Abstract: This short paper Dresents a new algorithm for minimun information Bayesian Inferencing within Expert Systems. This algorithm is as efficient in both time and space as previously reported work [3 3 but always provides a minimum information result. In addition to describing the new algorithm, we will prove that it does indeed satisfy minimum information criteria. Since both algorithms are sub stantially different from the "Bayesian" approaches in well known expert systems such as the original Prospector [1], AL/X [8], and MYCIN [9 3, and from the approach of Kulikowski [5], background is provided to show the motivation for using the minimum information approach to Bayesian updating.

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.

Abstract: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

Abstract: : This report describes a method of computing confidences in rule-based inference systems by using the Dempster-Shafter theory. The theory is applicable to tactical decision problems which can be formulated in terms of sets of exhaustive and mutually exclusive propositions. Dempster's combining procedure, a generalization of Bayesian inference, can be used to combine probability mass assignments supplied by independent bodies of evidence. This report describes the use of Dempster's combining method and Shafer's representation framework in rule-based inference systems. It is shown that many kinds of data fusion problems can be represented in a way such that the constraints are met. Although computational problems remain to be solved, the theory should provide a versatile and consistent way of combining confidences for a large class of inferencing problems. (Author)

Abstract: In this paper we present an alternative procedure for establishing an upper precision limit (UPL) for the rate of noncompliance of an internal control attribute. This procedure is based on Ericson's [1969] pioneering work in which he applied Bayesian methods to finite population sampling. We refer to our proposed procedure as the finite Bayesian procedure (FBP). The FBP is presented as an alternative to both the classical procedure (CP) and the infinite Bayesian procedure (IBP). The classical procedure for computing a UPL uses any one of the three well-known sampling distributions for number of errors: the hypergeometric, binomial, or Poisson. The CP does not use prior information, and the only variability that is accounted for is that which is due to the sampling design. Felix and Grimlund [1977] introduced the IBP as the first alternative to the classical procedure. The IBP utilizes prior information through a prior distribution and makes a posterior probability statement about the process error rate. The IBP does not account for the fact that most audit populations of attributes are finite. This is also true of the CP when it uses the binomial or Poisson distribution. The FBP correctly assumes a finite population, but in addition it exploits the auditor's prior information through a prior distribution. In contrast to the IBP, the FBP makes a conclusion about the value of interest, the finite population error rate, rather than the process error rate. Our main conclusions are: (1) the finite Bayesian model emulates the

Abstract: Bayesian inference modeling may be applied to empirical stochastic prediction in geomorphology where outcomes of geomorphic processes can be expressed by probability density functions. Natural variations in process outputs are accommodated by the probability model. Uncertainty in the values of model parameters is reduced by considering statistically independent prior information on long-term, parameter behavior. Formal combination of model and parameter information yields a Bayesian probability distribution that accounts for parameter uncertainty, but not for model uncertainty or systematic error which is ignored herein. Prior information is determined by ordinary objective or subjective methods of geomorphic investigation. Examples involving simple stochastic models are given, as applied to the prediction of shifts in river courses, alpine rock avalanches, and fluctuating river bed levels. Bayesian inference models may be applied spatially and temporally as well as to functions of a random variable. They provide technically superior forecasts, for a given shortterm data set, to those of extrapolation or stochastic simulation models. In applications the contribution of the field geomorphologist is of fundamental quantitative importance.

Abstract: Probability updating via Bayes' rule often entails extensive informational and computational requirements. In consequence, relatively few practical applications of Bayesian adaptive control techniques have been attempted. This paper discusses an alternative approach to adaptive control, Bayesian in spirit, which shifts attention from the updating of probability distributions via transitional probability assessments to the direct updating of the criterion function, itself, via transitional utility assessments. Re- suits are illustrated in terms of an adaptive reinvestment two-armed bandit problem.

