Showing papers on "Bayes' theorem published in 1992"

Bayesian statistics without tears: A sampling-resampling perspective

Abstract: Even to the initiated, statistical calculations based on Bayes's Theorem can be daunting because of the numerical integrations required in all but the simplest applications. Moreover, from a teaching perspective, introductions to Bayesian statistics—if they are given at all—are circumscribed by these apparent calculational difficulties. Here we offer a straightforward sampling-resampling perspective on Bayesian inference, which has both pedagogic appeal and suggests easily implemented calculation strategies.

Bayesian Statistics without Tears: A Sampling–Resampling Perspective

Abstract: Even to the initiated, statistical calculations based on Bayes's Theorem can be daunting because of the numerical integrations required in all but the simplest applications. Moreover, from a teaching perspective, introductions to Bayesian statistics—if they are given at all—are circumscribed by these apparent calculational difficulties. Here we offer a straightforward sampling-resampling perspective on Bayesian inference, which has both pedagogic appeal and suggests easily implemented calculation strategies.

Learning classification trees

Abstract: Algorithms for learning classification trees have had successes in artificial intelligence and statistics over many years. This paper outlines how a tree learning algorithm can be derived using Bayesian statistics. This introduces Bayesian techniques for splitting, smoothing, and tree averaging. The splitting rule is similar to Quinlan's information gain, while smoothing and averaging replace pruning. Comparative experiments with reimplementations of a minimum encoding approach,c4 (Quinlanet al., 1987) andcart (Breimanet al., 1984), show that the full Bayesian algorithm can produce more accurate predictions than versions of these other approaches, though pays a computational price.

Bayesian Analysis of Linear and Non-Linear Population Models by Using the Gibbs Sampler

Abstract: : A fully Bayesian analysis of linear and nonlinear population models has previously been unavailable, as a consequence of the seeming impossibility of performing the necessary numerical Integrations in the complex multi- parameter structures typically arising in such models It is demonstrated that, for a variety of linear and nonlinear population models, a fully Bayesian analysis can be implemented in a straightforward manner using the Gibbs sampler The approach is illustrated with examples involving challenging problems of outliers and mean-variance relationships in population modelling

Estimating the probability of failure when testing reveals no failures

Abstract: Formulas for estimating the probability of failure when testing reveals no errors are introduced. These formulas incorporate random testing results, information about the input distribution; and prior assumptions about the probability of failure of the software. The formulas are not restricted to equally likely input distributions, and the probability of failure estimate can be adjusted when assumptions about the input distribution change. The formulas are based on a discrete sample space statistical model of software and include Bayesian prior assumptions. Reusable software and software in life-critical applications are particularly appropriate candidates for this type of analysis. >

Empirical Bayes versus fully Bayesian analysis of geographical variation in disease risk.

Abstract: This paper reviews methods for mapping geographical variation in disease incidence and mortality. Recent results in Bayesian hierarchical modelling of relative risk are discussed. Two approaches to relative risk estimation, along with the related computational procedures, are described and compared. The first is an empirical Bayes approach that uses a technique of penalized log-likelihood maximization; the second approach is fully Bayesian, and uses an innovative stochastic simulation technique called the Gibbs sampler. We chose to map geographical variation in breast cancer and Hodgkin's disease mortality as observed in all the health care districts of Sardinia, to illustrate relevant problems, methods and techniques.

The identification of concealed memories using the event-related potential and implicit behavioral measures: a methodology for prediction in the face of individual differences.

Abstract: The development and validation of an event-related potential (ERP) memory assessment procedure is detailed. The procedure identifies learned material with high rates of accuracy, whether or not subjects give intentional responses indicating they had previously learned it. Because the traditional analysis of variance approach fails to provide probabilistic conclusions about any given individual, Bayesian posterior probabilities were computed, indicating the probability for each and every person that material was learned. The method was developed on a sample of 20 subjects, and then cross-validated on two additional samples of 20 subjects each. Across the three samples, the method correctly defined over 94% of learned material as learned, and misclassified 4% of the unlearned material. Additionally, in a simple oddball task performed by the same subjects, the method classified rare and frequent material with perfect accuracy. Finally, combining two implicit behavioral measures--mean reaction time and the number of incorrect responses--in Bayesian fashion yielded classification accuracy that actually exceeded that of the ERP-based procedure overall, but the two methods provided identical accuracy in classifying the most critical material as recognized.

