Showing papers on "Bayes' theorem published in 1982"

Random-effects models for longitudinal data

Abstract: Models for the analysis of longitudinal data must recognize the relationship between serial observations on the same unit. Multivariate models with general covariance structure are often difficult to apply to highly unbalanced data, whereas two-stage random-effects models can be used easily. In two-stage models, the probability distributions for the response vectors of different individuals belong to a single family, but some random-effects parameters vary across individuals, with a distribution specified at the second stage. A general family of models is discussed, which includes both growth models and repeated-measures models as special cases. A unified approach to fitting these models, based on a combination of empirical Bayes and maximum likelihood estimation of model parameters and using the EM algorithm, is discussed. Two examples are taken from a current epidemiological study of the health effects of air pollution.

Reverend bayes on inference engines: a distributed hierarchical approach

Abstract: This paper presents generalizations of Bayes likelihood-ratio updating rule which facilitate an asynchronous propagation of the impacts of new beliefs and/or new evidence in hierarchically organized inference structures with multi-hypotheses variables. The computational scheme proposed specifies a set of belief parameters, communication messages and updating rules which guarantee that the diffusion of updated beliefs is accomplished in a single pass and complies with the tenets of Bayes calculus.

Conservatism in human information processing

Abstract: … An abundance of research has shown that human beings are conservative processors of fallible information. Such experiments compare human behavior with the outputs of Bayes's theorem, the formally optimal rule about how opinions (that is, probabilities) should be revised on the basis of new information. It turns out that opinion change is very orderly, and usually proportional to numbers calculated from Bayes's theorem – but it is insufficient in amount. A convenient first approximation to the data would say that it takes anywhere from two to five observations to do one observation's worth of work in inducing a subject to change his opinions. A number of experiments have been aimed at an explanation for this phenomenon. They show that a major, probably the major, cause of conservatism is human misaggregation of the data. That is, men perceive each datum accurately and are well aware of its individual diagnostic meaning, but are unable to combine its diagnostic meaning well with the diagnostic meaning of other data when revising their opinions. … Probabilities quantify uncertainty. A probability, according to Bayesians like ourselves, is simply a number between zero and one that represents the extent to which a somewhat idealized person believes a statement to be true. The reason the person is somewhat idealized is that the sum of his probabilities for two mutually exclusive events must equal his probability that either of the events will occur.

Combining site‐specific and regional information: An empirical Bayes Approach

Abstract: Empirical Bayes theory, adapted to a hydrologic context, is used to develop procedures for inferring hydrologic quantities by combining site-specific and regional information. It ‘borrows strength’ from ‘similar’ basins to improve upon inference at a particular basin. The superpopulation is a key concept in the empirical Bayes approach. It is a probability distribution from which basin parameters are randomly assigned, a conceptualization closely related to regionalization models. It is inferred from observable regional data and expresses the degree of basin ‘similarity’ in a region. This approach treats regionalized estimators as a special case and leads to procedures similar to James-Stein estimators. Empirical Bayes procedures can lead to substantial improvements in performance over site-specific procedures. However, for basins which are very different from the majority, site-specific procedures may perform better. The method of moments approach to inferring the superpopulation is considered in detail. Finally, two examples in flood frequency analysis are presented to illustrate various facets of empirical Bayes procedures.

Bayes-like Decision Making With Upper and Lower Probabilities

Abstract: We consider the use of interval-valued probabilities to represent the support lent to the hypothesis that the parameter value θ lies in a subset A of the parameter set Θ when we observe x, know the likelihoods {fΘ: θeΘ}, and have some prior information concerning the parameter. Our model for prior information is that of a salient prior distribution in which we have little confidence, although we have much less confidence in any alternative prior. We consider notions of acceptable and coherent decision making as well as notions of being able to achieve a Bayes rule and least commitment. Throughout we are motivated to preserve some of the elements of Bayesian decision making without thereby committing ourselves to unwarranted claims of knowledge.

