Showing papers on "Bayes' theorem published in 1974"

Empirical Bayes Methods Applied to Estimating Fire Alarm Probabilities

Abstract: An empirical Bayes approach is used to derive a Stein-type estimator of a multivariate normal mean when the components have unequal variances. This estimator is applied to estimating the probability that a fire alarm reported from a particular street box signals a structural fire rather than a false alarm or other emergency. The approach is to group alarm boxes into relatively homogeneous neighborhoods and to make empirical Bayes estimates of the “probability structural” for each box in the neighborhood from yearly (1967–1969) Bronx data. A dispatching rule based on the estimates is evaluated on 1970 data.

On a New Class of Bounds on Bayes Risk in Multihypothesis Pattern Recognition

Abstract: An important measure concerning the use of statistical decision schemes is the error probability associated with the decision rule. Several methods giving bounds on the error probability are presently available, but, most often, the bounds are loose. Those methods generally make use of so-cailed distances between statistical distributions. In this paper a new distance is proposed which permits tighter bounds to be set on the error probability of the Bayesian decision rule and which is shown to be closely related to several certainty or separability measures. Among these are the nearest neighbor error rate and the average conditional quadratic entropy of Vajda. Moreover, our distance bears much resemblance to the information theoretic concept of equivocation. This relationship is discussed. Comparison is made between the bounds on the Bayes risk obtained with the Bhattacharyya coefficient, the equivocation, and the new measure which we have named the Bayesian distance.

Classification Based on Dichotomous and Continuous Variables

Abstract: A Bayes procedure for classifying an observation consisting of one dichotomous variable (X) and a continuous vector Y is applied to a model assuming that the conditional distribution of Y given X is normal. The procedure reduces to two linear discriminant functions, one for each value of X. An example utilizing data on critically ill patients is given. Extension to one polytomous variable or several dichotomous variables is discussed.

The History of Likelihood

Abstract: One of R. A. Fisher's most influential papers was "On the mathematical foundations of theoretical statistics" (1922), in which he propounded the method of maximum likelihood as a means of point estimation, and hence established a whole branch of statistical reasoning. Yet this paper does not contain his original statement of the method, which was published in 1912, nor does it contain his original definition of likelihood, which appeared in 1921. The great innovation of the 1922 paper was, rather, the clear specification of one approach to the problem of estimation, and the elucidation of the properties of maximum-likelihood estimators. Methods similar to the method of maximum likelihood have a history prior to the work of Fisher, but the definition of likelihood itself, and the realization that it could be used independently as a measure of relative support for hypotheses, appears to be entirely his own, and the main purpose of the present paper is to investigate the background to his introduction of the new concept. With the decline in the esteem with which repeated-sampling methods are held, in the face of the Bayesian revival, the concept of likelihood has come to the fore as offering an approach to statistical inference that is neither repeated-sampling nor Bayesian (see Edwards, 1972). Although it is too early to predict the extent to which pure likelihood methods can supersede other approaches, the concept of likelihood is, in addition, so fundamental to both Bayesian inference and Neyman-Pearson methods that an account of its origins needs no further justification. In the former, the likelihood is the vehicle which carries the observational or experimental results in Bayes' Theorem, whilst in the latter, likelihood was one of the key concepts in the original formulation by Neyman and Pearson (1928). The present paper traces the history of both likelihood and the method of maximum likelihood; it is essential to keep the distinction between the two clearly in mind.

The Bayes/Non-Bayes Compromise and the Multinomial Distribution

Abstract: Compromises between Bayesian and non-Bayesian significance testing are exemplified by examining distributions of criteria for multinominal equiprobability. They include Pearson's X2, the likelihood-ratio, the Bayes factor F, and a statistic G that previously arose from a Bayesian model by “Type II Maximum Likelihood.” Its asymptotic distribution, implied by the theory of the “Type II Likelihood Ratio,” is remarkably accurate into the extreme tail. F too can be treated as a non-Bayesian criterion and is almost equivalent to G. The relationship between F and its own tail area sheds further light on the relationship between Bayesian and “Fisherian” significance.