Abstract: : Bayesian inference provides a general framework for testing hypotheses It is a normative method, in the sense of prescribing how hypotheses should be tested However, it may also be used descriptively, by characterizing people's actual hypothesis-testing behavior in terms of its consistency with or departures from the model Such a characterization may facilitate the development of psychological accounts of how that behavior is produced (ie, as the result of failed attempts to act in a Bayesian fashion, as the result of attempts of process information in non-Bayesian ways This essay exploits the descriptive potential of Bayesian inference First, it identifies a set of logically possible forms of non-Bayesian behavior Second, it reviews existing research in a variety of areas in order to see whether these possibilities are ever realized The analysis shows that in some situations, several apparently distinct phenomena are usefully viewed as special cases of the same kind of behavior, whereas in other situations previous investigations have conferred a common label (eg, confirmation bias) to several distinct phenomena It also calls into question a number of attributions of judgmental bias, suggesting that in some cases the bias is different than what has previously been claimed, whereas in others, there may be no bias at all (Author)

Abstract: SUMMARY This paper is a review of the contributions of Jerome Cornfield to the theory of statistics. It discusses several highlights of his theoretical work as well as describing his philosophy relating theory to application. The three areas discussed are: linear programming, urn sampling and its generalizations to the analysis of variance, and Bayesian inference. It is not widely known that Jerome Cornfield was perhaps the first to formulate and approximately solve the lirlear programming problem in 1941. His formulation was made for the famous "Diet Problem". An early publication introduced the method of indicator random variables in the context of urn sampling. This simple method allowed straightforward calculations of the low order moments for estimates arising from sampling finite populations and was later generalized to the two-way analysis of variance. The application of the urn sampling model to the analysis of variance served to illuminate how one chooses proper error terms for making tests in the analysis of variance table. Jerome Cornfield's philosophy on applications of statistics was dominated by a Bayesian outlook. His theoretical contributions in the past two decades were mainly concerned with the development of Bayesian ideas and methods. A brief survey is made of his main contributions to this area. A particularly noteworthy result was his demonstration that for the twosample slippage problem of location, the likelihood function under a permutation setting is uninformative for the slippage parameter. However, the posterior distribution differs from the prior distribution despite the fact that the likelihood is uninformative.

Abstract: In designing a clinical trial, the investigator must often decide whether or not to exclude certain patients from randomization. Using Bayesian analysis this paper proposes a criterion for making this decision, based on an extension of Colton's model for the choice between two medical treatments.

Abstract: This article presents a Bayesian approach to estimation, in a sample survey setting, of a finite population total for some characteristic of interest when nonresponse is present. A general model is described for situations where a Hansen-Hurwitz sampling plan is used. The model makes use of concomitant information that is related both to the characteristic of interest and to the probability of response. A simple example is included to illustrate the usefulness of the approach.

Abstract: Statistical methods for evaluating the effectiveness of treatment at black spots are important for two reasons. First, they allow the engineer to determine how successful his efforts have been, and hence to estimate the costs and benefits of different types of treatment applied to different types of site. Second, they enable the researcher to test new treatment measures objectively. However, the methods which are currently used both in research and in local authority practice are not very suitable as a basis for making decisions. Bayesian methods, on the other hand, are ideally suited to this type of work. In this paper, a Bayesian approach to evaluation is briefly outlined. Some further developments in the Bayesian model, including an analysis based on a bivariate negative binomial distribution are discussed, and suggestions made for future research. (TRRL)

Abstract: This paper examines a quality control problem where testing is destructive. A skip-lot model is developed using a Bayesian approach to infer the process state between inspections. The model is then used to 1) maximize the s-expected return per lot produced and 2) determine the inspection interval, sample size, and acceptance number. Numerical evaluation is used to compare this model with a previous model of the situation. Examples suggest that this formulation of the skip-lot problem which accounts for the posterior distribution of process state for each lot and the revenue received appreciably reduces destructive sampling.

Abstract: Publisher Summary This chapter explains that Laplace's principle of insufficient reason is the most familiar and the most general attempted solution. In the absence of evidence relevant to the assessment of these alternatives, ignorance goes hand-in-hand with symmetry of credal probabilities. The chapter also reviews the distinction between direct and inverse statistical inference and a rehearsal of the role the Laplacean principle plays in solving inverse inference. Two interpretations of the probability calculus are used: a credal probability and a stochastic probability. In direct inference, inference is from the knowledge of chances to hypotheses about some (particular) outcome of the stochastic process, inference from “population” to “sample.” In inverse inference, inference is from the knowledge of particular outcomes to hypotheses about the unknown chances on that kind of trial.

