributions on the space of increasing failure rate functions. The jump processes are intui-tively appealing and have a meaningful physical interpretation. Examples are given and the estimates are compared with the maxi-mum likelihood estimates. In addition, the Bayesian nonpara-metric estimators are presented for arbitrarily right-censored observations.

Monotonicity of Bayes Sequential Tests

Abstract: Consider the problem of sequential testing of a one sided hypothesis when the risk function is a linear combination of a probability of an error component and an expected cost component. Sobel's results on monotonicity of Bayes procedures and essentially complete classes are extended. Sufficient conditions are given for every Bayes test to be monotone. The conditions are satisfied when the observations are from an exponential family. They are also satisfied for orthogonally invariant tests of a mean vector of a multivariate normal distribution and for scale invariant tests of two normal variances. Essentially complete classes of tests are the monotone tests for all situations where these sufficient conditions are satisfied.

Use of a Bayesian reasoning in safety and reliability decisions - three examples

Abstract: Bayes' theorem is used to quantify the impact of new evidence in three energy-related decision problems. The first problem concerns the risk of radioactivity release during the railroad transport of spent nuclear fuel. This history of shipments thus far is shown to make it highly unlikely that the frequency of release is on the order of 0.001 or greater per shipment. The second and third applications involve predicting the availability performance of new generations of turbine blades. Bayes' theorem is demonstrated as a means for incorporating in the prediction the limited operational data on the new blades along with the experience of the earlier generation and the knowledge of the design changes.

Gamma minimax robustness of bayes rules

Abstract: Bayes rules are shown to be robust in multiple decision problems in the sense that they retain an optimality property, r-minimaxity, when the original distributions are replaced by families of e-contaminated versions of themselves and the prioris replaced by a family of priors on these -contaminations. Thus, the Bayes rule is robust against inaccurately specifiedparameter spaces and, hence, inaccurately specified priors. Bounds are obtained on the amount of contamination which can be present with the Bayes rule remaining ρ-minimax.

Credibility and the Dirichlet process

Abstract: Credibilists usually side-step the problem of specifying the structure function for a collective of risk experiences by finding certain projections in certain Hilbert spaces. These projections represent approximations to the Bayes rule. However, since both the credibility mean and credibility distribution function are proper Bayes relative to a Dirichlet process prior it is natural for the present paper to investigate the suitability of this process as a structure function. The proposed nonparametric models provide some increased insight into the credibility factor Z and the manual premium m; in particular, novel estimators for them are suggested.

An emprical bayes approach to two-sample problems with censored data

Abstract: Empirical Bayes methods are used in estimating the probability based on randomly right-censored samples. The estimator is shown to be asymptotically optimal. Thus, in a way, this uork is similar to the results of Hollander and Korwar (1976) who used a similar approach in estimating A in the case of non-censored data. We also give hero a shorter proof to their rate result. In addition, a. resting procedure is obtained to test the hypothesis against on the basis of censored data. It is shown that this procedure is asymptotically optimal with rate of convergence n . Tnis result is analogous to our earlier result for the uncensored case (1970) The empirical Hayes procedure has been illustrated by means of a practical example.

Multiclass Bayes error estimation by a feature space sampling technique

Abstract: A general Gaussian M-class N-feature classification problem is defined. An algorithm is developed that requires the class statistics as its only input and computes the minimum probability of error through use of a combined analytical and numerical integration over a sequence simplifying transformations of the feature space. The results are compared with those obtained by conventional techniques applied to a 2-class 4-feature discrimination problem with results previously reported and 4-class 4-feature multispectral scanner Landsat data classified by training and testing of the available data.

Modified Method-of-Moments in Empirical Bayes Estimation

Abstract: A unit is put on test for a fixed time and the number of failures is observed. The probability distribution of the number of failures is assumed to be Poisson, and the Poisson failure intensity is assumed to be a stochastic variable with gamma prior distribution. Schafer & Feduccia introduced an empirical procedure for estimating the parameters of the prior based on method of moments. We investigate the s-efficiencies of empirical Bayes estimates of Poisson failure intensity and reliability when the prior is estimated by the Schafer & Feduccia method. Mean square errors (MSEs) are compared for a range of parameters which typifies certain military equipment failure data. The empirical Bayes estimates have high s-efficiencies for sample size more than 40. A modification of the Schafer & Feduccia procedure substantially improves s-efficiencies for small sample sizes.

Further Study of Robustification via A Bayesian Approach.

Abstract: : A Bayesian model has been proposed which describes the generation of an observation by a process whereby with prior probability 1-alpha the usually assumed statistical structure is correct but with small probability alpha it is incorrect (for example, the observation has a very large variance). For a simple location estimate the nature of the down weighting of outlying observations produced by this model is studied and is compared with that of the presently popular M-estimators.

On the optimum choice of categories for the classification of biomedical data patterns.

Abstract: The optimum choice of categories for pattern recognition problems is on the one hand determined by the requirement of a low rate of misclassification. On the other hand, the classification of a pattern should result in an information gain as high as possible. A criterion for an optimum choice of categories which is the best compromise between the demands mentioned above is worked out. The Bayes rule is used as a decision function. The alteration of the Bayes risk as indicator for the rate of malrecognition is examined for different choice of categories concerning the very same classification problem. As the calculation of the Bayes risk is commonly difficult, an estimation using the Bhattacharyya coefficient is given. The information content of a choice of categories is defined using Shannon’s information measure. The alteration of the information contents is analyzed by putting together certain categories, i.e. a coarser choice of categories. With the aid of the relative information loss and the relative reduction of the Bayes risk coefficient, a criterion on the goodness of a choice of categories can be given. The criterion also serves as an optimum choice of classes. The extension of the latter criterion to a generalized decision rule is possible.