Diagnostic accuracy of cardiologists compared with probability calculations using Bayes' rule

Abstract: Probability analysis has provided insights into the use of diagnostic tests in coronary artery disease, and recent developments may permit clinical application of individual patients. To validate independently two available methods of probability calculation, their diagnostic accuracy was compared with that of cardiologists. Ninety-one cardiologists participated in the study; each evaluated the clinical summaries of eight randomly selected patients. For each patient, the cardiologist assessed the probability of coronary artery disease after reviewing the clinical history, physical examination and laboratory data, including complete results from a treadmill exercise test. The probability of coronary artery disease was also obtained for each patient, using the identical information, from two methods employing Bayes' rule: (1) from a published table of data based on the patient's age, sex, symptoms and degree of S-T segment change during exercise; and (2) from a computer program using the age, sex, risk factors, resting electrocardiogram and multiple exercise measurements. Diagnostic accuracy was assessed on a scale from 0 to 100 with the coronary angiogram as the diagnostic standard. The average diagnostic accuracy on this scale was: 80.2 for the cardiologists' estimates, 78.0 for the estimates based on tables (difference from cardiologists' estimates p less than 0.05) and 83.1 for the estimates based on computer calculations (p less than 0.01). Thus probability analysis incorporating sufficient detail can achieve a diagnostic accuracy comparable with that of cardiologists. Studies of the efficacy of probability analysis in patient care are warranted.

An Automated Approach to the Design of Decision Tree Classifiers

Abstract: The classification of large dimensional data sets arising from the merging of remote sensing data with more traditional forms of ancillary data causes a significant computational problem. Decision tree classification is a popular approach to the problem. This type of classifier is characterized by the property that samples are subjected to a sequence of decision rules before they are assigned to a unique class. If a decision tree classifier is well designed, the result in many cases is a classification scheme which is accurate, flexible, and computationally efficient. This correspondence provides an automated technique for effective decision tree design which relies only on a priori statistics. This procedure utilizes canonical transforms and Bayes table look-up decision rules. An optimal design at each node is derived based on the associated decision table. A procedure for computing the global probability of correct classification is also provided. An example is given in which class statistics obtained from an actual Landsat scene are used as input to the program. The resulting decision tree design has an associated probability of correct classification of 0.75 compared to the theoretically optimum 0.79 probability of correct classification associated with a full dimensional Bayes classifier. Recommendations for future research are included.

15 Intrinsic dimensionality extraction

Abstract: Publisher Summary This chapter discusses the computation of the intrinsic dimensionality in the context of representation, that is, the structure of a data distribution was to be preserved by the mapping and intrinsic dimensionality in the context of classification. In classification, it is well known that the Bayes classifier is the best classifier for given distributions. The resulting classification error, the Bayes error, is the minimum probability of error. As for criteria, the Bayes error is the best. However, since the Bayes error is not easily expressed in an explicit form that may be readily manipulated, many alternatives have been proposed. These include the asymptotic nearest neighbor error, the equivocation, the Chernoff bound, the Bhattacharyya bound and so on. These criteria give upper bounds for the Bayes error. However, the optimization of these criteria does not give the posterior probability functions as the solutions.

Bayes-cost reduction algorithm in quantum hypothesis testing (Corresp.)

Abstract: An iterative procedure is described for reducing the Bayes cost in decisions among M>2 quantum hypotheses by minimizing the average cost in binary decisions between all possible pairs of hypotheses: the resulting decision strategy is a projection-valued measure and yields an upper bound to the minimum attainable Bayes cost. From it is derived an algorithm for finding the optimum measurement states for choosing among M linearly independent pure states with minimum probability of error. The method is also applied to decisions among M unimodal coherent quantum signals in thermal noise.

