Showing papers on "Bayes' theorem published in 1983"

Parametric Empirical Bayes Inference: Theory and Applications

Abstract: This article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical Bayes viewpoint, particularly in the case of parametric prior distributions. Some successful applications of major importance are considered. Recent results concerning estimates of error and confidence intervals are described and illustrated with data.

Contextual analysis through the multilevel linear model.

Abstract: A general linear multilevel model and its estimation are described and illustrated empirically. The specification of a multilevel linear model within the covariance component framework was rendered. This model is appropriate for a wide range of applications in contextual analysis because it allows for macro as well as micro errors. Then an estimation procedure the restricted maximum likelihood/Bayes (REML/Bayes) was proposed and described for this multilevel model. Finally an extended empirical example was presented with the primary purpose of illustrating the use of the statistical methodology in conjunction with a meaningful substantive problem not to carry out a critical test of the underlying substantive theory and not to demonstrate the general superiority of the proposed estimation procedure. The example suggests that the REML/Bayes estimation procedure may be appropriate inasmuch as the results extracted with REML/Bayes are more consistent with theoretical anticipations than those extracted with ordinary least squares or estimated generalized least squares. The methodology presented by no means exhausts the subject of multilevel estimation. There is a need for estimation procedures to handle discrete micro response variables systems of micro-structural equations and other generalizations. Work is currently proceeding on some of these extensions of the multilevel linear model as the goal of sound estimation procedures for generalized multilevel models is worthwhile and by no means esoteric. It may be hard for researchers to find contextual effects unless they use efficient and appropriate estimation techniques.

Base Rates in Bayesian Inference: Signal Detection Analysis of the Cab Problem

Abstract: Several investigators concluded that humans neglect base rate information when asked to solve Bayesian problems intuitively. This conclusion is based on a comparison between normative (calculated) and subjective (responses by naive judges) solutions to problems such as the cab problem. The present article shows that the previous normative analysis was incomplete. In particular, problems of this type require both a signal detection theory and a judgment theory for their proper Bayesian analysis. In Bayes' theorem, posterior odds equals prior odds times the likelihood ratio. Previous solutions have assumed that the likelihood ratio is independent of the base rate, whereas signal detection theory (backed up by data) implies that this ratio depends on base rate. Before the responses of humans are compared with a normative analysis, it seems desirable to be sure that the normative analysis is accurate.

Bayesian inference: Combining base rates with opinions of sources who vary in credibility.

Abstract: Michael H. BirnbaumUniversity of Illinois at Urbana-ChampaignBarbara A. MellersUniversity of California, BerkeleySubjects made judgments of the probability of an event given base-rate informationand the opinion of a source. Base rate and the source's hit and false-alarm rateswere manipulated in a within-subjects design. Hit rate and false-alarm rate weremanipulated to produce sources of varied expertise and bias. The base rate, thesource's opinion, and the source's expertise and bias all had large systematic effects.Although there was no evidence of a "base-rate fallacy," neither Bayes' theoremnor a subjective Bayesian model that allows for "conservatism" due to misperceptionor response bias could account for the data. Responses were consistent with a scale-adjustment averaging model developed by Birnbaum & Stegner (1979). In thismodel, the source's report corresponds to a scale value that is adjusted accordingto the source's bias. This adjusted value is weighted as a function of the source'sexpertise and averaged with the subjective value of the base rate. These results areconsistent with a coherent body of experiments in which the same model couldaccount for a variety of tasks involving the combination of information fromdifferent sources.The question, How should humans revisetheir beliefs? has been studied by philosophersand mathematicians, and the question, Howdo humans form opinions and revise them?has been investigated by psychologists. Earlyresearch that compared the two questionsconcluded that Bayes' theorem was a usefulstarting point for the description of humaninference but that humans are "conservative,"or revise their probability judgments in amanner less extreme than implied by Bayes'theorem (Edwards, 1968; Peterson & Beach,1967; Slovic & Lichtenstein, 1971).Edwards (1968) discussed three interpre-tations of conservatism: misperception, mis-aggregation, and response bias. Misperceptionincludes the possibility that objective proba-bilities are transformed to subjective proba-bilities by a psychophysical function. Misag-gregation refers to use of a non-Bayesian ruleto combine evidence. Response bias refers tononlinearity in the judgment function relatingjudged probabilities to subjective likelihoods.Early experimental work attempted to separate

Empirical Bayes estimation of individual growth-curve parameters and their relationship to covariates.

