Showing papers on "Bayes' theorem published in 1987"

Reconciling Bayesian and Frequentist Evidence in the One-Sided Testing Problem

Abstract: For the one-sided hypothesis testing problem it is shown that it is possible to reconcile Bayesian evidence against H 0, expressed in terms of the posterior probability that H 0 is true, with frequentist evidence against H 0, expressed in terms of the p value. In fact, for many classes of prior distributions it is shown that the infimum of the Bayesian posterior probability of H 0 is equal to the p value; in other cases the infimum is less than the p value. The results are in contrast to recent work of Berger and Sellke (1987) in the two-sided (point null) case, where it was found that the p value is much smaller than the Bayesian infimum. Some comments on the point null problem are also given.

Shafer-dempster reasoning with applications to multisensor target identification systems

Abstract: Bayes theory is provably optimal whenever all the sensor sources are contributing information at a single Bayesian level of abstraction. Often, however, this is not the case. Shafer-Dempster reasoning is a generalization of Bayes reasoning that offers a way to combine uncertain information from disparate sensor sources with different levels of abstraction. Shafer-Dempster logic is discussed and described, and realistic examples are provided of its use drawn from the field of multisensor target identification systems and on simulating its operation.

The Probability Approach to the Treatment of Uncertainty in Artificial Intelligence and Expert Systems

Abstract: Arguments are adduced to support the claim that the only satis- factory description of uncertainty is probability. Probability is described both mathematically and interpretatively as a degree of belief. The axio- matic basis and the use of scoring rules in developing coherence are discussed. A challenge is made that anything that can be done by alternative methods for handling uncertainty can be done better by probability. This is demonstrated by some examples using fuzzy logic and belief functions. The paper concludes with a forensic example illustrating the power of probability ideas.

Robust empirical bayes analyses of event rates

Abstract: A collection of I similar items generates point event histories; for example, machines experience failures or operators make mistakes. Suppose the intervals between events are modeled as iid exponential (λ i , or the counts as Poisson (λ i t i ,) for the ith item. Furthermore, so as to represent between-item variability, each individual rate parameter, λ i , is presumed drawn from a fixed (super) population with density g λ (·; θ), θ being a vector parameter: a parametric empirical Bayes (PEB) setup. For g λ, specified alternatively as log-Student t(n) or gamma, we exhibit the results of numerical procedures for estimating superpopulation parameters ll and for describing pooled estimates of the individual rates, λ i , obtained via Bayes's formula. Three data sets are analyzed, and convenient explicit approximate formulas are furnished for λ i estimates. In the Student-t case, the individual estimates are seen to have a robust quality.

Practical issues in constructing a Bayes' belief network

Abstract: Bayes belief networks and influence diagrams are tools for constructing coherent probabilistic representations of uncertain expert opinion. The construction of such a network with about 30 nodes is used to illustrate a variety of techniques which can facilitate the process of structuring and quantifying uncertain relationships. These include some generalizations of the "noisy OR gate" concept. Sensitivity analysis of generic elements of Bayes' networks provides insight into when rough probability assessments are sufficient and when greater precision may be important.

Examining Correlates of Diversity

Abstract: Statistical methods are presented for studying “correlates of diversity”: characteristics of educational organizations which predict dispersion on the dependent variable. The conceptual framework for these methods distinguishes between variance heterogeneity that arises from educational program effects and heterogeneity that merely reflects heterogeneity of variance of inputs. The estimation theory is empirical Bayes, requiring probabilistic models both for the data and for the random dispersion parameters from each of many groups. Two strategies are considered, one based on exact distribution theory and the second based on an asymptotic normal approximation. The accuracy of the approximation is evaluated analytically and its use illustrated by an analysis of mathematics achievement data from a random sample of U.S. high schools.

Empirical Bayes estimation of parameters for n polygenic binary traits

Abstract: The conditional probability of an observation in a subpopulation i (a combination of levels of explanatory variables) falling into one of 2\" mutually exclusive and exhaustive categories is modelled using a normal integral in n-dimensions. The mean of subpopulation i is written as a linear combination of an unknown vector 8 which can include « fixed >> effects (e.g., nuisance environmental effects, genetic group effects) and « random » effects such as additive genetic value

Bug Distribution and Statistical Pattern Classification.

Abstract: A model (called the rule space model) which permits measuring cognitive skill acquisition, diagnosing cognitive errors, detecting the weaknesses and strengths of knowledge possessed by individuals was introduced earlier. This study further discusses the theoretical foundation of the model by introducing “bug distribution” and hypothesis testing (Bayes' decision rules for minimum errors) for classifying subjects into their most plausible latent state of knowledge. The model is illustrated with the domain of fraction arithmetic and compared with the results obtained from a conventional artificial intelligence approach.

Inference and Prediction for a General Order Statistic Model with Unknown Population Size.