The Improvement of Probability Judgements

Abstract: SUMMARY A subject S assesses the probability for an event A as q. In this paper, it is shown how the assessment can be improved. Improvement requires two functions fi(q) (i = 0, 1); the probability for q when A is true (i = 1) and when A is false (i = 0). It is shown that data on S's probability assessments can be used to update these functions. Specific, normal forms for them are examined. The connection with calibration is explored. 1. THE TRANSFORMATION OF PROBABILITY JUDGEMENTS CONSIDER a subject, S, who has provided his personal probability for an event A. In this paper we address the problem of whether his probability assessment can be improved, not by providing information additional to that which S already has, or even by S remembering something he had temporarily forgotten, a process Brown and Lindley (1980) have termed "digging in S's psychological field"; but by a better understanding of the assessment mechanism. It is shown that improvement is possible and the method of making it is described. Let the event A under consideration initially have probability y. Because Bayes' theorem in log-odds form is linear, it is convenient to suppose S's statement of probability assessment is that the log-odds for A is q. Our question is: can we transform q to a new value, q* say, which, in some sense, is a better assessment of the log-odds? Two examples may be illuminating. Example 1 S is given an almanac question, with two possible answers, say an upper and a lower one. Data set (a) below provides an example. Before seeing the question he is told that one, and only one, of the answers is correct and that they are equally likely to be correct. On seeing the question he is asked for the probability that the upper answert is correct. Here y =-. The

Bayesian Robustness and the Stein Effect

Abstract: In simultaneous estimation of normal means, it is shown that through use of the Stein effect surprisingly large gains of a Bayesian nature can be achieved, at little or no cost, if the prior information is misspecified. This provides a justification, in terms of robustness with respect to mis-specification of the prior, for employing the Stein effect, even when combining a priori independent problems (i.e., problems in which no empirical Bayes effects are obtainable). To study this issue, a class of minimax estimators that closely mimic the conjugate prior Bayes estimators is introduced.

On Bayes Risk Consistent Pattern Recognition Procedures in a Quasi-Stationary Environment

Abstract: Van Ryzin and Greblicki showed that pattern recognition procedures derived from orthogonal series estimates of a probability density function are Bayes risk consistent. In this note it is proved that these procedures do not lose-under some additional conditions-their asymptotic properties even if the random environment is nonstationary.

Bayesian approach for a nonlinear growth model.

Abstract: Nonlinear least squares methods are currently used for fitting a well-known growth model, namely the Jenss model, to the length measurements of a child followed throughout the first six years of life. An empirical Bayes approach is developed for fitting the model, and the prior distribution of the growth-model parameters is estimated from a large sample of least squares parameters. An expression which is proportional to the posterior distribution is derived so that the posterior mode can be estimated. Given the observations on a child, this posterior mode provides Bayes estimates of the Jenss curve parameters for the child.

Bayes's Two Arguments for The Rule of Conditioning

Abstract: The introductory section of Thomas Bayes's famous essay on probability contains two arguments for what we now call the rule of conditioning. The first argument, which leads to Bayes's third proposition, can be made rigorous if we use rooted trees to represent the step-by-step determination of events. The second argument, which leads to Bayes's fifth proposition, does not stand up to scrutiny.

Classical and Bayesian Approaches to the Change-Point Problem: Fixed Sample and Sequential Procedures.

Abstract: : The change-point problem can be considered one of the central problems of statistical inference, linking together statistical control theory, theory of estimation and testing hypotheses, classical and Bayesian approaches, fixed sample and sequential procedures. It is very often the case that observations are taken sequentially over time, or can be intrinsically ordered in some other fashion. The basic question is, therefore, whether the observations represent independent and identically distributed random variables, or whether at least one change in the distribution law has taken place. This is the fundamental problem in the statistical control theory, testing the stationarity of stochastic processes, estimation of the current position of a time-series, etc. Accordingly, a survey of all the major developments in statistical theory and methodology connected with the very general outlook of the change-point problem, would require review of the field of statistical quality control, the switching regression problems, inventory and queueing control, etc. The present review paper is therefore focused on methods developed during the last two decades for the estimation of the current position of the mean function of a sequence of random variables (or of a stochastic process); testing the null hypothesis of no change among given n observations, against the alternative of at most one change; the estimation of the location of the change-point(s) and some sequential detection procedures.