Abstract: SUMMARY The analysis of growth curves has long been important in biostatistics. Work has focused on two problems: the estimation of individual curves based on many data points, and the estimation of the mean growth curve for a group of individuals. This paper extends a recent approach that seeks to combine data from a group of individuals in order to improve the estimates of individual growth parameters. Growth is modeled as polynomial in time, and the group model is also linear, incorporating growth-related covariates into the model. The estimation used is empirical Bayes. The estimation formulas are illustrated with a set of data on rat growth, originally presented by Box (1950, Biometrics 6, 362-389).

Bayes Methods for Combining the Results of Cancer Studies in Humans and other Species

Abstract: We propose a class of Bayesian statistical methods for interspecies extrapolation of dose-response functions. The methods distinguish formally between the conventional sampling error within each dose-response experiment and a novel error of uncertain relevance between experiments. Through a system of hierarchical prior distributions similar to that of Lindley and Smith (1972), the dose-response data from many substances and species are used to estimate the interexperimental error. The data, the estimated error of interspecies extrapolation, and prior biological information on the relations between species or between substances each contribute to the posterior densities of human dose-response. We apply our methods to an illustrative problem in the estimation of human lung cancer risk from various environmental emissions.

Asymptotic optimal inference for non-ergodic models

Abstract: 0. An Over-view.- 1. Introduction.- 2. The Classical Fisher-Rao Model for Asymptotic Inference.- 3. Generalisation of the Fisher-Rao Model to Non-ergodic Type Processes.- 4. Mixture Experiments and Conditional Inference.- 5. Non-local Results.- 1. A General Model and Its Local Approximation.- 1. Introduction.- 2. LAMN Families.- 3. Consequences of the LAMN Condition.- 4. Sufficient Conditions for the LAMN Property.- 5. Asymptotic Sufficiency.- 6. An Example (Galton-Watson Branching Process).- 7. Bibliographical Notes.- 2. Efficiency of Estimation.- 1. Introduction.- 2. Asymptotic Structure of Limit Distributions of Sequences of Estimators.- 3. An Upper Bound for the Concentration.- 4. The Existence and Optimality of the Maximum Likelihood Estimators.- 5. Optimality of Bayes Estimators.- 6. Bibliographical Notes.- 3. Optimal Asymptotic Tests.- 1. Introduction.- 2. The Optimality Criteria: Definitions.- 3. An Efficient Test of Simple Hypotheses: Contiguous Alternatives.- 4. Local Efficiency and Asymptotic Power of the Score Statistic.- 5. Asymptotic Power of the Likelihood Ratio Test: Simple Hypothesis.- 6. Asymptotic Powers of the Score and LR Statistics for Composite Hypotheses with Nuisance Parameters.- 7. An Efficient Test of Composite Hypotheses with Contiguous Alternatives.- 8. Examples.- 9. Bibliographical Notes.- 4. Mixture Experiments and Conditional Inference.- 1. Introduction.- 2. Mixture of Exponential Families.- 3. Some Examples.- 4. Efficient Conditional Tests with Reference to L.- 5. Efficient Conditional Tests with Reference to L?.- 6. Efficient Conditional Tests with Reference to LC: Bahadur Efficiency.- 7. Efficiency of Conditional Maximum Likelihood Estimators.- 8. Conditional Tests for Markov Sequences and Their Mixtures.- 9. Some Heuristic Remarks about Conditional Inference for the General Model.- 10. Bibliographical Notes.- 5. Some Non-local Results.- 1. Introduction.- 2. Non-local Behaviour of the Likelihood Ratio.- 3. Examples.- 4. Non-local Efficiency Results for Simple Likelihood Ratio Tests.- 5. Bibiographical Notes.- Appendices.- A.1 Uniform and Continuous Convergence.- A.2 Contiguity of Probability Measures.- References.

Some Thoughts on Empirical Bayes Estimation

Abstract: Examples are given to illustrate the "linear" and "general" empirical Bayes approaches to estimation. A final example concerns testing the null hypothesis that a treatment has had no effect.