Bayesian estimation and experimental design in linear regression models

Abstract: Estimation and Design as a Bayesian Decision Problem Choice of a Prior Distribution Conjugate Prior Distributions Bayes Estimation of the Regression Parameter Optimality and Robustness of the Bayes Estimator Bayesian Interpretation of Estimators Using Non-Bayesian Prior Knowledge Bayes Estimation in Case of Prior Ignorance Further Problems The Design Problem for the Linear Bayes Estimator Characterization of Optimal Designs Construction of Optimal Continuous Designs Construction of Exact Optimal Designs.

Hierarchical Bayes Models for the Progression of HIV Infection Using Longitudinal CD4 T-Cell Numbers

Abstract: Taking the absolute number of CD4 T-cells as a marker of disease progression for persons infected with the human immunodeficiency virus (HIV), we model longitudinal series of such counts for a sample of 327 subjects in the San Francisco Men's Health Study (Waves 1–8, excluding zidovudine cases). We conduct a fully Bayesian analysis of these data. We employ individual level nonlinear models incorporating such critical features as incomplete and unbalanced data, population covariates (age at study entry and an indicator of self-reported herpes simplex virus infection), unobserved random change points, heterogeneous variances, and errors in variables. We construct prior distributions using results of previously published work from several different sources and data from HIV-negative men in this study. We also develop an approach to Bayesian model choice and individual prediction. Our analysis provides marginal posterior distributions for all population parameters in our model for this cohort. Using ...

A new method for the detection of a periodic signal of unknown shape and period

Abstract: We present a new method for the detection and measurement of a periodic signal in a data set when we have no prior knowledge of the existence of such a signal or of its characteristics. It is applicable to data consisting of the locations or times of individual events. To address the detection problem, we use Bayes’ theorem to compare a constant rate model for the signal to models with periodic structure. The periodic models describe the signal plus background rate as a stepwise distribution in m bins per period, for various values of m. The Bayesian posterior probability for a periodic model contains a term which quantifies Ockham’s razor, penalizing successively more complicated periodic models for their greater complexity even though they are assigned equal prior probabilities. The calculation thus balances model simplicity with goodness-of-fit, allowing us to determine both whether there is evidence for a periodic signal, and the optimum number of bins for describing the structure in the data. Unlike the results of traditional “frequentist” calculations, the outcome of the Bayesian calculation does not depend on the number of periods examined, but only on the range examined. Once a signal is detected, we again use Bayes’ theorem to estimate the frequency of the signal. The probability density for the frequency is inversely proportional to the multiplicity of the binned events and is thus maximized for the frequency leading to the binned event distribution with minimum combinatorial entropy. The method is capable of handling gaps in the data due to intermittent observing or dead time.

Empirical Bayes Methods.

Abstract: Empirical Bayes Methods. By J. Maritz and T. Lwin. ISBN 0 412 27760 3. Chapman and Hall, London, 1989. xii + 284 pp. £27.50.

The Bayes/Non-Bayes Compromise: A Brief Review

Abstract: Various compromises that have occurred between Bayesian and non-Bayesian methods are reviewed. (A citation is provided that discusses the inevitability of compromises within the Bayesian approach.) One example deals with the masses of elementary particles, but no knowledge of physics will be assumed.

An inductive modelling procedure based on Bayes' theorem for analysis of pattern in spatial data

Abstract: This paper describes an inductive modelling procedure integrated with a geographical information system for analysis of pattern within spatial data. The aim of the modelling procedure is to predict the distribution within one data set by combining a number of other data sets. Data set combination is carried out using Bayes’ theorem. Inputs to the theorem, in the form of conditional probabilities, are derived from an inductive learning process in which attributes of the data set to be modelled are compared with attributes of a variety of predictor data sets. This process is carried out on random subsets of the data to generate error bounds on inputs for analysis of error propagation associated with the use of Bayes’ theorem to combine data sets in the GIS. The statistical significance of model inputs is calculated as part of the inductive learning process. Use of the modelling procedure is illustrated through the analysis of the winter habitat relationships of red deer in Grampian Region, north-ea...