Statistical Properties of Error Estimators in Performance Assessment of Recognition Systems

Abstract: The problem of estimating the error probability of a given classification system is considered. Statistical properties of the empirical error count (C) and the average conditional error (R) estimators are studied. It is shown that in the large sample case the R estimator is unbiased and its variance is less than that of the C estimator. In contrast to conventional methods of Bayes error estimation the unbiasedness of the R estimator for a given classifier can be obtained only at the price of an additional set of classified samples. On small test sets the R estimator may be subject to a pessimistic bias caused by the averaging phenomenon characterizing the functioning of conditional error estimators.

Bayes or Laplace? An examination of the origin and early applications of Bayes' theorem

Abstract: Maistrov (1974) in fact goes so far as to say "Bayes' formula appears in all texts on probability theory" (p. 87), a statement which is perhaps a little exaggerated (unless, of course, one is perverse enough to make this result's presence a sine qua non for a book to be so described!). The fame (or notoriety, rather, in some statistical circles) of this "Bayes' Theorem" is such that it comes as something of a supriseif not a shockto discover that this proposition is nowhere to be found in Bayes' Essay! Yet one might in fact at least be thankful that mutuus consensus has to a large measure prevailed as to the formula to which this name is applied (mistaken though this may be), for such has by no means always been the case.

Flood Probability Estimation

Abstract: A procedure is developed for estimating design floods in situations where data are insufficient for a conventional frequency analysis. It involves the use of a compound distribution which is a weighted combination of individual two parameter distributions. Initial values are assigned to the parameters of the component distributions and their weights on the basis of subjective estimates of the mean and standard deviation of the flood peaks. The weights of the component distributions are updated with Bayes' theorem in the light of any additional information such as recorded floods, the largest flood in a number of years or a flow which has not been exceeded in a given number of years. Numerical methods are used. A computer program is developed to carry out the necessary computations and is described to illustrate how the method could be used by practicing hydrologists. Examples are presented to illustrate the procedure.

Bayesian Reliability & Availability - A Review

Abstract: Reliability/availability estimation in the classical sense has been well developed and widely discussed. The parameters in the classical reliability/availability distributions are considered to be unknown constants to be determined. If there is information about these parameters, besides that from some current experiment, it can be used as the basis of Bayesian inference. This paper illustrates procedures for using the Bayesian approach for the study of reliability/availability problems. References on the subject are collected and classified. Specifically, included in this paper are: 1) the statement of the classical estimate and Bayesian estimate, 2) the structure of the prior distribution, 3) the reliability life testing, 4) the empirical Bayes approach in reliability, and 5) Bayesian availability. A Bayesian parameter estimate is generally independent of the sampling stopping rule and has smaller variability than the classical estimate; however, a Bayesian estimate is usually biased and associated with difficulties of choosing a prior. The empirical Bayes approach eliminates some of these difficulties, but frequently at the cost of mathematical tractability. The Bayesian approach to reliability/availability is part of the general trend toward using comprehensive probabilistic methods for dealing with the uncertainties associated with modern engineering problems. This trend should also move toward some system effectiveness measures other than simply reliability or availability. For a large system the Bayesian approach could apply, especially when the test data are scarce and the testing procedure is expensive.

Simplified application of Bayesian analysis to multiple cardiologic tests.

Abstract: Bayes' theorem is an accepted method to integrate the results of multiple tests for coronary artery disease, but complicated calculations make its application awkward for the clinician. This report presents tables which allow the reader to serially apply Bayes' theorem to the results of stress electrocardiography, thallium scintigraphy, technetium blood-pool scintigraphy, and cardiac fluoroscopy.

A Useful Empirical Bayes Identity

Abstract: For any decision problem, one wishes to find that estimator which minimizes the expected loss. If the loss function is squared error, then the estimator is the mean of the Bayes posterior distribution. Unfortunately the prior distribution may be unknown, but in certain situations empirical Bayes methods can circumvent this problem by using past observations to estimate either the prior or the Bayes estimate directly. Empirical Bayes methods are particularly appealing when the Bayes estimate depends only on the marginal distribution of the observed variable, yielding what is known as a simple empirical Bayes estimate. The paper looks at the underlying circumstance of when a simple empirical Bayes estimator is available, and shows its occurrence not to be happenstance.

