Showing papers on "Bayesian probability published in 1987"
TL;DR: For the one-sided hypothesis testing problem, it is shown in this article that the infimum of the Bayesian posterior probability of H 0 is equal to the p value, while for some classes of prior distributions the infum is less than or equal to p value.
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.
390 citations
TL;DR: The impact of “uncertain evidence” can be (formally) represented by Dempster conditioning, in Shafer's framework, in the framework of convex sets of classical probabilities by classical conditionalization.
378 citations
01 Mar 1987
TL;DR: It is shown that the causal relationships in a general diagnostic domain can be used to remove the barriers to applying Bayesian classification effectively and provides insight into which notions of "parsimony" may be relevant in a given application area.
Abstract: The issue of how to effectively integrate and use symbolic causal knowledge with numeric estimates of probabilities in abductive diagnostic expert systems is examined. In particular, a formal probabilistic causal model that integrates Bayesian classification with a domain-independent artificial intelligence model of diagnostic problem solving (parsimonious covering theory) is developed. Through a careful analysis, it is shown that the causal relationships in a general diagnostic domain can be used to remove the barriers to applying Bayesian classification effectively (large number of probabilities required as part of the knowledge base, certain unrealistic independence assumptions, the explosion of diagnostic hypotheses that occurs when multiple disorders can occur simultaneously, etc.). Further, this analysis provides insight into which notions of "parsimony" may be relevant in a given application area. In a companion paper, Part Two, a computationally efficient diagnostic strategy based on the probabilistic causal model discussed in this paper is developed.
257 citations
TL;DR: The real challenge probability poses to artificial intelligence is to build systems that can design probability arguments, and the real challenge statisticians pose to statistics is to explain how statisticiansDesign probability arguments.
Abstract: Historically, the study of artificial intelligence has emphasized symbolic rather than numerical computation. In recent years, however, the practical needs of expert systems have led to an interest in the use of numbers to encode partial confidence. There has been some effort to square the use of these numbers with Bayesian probability ideas, but in most applications not all the inputs required by Bayesian probability analyses are available. This difficulty has led to widespread interest in belief functions, which use probability in a looser way. It must be recognized, however, that even belief functions require more structure than is provided by pure production systems. The need for such structure is inherent in the nature of probability argument and cannot be evaded. Probability argument requires design as well as numerical inputs. The real challenge probability poses to artificial intelligence is to build systems that can design probability arguments. The real challenge artificial intelligence poses to statistics is to explain how statisticians design probability arguments.
143 citations
TL;DR: The authors developed a Bayesian test of portfolio efficiency and derived a computationally convenient posterior-odds ratio, which indicates that significance levels higher than the traditional 0.05 level are recommended for many test situations.
118 citations
TL;DR: In this paper, a representative cross-sectional property value data set is used to translate a range of priors in covariate selection typical of hedonic property value studies into a rangeof posterior estimates.
Abstract: Hedonic price models are widely employed to estimate implicit prices for bundled attributes. Residential property value studies dominate these applications. Using a representative cross-sectional property value data set, we employ Bayesian methods to translate a range of priors in covariate selection typical of hedonic property value studies into a range of posterior estimates. We also formulate priors regarding measurement error in individual covariates and compute the ranges of resulting posterior means. Finally, we empirically demonstrate that a greater and more systematic use of prior information drawn from one's own data and from other studies can break the collinearity deadlock in this data.
91 citations
71 citations
69 citations
TL;DR: Bayesian methods are used on the results of the preliminary study to obtain a posterior distribution representing the state of knowledge of the parameters of interest that provides the probability that the experimental regimen is superior to the standard by any particular amount.
62 citations
Journal Article•
TL;DR: In this article, the posterior probability for a general comparative parameter is formulated as a finite sum of the beta-binomial type, which can be parameterized in terms of a difference and a ratio of two proportions as well, and the analysis is extended to concern the non-null values of the three usual parameters of association.
Abstract: Altham (1969) derived a relation between the cumulative posterior probability for association and the exact p-value in a 2 x 2 table. But she found that, in general, the exact posterior distribution of the chosen measure of association (odds ratio) was not easy to deal with. This paper covers generalizations of the Bayesian analysis in two directions. First, the posterior probability is formulated for a general comparative parameter, which implies that the analysis is not limited in application to problems involving odds ratio but can be parameterized in terms of a difference and a ratio of two proportions as well. Second, the formal analysis is extended to concern the non-null values of the three usual parameters of association. Under the model of a general beta (or a particular rectangular) prior distribution, the parameter-specific posterior functions are express- ible as finite sums of the beta-binomial type. The posterior distributions are immediately intelli- gible and provide for a uniform basis for the Bayesian analogues of interval estimation, point estimation and significance testing.