Capture—recapture estimation via Gibbs sampling

Abstract: Capture-recapture models are widely used in the estimation of population sizes. Based on data augmentation considerations, we show how Gibbs sampling can be applied to calculate Bayes estimates in this setting. As a result, formulations which were previously avoided because of analytical and numerical intractability can now be easily considered for practical application. We illustrate this potential by using Gibbs sampling to calculate Bayes estimates for a hierarchical capture-recapture model in a real example

Constrained Bayes Estimation with Applications

Abstract: Bayesian techniques are widely used in these days for simultaneous estimation of several parameters in compound decision problems. Often, however, the main objective is to produce an ensemble of parameter estimates whose histogram is in some sense close to the histogram of population parameters. This is for example the situation in subgroup analysis, where the problem is not only to estimate the different components of a parameter vector, but also to identify the parameters that are above, and the others that are below a certain specified cutoff point. We have proposed in this paper Bayes estimates in a very general context that meet this need. These estimates are obtained by matching the first two moments of the histogram of the estimates, and the posterior expectations of the first two moments of the histogram of the parameters, and minimizing, subject to these conditions, the posterior expectation of the Euclidean distance between the estimates and the parameters. Several applications of the m...

Prediction Theory for Finite Populations

Abstract: A large number of papers have appeared in the past 20 years on estimating and predicting characteristics of finite populations. This monograph is designed to present this modern theory in a systematic and consistent manner. The author's approach is that of superpopulation models in which values of the population elements are considered as random variables having joint distributions. Throughout, the emphasis is on the analysis of data rather than on the design of samples. Topics covered include: optimal predictors for various superpopulation models, Bayes, minimax, and maximum likelihood predictors, classical and Bayesian prediction internals, model robustness, and models with measurement errors. Each chapter contains numerous examples, and exercises which extend and illustrate the themes in the text. As a result, this book will be ideal for all those research workers seeking an up-to-date and well-referenced introduction to the subject.

Sample Size Determination in Bayesian Analysis

Abstract: The problem of sample size determination in the context of Bayesian analysis, decision theory and quality management is considered. For the familiar, and practically important, parameter of a binomial distribution with a beta prior, we present complete and exact solutions for determining the sample sizes when preset precision conditions are imposed on commonly used criteria such as posterior variance, Bayes risk and expected value of sample information. The results obtained here also permit a unifying treatment of several sample size problems of practical interest and an example shows how they can be used in a managerial situation. A computer program for a personal computer handles all computational complexities and is available upon request.

A semi-Bayes approach to the analysis of correlated multiple associations, with an application to an occupational cancer-mortality study.

Abstract: Thomas et al. presented the application of empirical-Bayes methods to the problem of multiple inference in epidemiologic studies. One limitation of their approach, which they noted, was the need to assume exchangeable log relative-risk parameters, and independent relative-risk estimates. Numerical integration was also required. Here I generalize their approach to allow for non-exchangeable parameters and non-independent estimates. The resulting method is Bayesian in so far as some feature of the prior distribution are specified from prior information, but is empirical Bayes in so far as some explicit parameters in the prior distribution are estimated from the data. Estimation is based on approximations to the posterior distribution; this allows one to implement the approach with standard software packages for matrix algebra. The method is illustrated in an occupational mortality study of 84 exposure-cancer associations.

Bayesian estimation of the parameters,reliability and failure rate functions of the burr type xii failure model

Abstract: Based on a type-2 censored sample of the life times from a two parameter Burr type-XII failure time model, the Bayes estimates of the two (unknown) parameters, the reliability and failure rate functions are obtained by using Bayes approximation form due to Lindley (1980). The estimated risks of the Bayes estimates are computed and compared with the corresponding estimated risks of the maximum likelihood estimates.

The topological fusion of bayes nets

Abstract: Bayes nets are relatively recent innovations. As a result, most of their theoretical development has focused on the simplest class of single-author models. The introduction of more sophisticated multiple-author settings raises a variety of interesting questions. One such question involves the nature of compromise and consensus. Posterior compromises let each model process all data to arrive at an independent response, and then split the difference. Prior compromises, on the other hand, force compromise to be reached on all points before data is observed. This paper introduces prior compromises in a Bayes net setting. It outlines the problem and develops an efficient algorithm for fusing two directed acyclic graphs into a single, consensus structure, which may then be used as the basis of a prior compromise.