Optimal choice of type and order of river flow time series models

Abstract: The performance of various types of models for river flows are compared by using a decision rule derived from the Bayes criterion. The decision rule has the property that it minimizes the probability of error. The best model among the autoregressive, autoregressive moving average, and moving average models of various orders for the annual flows of about 10 rivers is found by using this decision rule. For the monthly flows, not only the best seasonal autoregressive integrated moving average model but also the best type of transformation is determined. The models for the log transformed monthly data are superior to the models fitted to the observed data without transformation. The variability in the decisions caused by the different prior probability density functions is also discussed.

Invention and Appraisal

Abstract: Among the many important contributions made by Wesley Salmon to the explication of the logic of hypothesis-appraisal has been his insistence that considerations additional to the inductive relations between evidence-statements and hypothesis may bear upon the probability of the latter. In particular, he has urged that the prior (to test) probability of a hypothesis H, as well as the likelihoods of H and not-#, must be taken into account in assessing the posterior (to test) probability value of H. Salmon has argued that the relationships among these several probability-values are captured by a particular interpretation of Bayes’ Theorem — a theorem of the formal probability calculus (Salmon [1967a], [1970c]). If this proposal is adopted, the problem arises of how to assign a value to the prior probability of H. Since no frequencies directly relevant to H will be available prior to initial testing, Salmon, who adopts a frequency interpretation of probability, must appeal to other considerations in estimating this prior probability value ([1967a], pp. 124 ff). These are what he calls “plausibility considerations” for H. As well as formal and pragmatic criteria, they include material criteria, such as considerations of simplicity and symmetry. From plausibility considerations of these various types, a scientist would be enabled to judge the plausibility of his/her hypothesis prior to test, and to estimate a prior probability value for insertion in Salmon’s Bayesian schema (ibid., pp. 115 ff, and my pp. 76–8 below). Even if, in many cases, the prior probability of a hypothesis is difficult to quantify precisely, Salmon’s recognition that this factor plays an important role in the appraisal of the hypothesis has always seemed to me to be a valuable insight.

A Subjectivist View of Calibration.

Abstract: : Calibration concerns the relationship between subjective probabilities and the long-run frequencies of events. Theorems from the statistical and probability literature are reviewed to discover the conditions under which a coherent Bayesian expects to be calibrated. If the probability assessor knows the outcomes of all previous events when making each assessment, calibration is always expected. However, when such outcome feedback is lacking, the assessor expects to be well calibrated on an exchangeable set of events if and only if the events in question are viewed as independent. Although this strong condition has not been tested in previous research, we speculate that the past findings of pervasive overconfidence are not invalid. Although experimental studies of calibration hold promise for the development of cognitive theories of confidence, their value for the practice of probability assessment seems more limited. Efforts to train probability assessors to be calibrated may be misplaced. (Author)

Bayesian analysis of underground flooding

Abstract: An event-based stochastic model is used to describe the spatial phenomenon of water inrush into underground works located under a karstic aquifer, and a Bayesian analysis is performed because of high parameter uncertainty. The random variables of the model are inrush yield per event, distance between events, number of events per unit underground space, maximum yield, and total yield over mine lifetime. Physically based hypotheses on the types of distributions are made and reinforced by observations. High parameter uncertainty stems from the random characteristics of karstic limestone and the limited amount of observation data. Thus, during the design stage, only indirect data such as regional information and geological analogies are available; updating of this information should then be done as the construction progresses and inrush events are observed and recorded. A Bayes simulation algorithm is developed and applied to estimate the probability distributions of inrush event characteristics used in the design of water control facilities in underground mining. A real-life example in the Transdanubian region of Hungary is used to illustrate the methodology.