Clinical Trials and Statistical Verdicts: Probable Grounds for Appeal

Abstract: Conventional interpretation of clinical trials relies heavily on the classic p value. The p value, however, represents only a false-positive rate, and does not tell the probability that the investigator's hypothesis is correct, given his observations. This more relevant posterior probability can be quantified by an extension of Bayes' theorem to the analysis of statistical tests, in a manner similar to that already widely used for diagnostic tests. Reanalysis of several published clinical trials according to Bayes' theorem shows several important limitations of classic statistical analysis. Classic analysis is most misleading when the hypothesis in question is already unlikely to be true, when the baseline event rate is low, or when the observed differences are small. In such cases, false-positive and false-negative conclusions occur frequently, even when the study is large, when interpretation is based solely on the p value. These errors can be minimized if revised policies for analysis and reporting of clinical trials are adopted that overcome the known limitations of classic statistical theory with applicable bayesian conventions.

Empirical Bayes Estimation of Rates in Longitudinal Studies

Abstract: The usually irregular follow-up intervals in epidemiologic studies preclude the use of classical growth curve analysis. For short follow-up intervals, it is suggested that individual rates of change are useful for exploratory analysis. Empirical Bayes estimates of these rates of change are recommended and developed. An example of bone loss with age in women is also given.

The Bayes theorem

Abstract: An abstract definition of probability can be given by considering a set S, called the sample space, and possible subsets A, B,. .. , the interpretation of which is left open. The probability P is a real-valued function defined by the following axioms due to Kolmogorov [9]: From this definition and using the fact that A ∩ B and B ∩ A are the same, one obtains Bayes' theorem, From the three axioms of probability and the definition of conditional probability, one obtains the law of total probability,

Who Discovered Bayes's Theorem?

Abstract: (1983). Who Discovered Bayes's Theorem? The American Statistician: Vol. 37, No. 4a, pp. 290-296.

Parametric Empirical Bayes Confidence Intervals

Abstract: Publisher Summary This chapter outlines parametric empirical Bayes confidence intervals. Empirical Bayes modeling assumes the distributions π for the parameters θ= (θ 1 , …, θ k ) exist, with π taken from a known class Π of possible parameter distributions. Π is considered independent N (u, A) distributions on R k . It is called parametric empirical Bayes problem, because πɛ Π is determined by the parameters (u, A) and so is a parametric family of distributions. A simulation presented in the chapter was used to determine that the intervals ±s i and ±1.96s i contain the true values θ i in at least 68 percent and 95 percent of the cases. Empirical Bayes estimators, or Stein's estimator, can lead to misestimation of components that the statistician or his clients care about when exchangeability in the prior distribution is implausible. The term empirical Bayes, which is used for non-parametric empirical Bayes problems, actually fits the parametric empirical Bayes case too. The empirical Bayes methods in general and parametric empirical Bayes methods in particular provide a way to utilize this additional information by obtaining more precise estimates and estimating their precision.

On a ``Two-Stage'' Bayesian Procedure for Determining Failure Rates from Experimental Data

Abstract: A process is described for analyzing failure data in which Bayes' theorem is used twice, firstly to develop a "prior" or "generic" probability distribution and secondly to specialize this distribution to the specific machine or system in question. The process is shown by examples to be workable in practice as well as simple and elegant in concept.

Empirical bayes estimation of coefficients in the general linear model from data of deficient rank

Abstract: Empirical Bayes methods are shown to provide a practical alternative to standard least squares methods in fitting high dimensional models to sparse data. An example concerning prediction bias in educational testing is presented as an illustration.

Empirical Bayes Confidence Sets for the Mean of a Multivariate Normal Distribution

Abstract: Through the use of an empirical Bayes argument, a confidence set for the mean of a multivariate normal distribution is derived. The set is a recentered sphere, is easy to compute, and has uniformly smaller volume than the usual confidence set. An exact formula for the coverage probability is derived, and numerical evidence is presented which shows that the empirical Bayes set uniformly dominates the usual set in coverage probability.