Bayesian designs for maximizing information and outcome

Abstract: This article deals with a novel utility function to design experiments in a Bayesian framework. This utility function is a linear combination of the gain in Shannon information and of the total outcome of the experiment, defined as the sum of observed values in the dependent variable of a linear model. Thus the expected posterior utility to be maximized is a combination of the Bayes D-optimality criterion and the posterior expectation of the total output. Earlier studies have shown that Bayesian parallels of the classical D-, A-, and E-optimal designs can be obtained by considering utility or loss functions concerned with efficient estimation of the parameters of interest. A different requirement that might be desirable in applied problems is to combine the accuracy of parameter estimation with the maximization of experimental output. The utility function considered here does this. We look at the implications of using this utility in deriving designs in the context of hierarchical linear models. ...

Where to Look Next Using a Bayes Net: Incorporating Geometric Relations

Abstract: A task-oriented system is one that performs the minimum effort necessary to solve a specified task. Depending on the task, the system decides which information to gather, which operators to use at which resolution, and where to apply them. We have been developing the basic framework of a task-oriented computer vision system, called TEA, that uses Bayes nets and a maximum expected utility decision rule. In this paper we present a method for incorporating geometric relations into a Bayes net, and then show how relational knowledge and evidence enables a task-oriented system to restrict visual processing to particular areas of a scene by making camera movements and by only processing a portion of the data in an image.

Application of empirical bayes inference to estimation of rate of change in the presence of informative right censoring

Abstract: We apply parametric empirical Bayes inference of Morris7 to the estimation of rate of change from incomplete longitudinal studies where the right censoring process is considered informative, that is, the length of time the subjects participate in the study is associated with level of the study variable. Ignoring such an association can result in a biased estimate of rate of change. The proposed method provides estimates of rate of change for individual subjects as well as for the entire group, adjusted for informative right censoring. The method is considered more robust than those based on a specific parametric model for the censoring distribution. Under non-informative right censoring these estimators of slopes are equivalent to the Bayes estimators derived by Fearn.11 We illustrate the method with an example involving renal transplant data. We evaluate the method's performance through a simulation study.

Estimation in multi-path change-point problems

Abstract: The current literature deals with the change-point problem only in the context of the obser¬vation of a single sequence. In this paper, inference will be based on the observation of TV sequences of random variables, each sequence containing one change-point. This extension allows the effective use of bootstrap and empirical Bayes methods, both of which are not feasible in the single-path context. Two classes of these “multi-path” change-point problems are considered. If the change-point is assumed to occur at the the same position in each sequence, then the terminology “fixed-tau multi-path change-point” will be used. In other cases, one may expect the change-point to occur at random positions in each sequence, according to some distribution, a “random-tau multi-path change-point” problem. Examples and simulations are given.

Bayesian exchangeability, benefit transfer, and research efficiency

Abstract: We offer an economic model of the policymaker's site- or time-specific benefit estimate extrapolation problem when she must weigh the potential gains from an increase in the accuracy and the precision of her agents' estimates against the costs of conducting their assessments. If Bayesian exchangeability is treated as a maintained hypothesis, we suggest that empirical Bayes estimators offer a powerful way to increase the economic efficiency of extrapolation. Finally, we employ a hedonic study of pollution control benefits to illustrate a Bayesian diagnostic that allows the hypothesis of exchangeability to be tested rather than taken as maintained. The power of the diagnostic arises from its ability to identify those sources of parameter variability most likely to discourage extrapolations.

Off Bayes: effect of verification bias on posterior probabilities calculated using Bayes' theorem.

Abstract: Estimates of sensitivity and specificity can be biased by the preferential referral of patients with positive test responses or ancillary clinical abnormalities (the "concomitant information vector") for diagnostic verification. When these biased estimates are analyzed by Bayes' theorem, the resultant posterior disease probabilities (positive and negative predictive accuracies) are similarly biased. Accordingly, a series of computer simulations was performed to quantify the effects of various degrees of verification bias on the calculation of predictive accuracy using Bayes' theorem. The magnitudes of the errors in the observed true-positive rate (sensitivity) and false-positive rate (the complement of specificity) ranged from +11% and +23%, respectively (when the test response and the concomitant information vector were conditionally independent), to +16% and +48% (when they were conditionally non-independent). These errors produced absolute underestimations as high as 22% in positive predictive accuracy, and as high as 14% in negative predictive accuracy, when analyzed by Bayes' theorem at a base rate of 50%. Mathematical correction for biased verification based on the test response using a previously published algorithm significantly reduced these errors by as much as 20%. These data indicate 1) that selection bias significantly distorts the determination of predictive accuracies calculated by Bayes' theorem, and 2) that these distortions can be significantly offset by a correction algorithm.