61 citations
TL;DR: A normal logistic model is developed that accommodates the incidences of all tumor types or sites observed in the current experiment simultaneously as well as their historical control incidences.
Abstract: Statistical analyses of simple tumor rates from an animal experiment with one control and one treated group typically consist of hypothesis testing of many 2 X 2 tables, one for each tumor type or site. The multiplicity of significance tests may cause excessive overall false-positive rates. This paper presents a Bayesian approach to the problem of multiple significance testing. We develop a normal logistic model that accommodates the incidences of all tumor types or sites observed in the current experiment simultaneously as well as their historical control incidences. Exchangeable normal priors are assumed for certain linear terms in the model. Posterior means, standard deviations, and Bayesian P-values are computed for an average treatment effect as well as for the effects on individual tumor types or sites. Model assumptions are checked using probability plots and the sensitivity of the parameter estimates to alternative priors is studied. The method is illustrated using tumor data from a chronic animal experiment.
TL;DR: In this paper, the authors considered the problem of predicting the ordered failure times of n-k businesses, using the conditional probability density function (CPDF) with respect to a Pareto distribution.
Abstract: Suppose that the length of time in years for which a business operates until failure has a Pareto distribution. Let x1 ≤ x2 x3 ≤…≤zk denote the survival lifetimes of the first k of a random sample of n businesses. Bayesian predictions are to be made on the ordered failure times of t h e remaining (n-k) businesses, using the conditional probability density function. Examples are given to illustrate our results.
TL;DR: In this article, the posterior moments and predictive probabilities are proportional to ratios of B. C. Carlson's multiple hypergeometric functions, and closed-form expressions are developed for nested reported sets, when Bayesian estimates can be computed easily from relative frequencies.
Abstract: Bayesian methods are given for finite-category sampling when some of the observations suffer missing category distinctions. Dickey's (1983) generalization of the Dirichlet family of prior distributions is found to be closed under such censored sampling. The posterior moments and predictive probabilities are proportional to ratios of B. C. Carlson's multiple hypergeometric functions. Closed-form expressions are developed for the case of nested reported sets, when Bayesian estimates can be computed easily from relative frequencies. Effective computational methods are also given in the general case. An example involving surveys of death-penalty attitudes is used throughout to illustrate the theory. A simple special case of categorical missing data is a two-way contingency table with cross-classified count data xij (i = 1, …, r; j = 1, …, c), together with supplementary trials counted only in the margin distinguishing the rows, yi (i = 1, …, r). There could also be further supplementary trials report...
01 Jan 1987
TL;DR: The thorniest part of the Bayesian combination procedure is the assessment of a likelihood function by the decision maker to represent his beliefs regarding the quality of the information and the nature of the dependence among the sources.
Abstract: Imagine a decision maker who has heard from one or more information sources regarding the probability of some future event and who desires to use this information to revise his personal beliefs concerning the event. One approach to this problem involves the decision maker treating the probabilities as data in a Bayesian inferential problem, the output of which is an updated probability regarding the event in question. The thorniest part of the Bayesian combination procedure is the assessment of a likelihood function by the decision maker to represent his beliefs regarding the quality of the information and, in the case of multiple sources, the nature of the dependence among the sources.
10 Jun 1987
TL;DR: Taking the Bayesian approach in solving the discrete-time parameter estimation problem has two major results: the unknown parameters are legitimately included as additional system states, and the computational objective becomes calculation of the entire posterior density instead of just its first few moments.
Abstract: Taking the Bayesian approach in solving the discrete-time parameter estimation problem has two major results: the unknown parameters are legitimately included as additional system states, and the computational objective becomes calculation of the entire posterior density instead of just its first few moments. This viewpoint facilitates intuitive analysis, allowing increased qualitative understanding of the system behavior. With the actual posterior density in hand, the true optimal estimate for any given loss function may be calculated. While the computational burden may preclude on-line use, this provides a clearly justified baseline for comparison. These points are demonstrated by analyzing a scalar problem with a single unknown, and by comparing an established point estimator's performance to the true optimal estimate.
TL;DR: The purpose and environment of Bayesian forecasting systems are described and reviewed, stressing foundational concepts, component models, the discount concept and intervention, and interactive analyses using a purpose-built suite of APL functions.
Abstract: We describe and review the purpose and environment of Bayesian forecasting systems, stressing foundational concepts, component models, the discount concept and intervention, and interactive analyses using a purpose-built suite of APL functions.
TL;DR: In this paper, Dempster's rule is shown to be at best a special case of the rule derived in connection with second-order probabilities, which represents a restriction of a full Bayesian analysis.