Estimation Risk and Simple Rules for Optimal Portfolio Selection

Abstract: Elton, Gruber and Padberg (EGP) [6, 7] have recently simplified the process of constructing optimal portfolios by developing simple criteria for optimal portfolio selection which do not involve use of mathematical programming. Their simple decision rules permit one to determine easily which securities to include in an optimal portfolio and how much to invest in each. However, in practical applications of theoretical models, sample estimators are usually treated as if they were true values of unknown parameters. As a result, the effect of the standard errors of sample estimators on decision rules are completely ignored. Bawa, Brown and Klein [1] have shown that what is optimal in the absence of estimation risk is not necessarily optimal or even approximately optimal in the presence of estimation risk. Moreover, Brown [4] examined optimal portfolio choice under uncertainty for various portfolio selection procedures-the diffuse Bayes rule, the Markowitz Certainty Equivalent (CE) rule, the aggregation CE rule, and the equal weight rule,1 and found that the diffuse Bayes rule uniformly dominates the Markowitz CE rule in repeated samples for the quadratic utility case. As the sample size increases, the Bayes rule becomes superior to the aggregation CE and the equal weight rules. In addition, the result holds even where the probability distribution of returns is seriously misspecified. Thus, Brown's [4] study has clearly indicated that, without taking estimation risk into account, portfolio selection rules other than the Bayes rule can lead investors to select suboptimal portfolios. This paper shows by using the single index model for the return generating process that the simple decision rules for optimal portfolio selection derived by Elton, Gruber and Padberg [7] are not identical under the Bayesian and the traditional methods of analysis.2 Moreover, in the case where short sales are not

An evaluation of bayesian forecasting

Abstract: ‘Bayesian forecasting’ is a time series method of forecasting which (in the United Kingdom) has become synonymous with the state space formulation of Harrison and Stevens (1976). The approach is distinct from other time series methods in that it envisages changes in model structure. A disjoint class of models is chosen to encompass the changes. Each data point is retrospectively evaluated (using Bayes theorem) to judge which of the models held. Forecasts are then derived conditional on an assumed model holding true. The final forecasts are weighted sums of these conditional forecasts. Few empirical evaluations have been carried out. This paper reports a large scale comparison of time series forecasting methods including the Bayesian. The approach is two fold: a simulation study to examine parameter sensitivity and an empirical study which contrasts Bayesian with other time series methods.

Some Alternatives to Bayes' Rule.

Abstract: : This document focuses on two proposed alternatives to Bayes' rule for revising probability assessments in the face of new information: Richard Jeffrey's rule of conditioning and Arthur Dempster's rule of combination. Section 2 describes Jeffrey's rule. Section 3 describes upper and lower probabilities and Dempster's rule for their combination. Section 4 shows that the two rules are in fact closely connected: Jeffrey's rule is the additive version of Dempster's rule in those situations where the two rules are comparable.

An Apology for Ecumenism in Statistics.

Abstract: Publisher Summary This chapter presents an apology for Ecumenism in statics Bays model had the difficulty that by supposing all possible sets of assumptions known a priori , it discredits the possibility of new discovery It had been argued that if significance tests are to be employed to check the model, then it is necessary to state in advance the level of significance α that is to be used and that no rational basis exists for making such a choice The present proposals excluded the possibilities of attempts to base estimation on sampling theory, using point estimates and confidence intervals and attempts to base criticism and hypothesis testing entirely on Bayesian theory The Bayes' approach provides some assurance against incredibility as it requires that all assumptions of the model be clearly visible and available for criticism The scientific method employs and requires not one but two kinds of inference, namely, criticism and estimation; once this is understood, the statistical advances made in recent years in Bayesian methods, data analysis, robust and shrinkage estimators can be seen as a cohesive whole

Optimal Rates of Convergence to Bayes Risk in Nonparametric Discrimination

Abstract: Consider the multiclassification (discrimination) problem with known prior probabilities and a multi-dimensional vector of observations Assume the underlying densities corresponding to the various classes are unknown but a training sample of size $N$ is available from each class Rates of convergence to Bayes risk are investigated under smoothness conditions on the underlying densities of the type often seen in nonparametric density estimation These rates can be drastically affected by a small change in the prior probabilities, so the error criterion used here is Bayes risk averaged (uniformly) over all prior probabilities Then it is shown that a certain rate, $N^{-r}$, is optimal in the sense that no rule can do better (uniformly over the class of smooth densities) and a rule is exhibited which does that well The optimal value of $r$ depends on the smoothness of the distributions and the dimensionality of the observations in the same way as for nonparametric density estimation with integrated square error loss

On inconsistent bayes estimates in the discrete case

Abstract: Consider sampling from an unknown probability distribution on the integers. With a tail-free prior, the posterior distribution is consistent. With a mixture of a tail-free prior and a point mass, however, the posterior may be inconsistent. This is likewise true for a countable mixture of tail-free priors. Similar results are given for Dirichlet priors.