Showing papers on "Bayes' theorem published in 1975"

Simultaneous estimation of parameters in different linear models and applications to biometric problems

Abstract: Empirical Bayes procedure is employed in simultaneous estimation of vector parameters from a number of Gauss-Markoff linear models. It is shown that with respect to quadratic loss function, empirical Bayes estimators are better than least squares estimators. While estimating the parameter for a particular linear model, a suggestion has been made for distinguishing between the loss due to decision maker and the loss due to individual. A method has been proposed but not fully studied to achieve balance between the two losses. Finally the problem of predicting future observations in a linear model has been considered.

k-nearest-neighbor Bayes-risk estimation

Abstract: Nonparametric estimation of the Bayes risk R^\ast using a k -nearest-neighbor ( k -NN) approach is investigated. Estimates of the conditional Bayes error r(X) for use in an unclassified test sample approach to estimate R^\ast are derived using maximum-likelihood estimation techniques. By using the volume information as well as the class representations of the k -NN's to X , the mean-squared error of the conditional Bayes error estimate is reduced significantly. Simulations are presented to indicate the performance of the estimates using unclassified testing samples.

Bayesian dynamic programming

Abstract: We consider a non-stationary Bayesian dynamic decision model with general state, action and parameter spaces. It is shown that this model can be reduced to a non-Markovian (resp. Markovian) decision model with completely known transition probabilities. Under rather weak convergence assumptions on the expected total rewards some general results are presented concerning the restriction on deterministic generalized Markov policies, the criteria of optimality and the existence of Bayes policies. These facts are based on the above transformations and on results of Hindererand Schal.

Prediction Intervals and Empirical Bayes Confidence Intervals

Abstract: Approximate parametric prediction intervals are obtained for an unobserved random variable when the amount of data on which to base the estimation is large. Applications include the construction of approximate confidence intervals in empirical Bayes estimation.

Estimating Population Size with Exponential Failure

Abstract: We are given J observations obtained by truncated sampling of a population of N items which fail independently according to the exponential distribution, where both N and the scale parameter of the exponential are unknown. Estimates of N are developed, and compared. These are conditional and unconditional maximum likelihood estimates, and a class of Bayes modal estimates. On the basis of second-order asymptotic properties, one of the Bayes estimates is singled out as most desirable. This estimator is also good for estimating mean life, but for estimating failure rate, the maximum likelihood estimates are preferable.

Personal probabilities of probabilities

Abstract: By definition, the subjective probability distribution of a random event is revealed by the (‘rational’) subject's choice between bets — a view expressed by F. Ramsey, B. De Finetti, L. J. Savage and traceable to E. Borel and, it can be argued, to T. Bayes. Since hypotheses are not observable events, no bet can be made, and paid off, on a hypothesis. The subjective probability distribution of hypotheses (or of a parameter, as in the current ‘Bayesian’ statistical literature) is therefore a figure of speech, an ‘as if’, justifiable in the limit. Given a long sequence of previous observations, the subjective posterior probabilities of events still to be observed are derived by using a mathematical expression that would approximate the subjective probability distribution of hypotheses, if these could be bet on. This position was taken by most, but not all, respondents to a ‘Round Robin’ initiated by J. Marschak after M. H. De-Groot's talk on Stopping Rules presented at the UCLA Interdisciplinary Colloquium on Mathematics in Behavioral Sciences. Other participants: K. Borch, H. Chernoif, R. Dorfman, W. Edwards, T. S. Ferguson, G. Graves, K. Miyasawa, P. Randolph, L. J. Savage, R. Schlaifer, R. L. Winkler. Attention is also drawn to K. Borch's article in this issue.