Bayes' decision rule for rapid detection and adaptive estimation scheme with space applications

Abstract: This paper describes a suboptimal filter applicable to systems which have unknown impulses at unknown instances of time. The minimum variance estimate of the input and the Bayes' decision rule is explained. Computer results indicate the improvement over the standard Kalman filter.

Empirical Bayes estimation of the binomial parameter

Abstract: SUMMARY A smooth empirical Bayes estimator for a binomial parameter is derived. The risk in using this estimator is compared by simulation with that of eight other binomial estimators. For those estimators not having a simple closed form risk, suitable regression equations on the simulation parameters are obtained. These equations are used to identify the regions of superiority of each of the estimators. A numerical example is provided.

Computer diagnosis in psychiatry: a Bayes approach.

Abstract: Along with the recent work on standardizing definitions, interviews, and records in clinical psychiatry, there have been several attempts at designing computer models of the psychiatric diagnosis process (e.g., Spitzer's DIAGNO and Wing's CATEGO). Such models aid scientific evaluation of issues such as reliability and validity. Generally these models are judged upon how well they can replicate clinician's diagnoses when presented with unknown clinical cases. This paper presents a model based on applied probability theory, Bayes method, which has been tested successfully in other medical fields, but has only been proposed with sufficient testing in psychiatry (because of inadequate samples). It describes the logic of Bayes method, details the samples used to develop and test the model, and compares the results of the model with the performance of DIAGNO (a well established, different type of computer model), and discusses the implications of this. Bayes method requires estimates of the relative frequencies of relevant symptoms in specified diseases(e.g., the relative incidence of disturbed reality testing in paranoid schizophrenia as compared with that in alcoholism). In this study estimates of these frequencies are derived from a sample of patients and nonpatients interviewed by New York psychiatrists using spitzer's Current and Past Psychopathology scales (CAPPS). The formal mathematical procedure (Bayes method) which translates this information into predicted diagnoses is briefly described. The model is tested on a subset of this sample, and then on three completely separate samples: an inpatient group from Columbia and the Institute of Living, a group of women in a maternity clinic who were selected by a screening questionnaire for schizophrenia, and a mixed group of Italian inpatients and outpatients interviewed by Italian psychiatrists. The CAPPS records are processed by both Bayes method and DIAGNO, and the results compared. The agreement kamong clinician and computer varies between 40 and 70 per cent for Bayes method, and between 45 and 55 per cent for DIAGNO. The reasons for this difference are discussed. Finally a comparison of the advantages and disadvantages of the respective methods is presented.

An Empirical Bayes Approach to Scoring Multiple-Choice Tests in the Misinformation Model

Abstract: This article develops multiple-choice test scoring rules, concentrating on Bayes rules and their frequency theory analogs, empirical Bayes rules. Conditions are given for empirical Bayes estimates to lie in the probability simplex. The misinformation model is considered in detail. It is shown that ranking by raw scores is equivalent to ranking by Bayes scores when the loss function increases with error and the sampling distribution has the monotone likelihood ratio property. Application of the techniques is made to data from a multiple-choice test given to students of an elementary statistics course.

Bidding Models — The Final Resolution of a Controversy

Abstract: There has been a controversy over the correct formula to use in computing the probability of winning a contract by competitive bidding. Friedman and Gates have put forward differing views, and Rosenshine has attempted to reconcile them. In this paper the underlying structure of the bidding situation is examined in detail and a full probability model is developed. A revised procedure for computing the probability is set out, in which use is made of Bayes theorem, and this is illustrated by a simple worked example.

The bayes methods for seeking the extremal point

Abstract: A number of problems in engineering and in the sciences may be formulated as involving a search for the extremal point of some multivariable utility function f. The Bayes method may be used for achieving this aim. This method has recently been applied to some problems for planning “extremal” experiments in chemistry and in agrophysics, and for the study of random processes and other fields.