Abstract: Suppose that the first n order statistics from a random sample of N positive random variables are observed, where N is unknown. This, the general order statistic model, has been applied to the study of market penetration, capture—recapture, burn-in in repairable systems, software reliability growth, the estimation of the number of individuals exposed to radiation, and the estimation of the number of unseen species. Inference is to be made about the unknown parameters, especially N, and future observations are to be predicted. A Bayes empirical Bayes approach to inference is presented. This permits the comparison of competing, perhaps nonnested, models for the distribution of the random variables in a natural way. It also provides easily implemented inference and prediction procedures that avoid the difficulties of non-Bayesian methods. One such difficulty is that the maximum likelihood estimator of N may be infinite. Results are given for the case in which vague prior information about the model ...

Numerical specification of discrete least favorable prior distributions

Abstract: A broad class of statistical decision problems are solved by minimax procedures which are Bayes with respect to discrete least favorable prior distributions. A general algorithm for specifying such distributions is presented which exploits the statistical properties of minimax procedures. The algorithm is demonstrated by characterizing the procedure which simultaneously minimizes a Bayes risk and a maximum risk under different loss functions in a simple multi-objective decision problem.

Bayes Linear Estimators for Randomized Response Models

Abstract: Bayes linear estimators provide simple Bayesian methods and require a minimum of prior specification. In this article, Bayes linear estimators are derived for a variety of randomized response models. Randomized response aims to reduce false responses on sensitive questions, at the expense of some loss of information in each observation. In this context, Bayesian methods are attractive because they permit the incorporation of potentially useful prior information. The basic principle of randomized response is that an interviewee answers one of two or more different questions, depending on the outcome of some randomizing device. The interviewer does not know which question has been answered. In this way, it is hoped, the interviewee will feel able to answer sensitive questions honestly, where direct questioning might have produced false responses. Two versions of randomized response are examined: the original formulation of Warner (1965) and the unrelated question method of Simmons (Horvitz, Shah, a...

A multinomial Bayesian approach to the estimation of population and vocabulary size

Abstract: We approach estimation of the size of a population or a vocabulary through a Bayesian analysis of the multinomial distribution. We view the sample as being generated from such a distribution with an unknown number of cells and unknown cell probabilities, and develop a Bayesian procedure to estimate the number of cells and the coverage of the sample. The prior distribution of the number of cells is arbitrary. Given that number, the cell probabilities are assumed to follow a symmetric Dirichlet prior. A two-stage approach is developed for use when the flattening constant of the latter prior cannot be specified in advance. Our procedures are applied to samples of butterflies, insect species and alleles, to the works of Shakespeare and Joyce, and to Eldridge's sample of English words.

The theorem of Bayes as a clinical research tool.

Abstract: The theorem of Bayes is a powerful research tool that has a multitude of clinical applications. Its use has been somewhat restricted because of the intrinsic complexity of Bayesian theory and the need for computer support. We explain herein the Theorem of Bayes in practical terms that specifically apply to clinical research.

The probability of inevitability: the inherent impossibility of validating criteria for brain death or 'irreversibility' through clinical studies.

Abstract: A great need persists for diagnostic criteria for both brain death in young children and irreversible loss of consciousness at all ages. This article examines the inferences derived from a hypothetical confirmatory study in which all of the N patients who fulfilled the criterion did in fact experience brain death (irreversibility). A Bayesian methodology proves that, for N in the range of a large clinical study, estimations of prior probabilities are, for all practical purposes, irrelevant to the calculation of the posterior probabilities. The risk of a false positive diagnosis for the next patient who meets the criterion is approximately 1/(N + 2). The chance of at least one false positive diagnosis among the next (N + 1) patients who meet the criterion is around 50 per cent. Thus, achievement of the requisite moral certainty of a declaration of death (irreversibility) necessitates an impossibly large N for the study. This does not mean that one cannot diagnose death, but rather that the validity of the diagnostic criteria must be self-evident on a priori grounds, and that confirmatory studies are necessarily either inadequate or superfluous.

The Nearest Neighbor and the Bayes Error Rates

Abstract: The (k, l) nearest neighbor method of pattern classification is compared to the Bayes method If the two acceptance rates are equal then the asymptotic error rates satisfy the inequalities Ek,l + 1 ? E*(?) ? Ek,l dE*(?), where d is a function of k, l, and the number of pattern classes, and ? is the reject threshold for the Bayes method An explicit expression for d is given which is optimal in the sense that for some probability distributions Ek,l and dE* (?) are equal

On Analytic Empirical Bayes Estimation of Failure Rates

Abstract: The estimation of plant accident rates and component failure rates is addressed within the framework of a parametric empirical Bayes approach. The observables, the numbers of failures recorded in various similar systems, obey the Poisson probability law. The parameters of a common gamma prior distribution are determined by a special moment matching method such that the results are consistent with classical (fiducial) confidence limits. Relations between Bayesian, classical, and Stein's estimation are discussed. The theory of the method is fully developed, although the suggested procedure itself is relatively simple. Solutions exist and they are in allowed ranges for all practical cases, including small samples and clustered data. They are also unbiased for large samples. Numerical examples are analyzed to illustrate the method and to allow comparisons with other methods.