A bayesian model for portfolio selection and revision

Abstract: IN PORTFOLIO ANALYSIS, the basic setting is that of an individual or a group of individuals making inferences and decisions in the face of uncertainty about future security prices and related variables. Formal models for decision making under uncertainty require inputs such as probability distributions to reflect a decision maker's uncertainty about future events and utility functions to reflect a decision maker's preferences among possible consequences [30]. Moreover, when a series of interrelated decisions is to be made over time, the decision maker should (1) revise his probability distributions as new information is obtained and (2) take into account the effect of the current decision on future decisions. In terms of formal models of the decision-making process, probability revision can be accomplished by using Bayes' theorem and the interrelationships among the decisions can be taken into consideration by using dynamic programming to determine optimal decisions. Since portfolio selection and revision involves a series of interrelated decisions made over time, formal portfolio models should, insofar as possible, incorporate these features. A search of the extensive literature concerning portfolio models indicates, however, that such models have ignored one or both of these features. Since Markowitz [18] developed his original model of portfolio selection, a considerable amount of work has been conducted in the area of mathematical portfolio analysis, and much of this work is summarized by Sharpe [31] and Smith [33]. Although the emphasis in portfolio analysis has been primarily on single-period models and portfolio selection, multiperiod models and portfolio revision are investigated by Tobin [35], Smith [32], Mossin [21], Pogue [22], Chen, Jen, and Zionts [3], and Hakansson [13, 14]. In addition, general multiperiod models of consumption-investment decisions are developed by Hakansson [10, 11, 12], Merton [19], Samuelson [29], Fama [6], and Meyer [20]. However, it is generally assumed that the probability distributions of interest are completely specified and that they are unaffected by new information, implying that the portfolio revision models do not involve probability revision over time. Bayesian models have received virtually no attention in the portfolio literature. Mao and Siirndal [17] present a simple, discrete, single-period Bayesian model in which the returns from securities are related to the level of general business activity and information is obtained concerning business conditions. Kalymon [16] develops a model

A Note on Some Bayesian Nonparametric Estimates

Abstract: With respect to a general quadratic loss function, the Bayes rule for the mean of a probability distribution of unknown form is obtained, in the class of linear functions of the sample. The associated Bayes risk is also obtained. A number of recent results in the literature are shown to be direct corollaries of this result, and applications are given for the empirical distribution function of the sample.

Approximate Bayes Solutions to Some Nonparametric Problems

Abstract: The problem of making inferences about real functions of a probability distribution of unknown form is examined in a Bayesian nonparameteric framework. With respect to a general quadratic loss function, Bayes estimates within the class of linear combinations of a given set of functions on the sample space are obtained for general functions on the distribution space. The result is then used to derive Bayes polynomial estimates of the moments of the distribution.

A note on partially Bayes inference and the linear model

Abstract: Some results are outlined for estimation in a replicated linear model in which the variance changes from cell to cell in accordance with an inverse gamma prior density. The primary parameters of the linear model are, however, treated as fixed unknown parameters. The analysis thus exemplifies the treatment of problems in which the nuisance parameters have a prior distribution but the parameters of primary interest do not.

Generalized Measures of Information, Bayes' Likelihood Ratio and Jaynes' Formalism:

Abstract: This paper relates generalized measures of information to expected likelihood functions (ELFs) derived from Bayes' equation It then demonstrates that Jaynes' formalism may be extended to formulate a class of minimally-prejudiced models of which those derived from Shannon's measure are but a limiting and special case The role of probable inference and of information-minimizing models in design is commented on

Rates of Convergence in Empirical Bayes Estimation Problems: Continuous Case

Abstract: In this paper we construct sequences of estimators for a density function and its derivatives, which are not assumed to be uniformly bounded, using classes of kernel functions. Utilizing these estimators, a sequence of empirical Bayes estimators is proposed. It is found that this sequence is asymptotically optimal in the sense of Robbins (Ann. Math. Statist. 35 (1964) 1-20). The rates of convergence of the Bayes risks associated with the proposed empirical Bayes estimators are obtained. It is noted that the exact rate is $n^{-q}$ with $q \leqq \frac{1}{3}$. An example is given and an explicit kernel function is indicated.

A Bayesian Approach to the Identification of Children with Learning Disabilities

Abstract: It is suggested that identification of learning disabilities be divided into two stages, screening and clinical diagnosis, to permit more effective use of clinical resources. The sequential revision of the probability that a child has learning disability based on data about the child and implemented by Bayes' rule is proposed as an efficient screening method. Forty component disabilities, potential data for screening, were selected and classified. For want of objective data, subjective estimates of the relatedness of these disabilities and their statistical prevalence among children with and without learning disability (quantities needed for the Bayesian revision) were obtained from specialists in this field. Some of the disabilities were found to be potentially highly diagnostic and relatively independent and could be used for identification.

Minimum Bayes Risk t-Intervals for Multiple Comparisons

Abstract: The purpose is to develop interval estimates for differences between treatment means which, when used for testing, would be equivalent to the Waller-Duncan k-ratio rule for multiple comparisons. First the intervals are derived from a family of extended k-ratio Bayes testing rules (exchangeable priors and additive linear losses). Then the intervals are shown to be Bayes also (same priors and squared-error losses). Particularly striking is the dependence of width and location on the between-treatments F-ratio rather than the number of treatments. As F decreases the new k-ratio t-intervals shrink in width and shift toward zero.