Abstract: Operationally, we can distinguish between objective and subjective probabilities as follows. Some probabilities are assigned the same numerical values by virtually all expert observers: for example, virtually all experts will agree that a fair coin will show either side with the same probability 1/2; or that a fair die will show any one of its six sides with the same probability 1/6; etc. Other probabilities will typically be assessed differently by different people: for example, different experts on horse racing will usually assign different numerical probabilities to a particular horse coming in first in a given race. Probabilities of the former type may be called objective probabilities, while those of the latter type may be called subjective probabilities.

Abstract: Comparisons between observed behavior and predicted response from a mathematical representation are often not consistent. Through Bayesian inference, use can be made of the data and parameters of the model to arrive at a better correlation. The corresponding objective function is minimized in an iterative manner by means of a modified Newton-Raphson scheme requiring first order derivatives of the measured quantities with respect to the parameters.

Abstract: Computers are being investigated as diagnostic aids in many fields of medicine. Models employing Bayes' theorem, a statistical formula, commonly are used to supply valuable information on the likelihood of each disease in the differential diagnosis to help the clinician make the diagnosis. However, knowledge of elementary decision analysis is beneficial to help understand the current and potential uses of these models. We discussed Bayes' theorem as an introduction to decision analysis. Moreover, we described a Bayesian model for the differential diagnosis of leukocoria to illustrate the application of computers to ophthalmologic diagnosis.

Abstract: : Statistical inference is the activity of characterizing the parameters of mathematical models by utilizing available sampling data. This report discusses as a specific motivation the modeling of reliability problems and deals only with inference while avoiding the larger area of decision theory. The classical and Bayesian approaches to evaluating the parameter of the familiar exponential reliability model are compared. Classically, model parameters are unknown constants which can be estimated. From the Bayesian viewpoint model parameters are treated as distributed random variables. As is also ture of the classical maximum likelihood method, the determining or informational impact of the sampling data is represented completely by the likelihood function. Operationally, Bayesian inference involves applying Bayes theorem, a celecbrated consequence of conditional probability theory. The relevant probability background is developed and Bayes theorem derives. Bayesian inference has the very appealing capacity to incorporate previous information as well as current sampling inputs. Classical results are reproduced in the limiting forms of this involving noninformative prior distributions. Several application examples are discussed illustrating the use of both continuously and discretely distributed data and in one case emphasizing numerical methods.

Abstract: : Modern forecasting and estimation techniques provide not only point estimates of unknown variables, but also associated intervals which reflect the expected accuracy of those estimates. Often different real world forecasts, produce conflicting estimates and associated intervals of accuracy. This paper addresses the issue of how to make sure of such estimates. It is argued that to both Classical and Bayesian statisticians the problem is essentially trivial. However, it is demonstrated that the assumptions required for a formal Bayesian approach are so sensitive to small changes, that the Bayesian approach has dubious advantages over simple intuition. With the Classical attitude being unhelpful in practice, it is argued that techniques should be developed which combine formal Bayesian updating procedures with intuition. Two possible techniques are explored. The first uses Bayesian updating with parameterized likelihood functions. With suitable interpretation of the parameters, decision makers can use their intuition to choose appropriate parameters. The second technique allows for a number of alternate likelihood functions, combined probabilistically according to the decision maker's judgment. (Author)

Abstract: SUMMARY Bayesian inference in finite populations uses probability models at two stages: (i) to describe relationships among population units and (ii) to express uncertainty concerning the values of parameters appearing at stage (i). Here we consider the Bayes posterior distribution of the population total when a multivariate normal regression model is used at stage (i), with a diffuse prior distribution on the regression coefficients. We study the situation where the stage (i) model is in error because an important regressor is omitted, and we show that in balanced samples such errors do not affect the posterior distribution. Cases where the covariance matrix contains an unknown scale parameter or is itself misspecified are also considered.

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.

Abstract: A theory of statistical inference, applicable to both classical and quantum systems, is first presented in an original version based on von Neumann algebras, then in aC*-algebraic generalization due to S. Gudder.