Optimal Sampling Schemes for Estimating System Reliability by Testing Components—I: Fixed Sample Sizes

Abstract: A Bayesian decision theoretic approach is employed to compare sampling schemes designed to estimate the reliability of series and parallel systems by testing individual components. Quadratic loss is assumed and schemes are found which minimize Bayes risk plus sampling cost. Several kinds of initial information concerning the reliability of the individual components, all of which assume the components function independently, are considered. The case for which the initial information is in terms of the system's reliability is briefly considered and related to the aforementioned case

The estimation of ρ in the first‐order autoregressive model: A Bayesian Approach

Abstract: Three general approaches to derive marginal posterior probability density functions for the autocorrelation coefficient of the first-order normal autoregressive model are presented, from which Bayes estimators can be obtained for a given loss function. The different approaches are based on varying assumptions about the incidental parameters of the model and are shown numerically to be approximately equivalent with respect to their mean and variance. A comparison is made between the Bayes estimator and some classical estimators on the basis of the risk function and the expected risk. The risk functions are determined by Monte Carlo methods for quadratic, symmetric linear, and various asymmetric linear loss functions. The Bayes estimators are shown to be considerably advantageous, especially when the sample size is small. The Bayes estimators are also shown to be extremely robust under changes of the loss function.

Some formal bayes estimators of variance components in the balanced three-stage nested random effects model

Abstract: The problem of estimation of the variance components in the three-stage nested randomeffects model is considered from a Bayesian viewpoint.Under the usual assumptions of normality and independence of random effects some Bayes estimators of the variance components are developed using appropriate loss functions and adopting a non-informative reference prior distribution.

A Comparison of Maximum Likelihood Exponential Smoothing and Bayes Forecasting Procedures in Inventory Modeling

Abstract: This paper compares four major schemes used for forecasting mean demand to be used as input into an inventory model so that ‘ optimum ’ stockage levels can be obtained. The inventory model is the classical order up to S, infinite horizon model with carry-over from period to period and complete back-ordering. Maximum likelihood, exponential smoothing, standard Bayes and adaptive Bayes schemes are used and results, via Monte Carlo simulation, are obtained on the average costs per period for (1) stationary demand, (2) long-term trend and (3) ‘ shock ’ changes in mean demand.

Some New Error Bounds and Approximations for Pattern Recognition

Abstract: For the average error probability Pe associated with the Bayes recognition procedures for two possible patterns, using no context, new upper and lower bounds and approximations are obtained. Results are given in terms of simple functions of feature "reliability" and a priori probabilities of the patterns. Two kinds of feature "reliability" are considered, i.e., distance between probability distributions and error probabilities without the use of a priori probabilities. Computational advantages offered by those bounds and approximations are pointed out. The question as to how close they are to P e is examined. In some special cases, they are perfect. Numerical examples show that the differences are in general about 5-10 percent, and comparisons with certain known results are quite favorable. Possible applications are discussed. Extension is also made to m possible patterns arranged in a hierarchy with two elements at each branching.

Compound Sequential Probability Ratio Test for the Classification of Statistically Dependent Patterns

Abstract: A sequential test procedure for the classification of statistically dependent patterns is developed. The test is based on the optimum (Bayes) compound decision theory and the theory of Wald's sequential probability ratio test (SPRT). The compound sequential probability ratio (SPRT) is shown to be recursively computable at every instant of the decision process. A two-class recognition problem with first-order Markov dependence among the pattern classes is considered for the purpose of comparing the performance of the CSPRT with that of Wald's SPRT. It is shown that when the pattern classes are statistically dependent the CSPRT requires, on the average, fewer features per pattern than Wald's equally reliable SPRT. Finally, the results of computer simulated recognition experiments using CSPRT and other sequential and nonsequential decision schemes are discussed in detail.

Bayes estimate of the noncentrality parameter in multivariate analysis

Abstract: Madansky and Olkin (1969) derived approximate confidence intervals for the parameter,μ∑-1μ This paper deals with the estimation of the parameter using a Bayes technique. The general case with parameter Ω=∑-1MN is also considered and Bayes estimates of Ω and tr j Ω derived.

Asymptotic Behavior of Bayes Tests and Bayes Risk

Abstract: In this paper the asymptotic behaviors of the Bayes test and Bayes risk are studied in both the one-sided and two-sided testing problems, where the independent observations are taken from one member of a one-parameter exponential family. Precise asymptotic expressions are found which show the Bayes procedure to be relatively insensitive to the prior distribution as the sample size increases to infinity.