A Bayesian Reliability Assessment of Complex Systems for Binomial Sampling

Abstract: This paper presents a reliability assessment procedure that systematically combines complete system binomial test data with lower level binomial test data obtained from either partial system or component tests. The procedure uses beta prior distributions of all reliabilities, Bayes theorem, and probability moments. The result is a posterior distribution of system reliability that can be used to determine Bayes point and interval estimates. The beta prior distributions evolve from data on predecessor systems similar to the system in question and engineering knowledge about what the various test-alternatives measure.

Bayesian Confidence Limits For The Availability of Systems

Abstract: This paper presents a numerical procedure for computing Bayes confidence intervals for the availability of a series or parallel system consisting of several statistically independent 2-state subsystems each having exponential distributions of life and repair time. The present results 1) provide a useful generalization of a previous result in which test data were limited to ``snap-shot'' observations on the subsystems operating states; 2) allow conventional life-test data for estimating the exponential parameters. The methods are suited to numerical evaluation using electronic computers as shown by particular examples.

A Course in Bayesian Statistics

Abstract: For the past four years we have been developing a new course in Bayesian statistics within the College of Education of the University of Iowa. This course runs in parallel with a divisional statistics sequence that focuses largely on classicial methods. My aim here is to describe some unusual features of our new course that may be of interest to others and that could be easily adapted and integrated into approaches differing somewhat from our own. Students for this course have come largely from our College of Education. A majority are students of Educational Measurement and Statistics, though others are students of Educational Psychology, Counseling and Guidance, Student Personnel, Mathematics Education, Physical Education, Educational Administration or Mathematical Statistics. The course is listed as a second semester statistics course with a sole prerequisite of a one-semester course in descriptive and classical inferential statistics. Usually students have had this course from the Blommers-Lindquist (1960) or Hayes (1973) text. The course does not have a calculus prerequisite; however, students typically understand the interpretation of an integral as a symbolic notation for an area under a curve. Students are carefully selected through pretest and many are advised to obtain a second semester of preparation. Our purpose in this course is to provide coverage of Bayesian methods through the Behrens-Fisher comparison of means, simple linear regression, multinomial analysis, and some simple work with several arc-sine transformed proportions. We seldom get all of this done because we spend part of the semester motivating Bayesian methods by looking at educational data and formulating some basic problems in simple regression and contingency-table terms. Our language is that of elementary decision theory, with least squares being taught as the minimization of squared-error loss, and generally the choice of estimator being seen as dependent on the loss structure of the problem. The unifying concept through all of this data analysis is that of conditional probability, which is thought of as probability in a specified subpopulation. Regression is taught as conditional expectation and variance in the subpopulation defined by the value of the conditioning variable, which may or may not be a random variable. Similarly, contingency tables are discussed and referenced to joint, marginal and conditional probabilities. Thus the student receives a coverage of probability theory that provides a clear motivation for Bayesian, that is, conditional probability inference. As we work through our analysis of data structured in the context of very real educational problems, we do, in fact, cover the derivation of the F, t, x2, x-2, and x-l distributions as they naturally appear in Bayesian analysis. But this work is almost always done only after the need for such distributions and derivations have been motivated by real problems. This contrasts with treatments which proceed in a mathematical way to state and prove a theorem and then give an example of how the theorem is used in application (e.g., Lindley, 1965). The main reason for motivating our work through data analysis (teaching this first and only then teaching some structure and mathematics) is that our primary goal is to develop in our students an ability to analyze data pertinent to educational problems. We believe that this approach develops a sensitivity to data that is hard to foster in a more mathematics-centered presentation. On the other hand, some of our students develop mathematical skills quite comparable to those of students coming out of a mathematical statistics department. Others, of course, do not. And in fact, some of our students really learn very little about the mathematical structure of Bayesian inference other than to know that under certain circumstances prior and posterior distributions are "t", x-2, or F, and that these distributions are used in certain ways for inference and decision making. But they do know precisely why and how to use these distributions to answer specific scientific questions. The next perhaps unique feature is that throughout this course we insist on the use of proper prior distributions. A major part of any analysis is the fitting, evaluation, re-evaluation, and defense of a proper prior distribution by the data analyst. We view statistical analysis and scientific report writing as principally an adversary proceeding in which a scientific investigator is attempting to convince * College of Education, Div. of Educational Psychology, Measurement & Statistics, Univ. of Iowa, Iowa City, IA 52242. I am grateful to my colleagues Paul Blommers, William Coffman, Leonard Feldt, and Robert Forsyth of the Division of Educational Psychology, Measurement and Statistics and Robert Hogg and George Woodworth of the Division of Mathematical Sciences for their comments on an earlier draft of this paper. I should also like to express my gratitude to the faculties of each of these Divisions for their toleratiQn of a vigorous presentation of ideas that conflict with their established practice. In this course we propound the thesis that, when available, Bayesian methods provide deeper analyses than do classical methods. At the same time we point out that Bayesian methods are not available for many problems and that the usefulness of a data analysis depends primarily on the care and skill exhibited in the analysis. We believe that our students benefit from the directness of these parallel presentations and we leave to them the choice of what methods they will use in their work. Instructional materials discussed in this paper are available from the Psychometric Research Department of the Iowa Testing Programs, Lindquist Center for Measurement, The University of Iowa, Iowa City, Iowa 52242.