A Bayes Reliability Growth Model for A Development Testing Program

Abstract: The problem of estimating the reliability of a system during development is considered. The development process has several stages at each stage binomial test data are obtained by testing a number of such systems on a success/fail basis. Marginal posterior distributions are derived under the assumption that the development process constrains the reliabilities to be nondecreasing and that the prior distribution for reliability at each stage is uniform. Simulation models are designed to facilitate testing for the validity and computation of the Bayesian model with ordered reliabilities as well as to compare results with other reliability growth models.

R.A. Fisher-a True Bayesian?

Abstract: Summary R.A. Fisher's claim that his fiducial argument uses the term 'probability' in the same sense as used by the Rev. Thomas Bayes is fully justifiable. But, while probability statements concerning parameters can be made, these parameters cannot be regarded as random variables in the sense of Kolmogoroff. Fisher was not a 'Bayesian' in the main current sense of the word. In the first edition (1956) of Statistical Methods and Scientific Inference, Ch. V, ? 6, R.A. Fisher discusses the logical situation arising when data of two kinds are available, one kind such as to give a fiducial distribution for the unknown parameter, the other such as to yield only a likelihood function. He imagines a charged particle recorder capable of being switched on or off at precisely chosen times. The recorder can be set to record the time at which a particle passes through, or alternatively to record whether any particles pass through in a specific time interval. Assuming the particles form a Poisson process with unknown rate 0 particles per unit time, the time t elapsing between switching on and observing the first particle has cumulative probability P(t, 0) = exp {-tO}, while the

Empirical bayes methods in sequential estimation

Abstract: The problem of Bayes sequential point estimation when the prior is not completely known is considered. When auxiliary data are available, a general empirical Bayes approach to the problem is proposed. The empirical Bayes procedures are shown to be asymptotically non deficient in certain cases involving exponentially and normally distributed data, provided that the number of auxiliary observations becomes large as the cost per observation becomes small.

Bayesian log linear estimates for three-way contingency tables

Abstract: SUMMARY This paper presents a Bayesian method to estimate the expected proportions in a three-way contingency table appropriate when prior knowledge about the main, firstand second-order interaction effects can be described by a particular kind of exchangeability assumption. The proposed Bayes estimates are calculated by finding those values of the effects which maximize the resulting posterior distribution and can be used to explore the possibility that a nonsaturated submodel, such as independence or conditional independence, fits the data. This extends the work of Leonard (1975) for two-dimensional tables. We discuss numerical strategies to solve the estimating equations and point out how an incorrect choice of values for the 'indifference' case, as made by previous authors, can have serious effects on the convergence of the algorithms. The method is exemplified by a survey on skin cancer and data on voting transitions in British elections.

Human versus machine: a comparison of a computer 'expert system' with human experts in the diagnosis of vaginal discharge.

Abstract: A model system has been designed which generates ‘case’ of vaginal discharge. Each such case is presented to a human for diagnosis, and this is then compared with a computer diagnosis using two forms of Bayes' theorem. Six subjects (2 medical; 4 non-medical) participated in the trial and each examined 100 successive ‘cases’. When the human had forewarning of the trial and full access to the knowledge-base their performance was superior to that of Bayes' theorem using positive features only and equivalent to that using both positive and negative features. When the trial was repeated without forewarning the human performance was markedly inferior to that of the machine. It is concluded: (1) that human and computer-aided diagnosis can be of approximately equal efficiency for complex and non-definitive data; (2) that the imperfections of human memory give an obvious potential advantage to the machine in this type of situation.

Is Shafer general Bayes

Abstract: This paper examines the relationship between Shafer's belief functions and convex sets of probability distributions. Kyburg's (1986) result showed that belief function models form a subset of the class of closed convex probability distributions. This paper emphasizes the importance of Kyburg's result by looking at simple examples involving Bernoulli trials. Furthermore, it is shown that many convex sets of probability distributions generate the same belief function in the sense that they support the same lower and upper values. This has implications for a decision theoretic extension. Dempster's rule of combination is also compared with Bayes' rule of conditioning.

Bayesian control in mixture models

Abstract: The problem is to choose values for independent variables in a regression mixture model with the intent of controlling the output toward a specified target value. This problem is posed as a Bayesian decision problem. The calculation of Bayes rules is shown to require solution of a quadratic programming problem. An example based on an article by Snee (1981) concerning gasoline blends is discussed.

Software for Bayesian Analysis: Current Status and Additional Needs

Abstract: : This document makes an attempt to provide comprehensive information about the existing software for data analysis within the Bayesian paradigm. The paucity of programs seems to indicate that the Bayesian software available for widespread use is still in its infancy. We have a long way to go before a general purpose Bayesian Statistical Analysis Package is made available. Alternatives for reaching this goal quickly are presented in the concluding section. Keywords: Bayesian software; ADA(Computer Assisted Data Analysis Monitor); BRAP(Bayesian Regression Analysis Program).