Aligning of estimates: an alternative to empirical bayes methods

Abstract: Summary The data that are used in constructing empirical Bayes estimates can properly be regarded as arising in a two-stage sampling scheme. In this setting it is possible to modify the conventional parameter estimates so that a reduction in expected squared error is effected. In the empirical Bayes approach this is done through the use of Bayes's theorem. The alternative approach proposed in this paper specifies a class of modified estimates and then seeks to identify that member of the class which yields the minimum squared error. One advantage of this approach relative to the empirical Bayes approach is that certain problems involving multiple parameters are easily overcome. Further, it permits the use of relatively efficient methods of non-parametric estimation, such as those based on quantiles or ranks; this has not been achieved by empirical Bayes methods.

A survey and comparison of several empirical bayes estimators for the binomial parameter

Abstract: Eight empirical Bayes estimators for the binomial parameter p are summarized and compared according to their estimated risk based on the results of a Monte Carlo simulation. For those estimators not found to possess a closed form risk, a suitable regression on the parameters of the simulation is obtained. These equations are then used to identify the region in parameter space of mean squared error superiority of each of the estimators. The equations presented can be used in practice to decide which estimator should be used in a given situation.

On the properties of an optimal selection procedure

Abstract: A subset selection procedure is presented along with an investigation of certain of its properties Although the procedure is Bayes. non-iniormauvte pi U-i distributions arc emphasized. The two properties considered are the probability of selecting the largest parameter and the expected number of parameters selected.

Classification by linear and quadratic discriminant scores

Abstract: This program provides two alternative Bayes solutions to problems of classifying an individual into one of K mutually exclusive populations on the basis of measurements taken on p predictor variables. It is assumed that the individual must have come from one of the K populations and must be assigned to one of them. Two simplifying assumptions are made. First, the p measurements are assumed to have a multivariate normal distribution in each of the populations.' Secondly, all misclassification errors are considered equally costly. The Bayes decision rule minimizes the total probability of misclassification. In this procedure. an individual is classified by means of "discriminant scores," one for each of the K populations, resulting in the assignment of the individual to that population for which he has the largest posterior probability. Such a Bayes procedure requires the "prior probabilities" that an individual, drawn at random, belongs to a given population. This procedure does not, however, require that the covariance matrices of the K populations be equal (homogeneous); a test of the homogeneity assumption is made by the program. If they are equal, the discriminant scores can be reduced to linear functions of the predictor variables. They are therefore called "linear discriminant scores." When the covariance matrices are unequal, the discriminant scores are quadratic functions and are called "quadratic discriminant scores." Thus, the mathematical form of the discriminant scores differentiates two types of Bayee procedures-linear and quadratic-both of which are provided by the program. For detailed discussions, see Anderson (1958), Rao (1965), and Fu1comer (1970). Method. (1) Notation. The following terms are used in this program description: K = number of populations, p = number of variables, x' = (xj , "', Xp ) = vector of predictor scores, 1Tk = prior probability of the kt h population, JJ.k = mean vector of the k t h population, ~k = covariance matrix of the kt h population, rii =cost in assigning an individual who actually belongs to the it h population to the jth, Pi(X)=probability density at Xfor the it h population, § = sample space of all potential observations, Wi = classification region for the it h popula tion. Carets (e .g., Pk) indicate the use of sample estimates for corresponding parameters. (2) Decision Rule. The expected loss in applying a decision rule for an individual from the it h population is

An empirical Bayes rule for testing simple hypothesis against simple alternative

Abstract: Summary This paper gives a short introduction into the empirical Bayes approach. To illustrate this approach the classical problem of testing a simple hypothesis against a simple alternative is discussed. Some explicit results are given for testing Ho: N(-1, 1) against H,: N(1, 1). This paper is a further elaboration of section 3.1 of the author's dissertation Wan Houwelingen (1973)).

Multistage Inference Models for Intelligence Analysis

Abstract: : The report discusses inference models which provide a potentially meaningful and useful framework for the analysis of current modes of intelligence processing The paper describes the development and evaluation of a class of multistage Bayesian inference models