Sequential Estimation of p with Squared Relative Error Loss

Abstract: We consider the sequential estimation of p, the probability of success in an infinite sequence of Bernoulli trials, when the loss incurred in stopping after n trials and estimating p by some function δn of the first n outcomes is taken to be [Formula: see text] The loss due to error of estimation is thus the symmetrized relative squared error, which is appropriate in applications when p may be near 0 or 1. We begin by finding a heuristic procedure for sequentially determining the sample size n which, with the usual terminal estimator of p, performs well for any fixed 0 0, however, this procedure is poor as p → 0 or 1. To remedy this defect, the uniform prior on p is introduced. The corresponding Bayes procedure is found and is shown to have a Bayes risk ∼2π√c as c → 0.

HEME: a self-improving computer program for diagnosis-oriented analysis of hematologic diseases

Abstract: HEME, a computer program for diagnosis-oriented analysis of hematologic diseases, accepts as input information about a patient and provides as output an ordered list of suggested diagnoses, an analysis of the logic behind these diagnoses, and a list of tests relevant to these diagnoses and not yet performed. The decision algorithm is based on Bayes' Theorem. Each disease in the system is individually analyzed, and the probability that the patient has the disease vs the probability that he does not is calculated. Bayesian methods of statistical inference areu tilized in that thep rior probabilities of the diseases and thper obabilities of findings in given diseases were initially estimated from the judgment of experienced hematologists with the intention that they be modified automatically as data are accumulated. This program is intended for use in teaching hematology, as an aid to diagnosis, and as a means for studying the diagnostic process.

Reliability after inspection

Abstract: The investigation is concerned with the derivation of relationships between the probability of having manufacturing defects, the probability of detecting a flaw, and the final reliability. Equations for the simple situation in which only one flaw can be present are used to introduce the relationships in a Bayes' theorem approach to the assessment of the final reliability. Situations which are prevalent in composites manufacturing are considered. Attention is given to a case involving the random occurrence of flaws on a laminate surface.

Comparison of a bayesian and a least squares method of educational prediction1

Abstract: The prediction systems under discussion apply where the following conditions obtain: Predictor data are given on the same scale, criterion scores may be given on different scales, and it is necessary to pool data even though criterion scale differences exist Such a system may be needed for minority group or graduate student prediction where the group sizes are small Least squares and Bayes methods are used in a cross-validation study conducted for comparison purposes Data for the study were taken from the files of the Validity Study Service of the College Entrance Examination Board A very limited amount of data were supplied by a few American graduate schools The Bayes method was better, but it was found that both methods yield negative regression weights; when the absolute values of the weights were used, the methods were both improved and yielded results which were very similar in terms of evaluative statistics computed in the cross sample

Methods for evaluating empirical bayes point estimates of latent trait scores

Abstract: Empirical Bayes point estimates of latent trait scores, derived under the assumptions of one of several test theory models, display a certain degree of instability unless the sample size is sufficiently large. A measure of this instability over repeated sampling is the distribution of the overall expected squared error loss which converges, both in probability and in the mean, to the minimum (Bayes) overall expected loss as sample size increases. An asymptotic distribution theory is developed, and the resulting large sample approximation is compared with results obtained from simulated data. Attention is also given to the effects of using a smoothing procedure.

Pooling Life-Test Data by Means of the Empirical Bayes Method

Abstract: The Empirical Bayes approach in reliability is reviewed non-technically by discussing the questions: What is Empirical Bayes; how does it differ from Bayes; when can I use it; how do I use it; and what do I gain by using it? The Empirical Bayes method is applied to the problem of estimating the outage rates for two 115 kV transmission circuits of a regional power company.

An Approach to Unsupervised Learning Classification

Abstract: In this correspondence, an approach to unsupervised pattern classifiers is discussed. The classifiers discussed here have the ability of obtaining the consistent estimates of unknown statistics of input patterns without knowing the a priori probability of each category's occurrence where the input patterns are of a mixture distribution. An analysis is made about their asymptotic behavior in order to show that the classifiers converge to the Bayes' minmum error classifier. Also, some results of a computer simulation on learning processes are shown.