Abstract: A second-order probability Q(P) may be understood as the probability that the true probability of something has the value P. “True” may be interpreted as the value that would be assigned if certain information were available, including information from reflection, calculation, other people, or ordinary evidence. A rule for combining evidence from two independent sources may be derived, if each source i provides a function Q i (P). Belief functions of the sort proposed by Shafer (1976) also provide a formula for combining independent evidence, Dempster's rule, and a way of representing ignorance of the sort that makes us unsure about the value of P. Dempster's rule is shown to be at best a special case of the rule derived in connection with second-order probabilities. Belief functions thus represent a restriction of a full Bayesian analysis.
TL;DR: In this article, the generalized least squares estimator was used to test linear hypotheses confidence regions for linear parameters and regression functions using Bayesian methods and structural inference experimental design methods, and the confidence regions were used to model causal relationships.
Abstract: Statistical problems in modelling causal relationships estimating linear parameters estimating linear parameters using additional information admissibility and improvements of the generalized least squares estimator testing linear hypotheses confidence regions for linear parameters and regression functions Bayesian methods and structural inference experimental design methods.
TL;DR: Bayesian image processing formalisms which incorporatea priori information about valued-uncorrelated and valued-correlated (patterned) source distributions are introduced and the corresponding iterative algorithms are derived using the EM technique.
01 Jan 1987
TL;DR: This paper presents a combination Bayesian/Item Response Theory procedure for pooling performance on a particular objective with information about an examinee’s overall test performance in order to produce more stable objective scores.
Abstract: This paper presents a combination Bayesian/Item Response Theory procedure for pooling performance on a particular objective with information about an examinee’s overall test performance in order to produce more stable objective scores. The procedure, including the calculation of a posterior distribution, is described. A split-half cross validation study finds that a credibility interval based on the posterior distribution is sufficiently accurate to be useful for scoring reports for teachers.
TL;DR: In this paper, the authors discuss two solutions to this problem: one proposed by several philosophers but criticised in detail by Clark Glymour; another set forth by Daniel Garber and developed by Ellery Eells.
Abstract: I begin this paper with a problem that Clark Glymour has posed for Bayesians, involving the discovery that some old evidence is found to support a new theory. I discuss two solutions to this problem: one proposed by several philosophers but criticised in detail by Glymour; another set forth by Daniel Garber and developed by Ellery Eells. I then go on to describe a situation in which a new theory is found to be supported by some old data-a discovery which produces a rise in confidence in the new theory, but which does not seem to be adequately analysable by classical Bayesian theory. It is the position of this paper that this difficulty for classical Bayesian theory is not resolvable by the Garber-Eells approach and that the aspect of Bayesian theory giving rise to this problem is quite different from the one on which Garber and Eells have focused.
Journal Article•
TL;DR: The Theorem of Bayes is explained in practical terms that specifically apply to clinical research.
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.
TL;DR: In this article, the authors developed rules based on the concept of a tolerance region R, such that a vector of parameters of interest, 0 say, is contained in R with a specified probability 1 -a which holds on average over all possible samples.
Abstract: Bayesian rules are developed for the determination of sample sizes for sampling from multinomial distributions. We develop rules based on the concept of a tolerance region R, such that a vector of parameters of interest, 0 say, is contained in R with a specified probability 1 -a which holds on average over all possible samples. We consider two classes of region (i) ellipsoids which are centred at the posterior expectation of 0 conditional on a sample matrix X and whose shape and orientation are determined by a given symmetric matrix A, and (ii) hyper-cubic regions consisting of intervals which are symmetric about the individual posterior means 6i the elements of 0. These procedures are applied to the sizing of market research surveys, for which the multinomial distribution is an appropriate model.
TL;DR: In this article, the estimation of plant accident rates and component failure rates is addressed within the framework of a parametric empirical Bayes approach, where the observables, the number of failures recorded in various similar systems, obey the Poisson probability law.
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.
TL;DR: Two examples of the use of a method based on Gauss-Hermite product rules proposed by Naylor & Smith (1982) using an IBM PC AT02 using a program which can treat up to six parameters.
Abstract: The routine use of Bayesian methods in scientific research has been hindered by the absence of appropriate, efficient software for the calculation and graphical presentation of posterior distributions for different combinations of likelihood and prior. In this paper we give two examples of the use of a method based on Gauss-Hermite product rules proposed by Naylor & Smith (1982). The analyses were carried out on an IBM PC AT02 using a program which can treat up to six parameters.
TL;DR: The application of the widely used Box-Jenkins approach to explore and model the structures present in diverse reliability data is considered and it is concluded that the existing time series approaches require further specialisation in order to better model the characteristics of reliability problems.
TL;DR: 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.
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.
TL;DR: In this paper, 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.
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