Showing papers on "Bayes' theorem published in 1978"

Estimating the Dimension of a Model

Abstract: The problem of selecting one of a number of models of different dimensions is treated by finding its Bayes solution, and evaluating the leading terms of its asymptotic expansion. These terms are a valid large-sample criterion beyond the Bayesian context, since they do not depend on the a priori distribution.

Bayesian classification in a time-varying environment

Abstract: The problem of classifying a pattern based on multiple observation made in a time-varying environment is analyzed. The identity of the pattern may itself change. A Bayesian solution is derived, after which the conditions of the physical situation are invoked to produce a cascade classifier model. Experimental results based on remote sensing data demonstrate the effectiveness of the classifier.

A Quasi‐Bayes Sequential Procedure for Mixtures

Abstract: SUMMARY Coherent Bayes sequential learning and classification procedures are often useless in practice because of ever-increasing computational requirements. On the other hand, computationally feasible procedures may not resemble the coherent solution, nor guarantee consistent learning and classification. In this paper, a particular form of classification problem is considered and a "quasi-Bayes" approximate solution requiring minimal computation is motivated and defined. Convergence properties are established and a numerical illustration provided.

Empirical Bayes Estimation of a Distribution (Survival) Function from Right Censored Observations

Abstract: : This paper provides an empirical Bayes approach to the problem of nonparametric estimation of a distribution (or survival) function when the observations are censored on the right. The results use the notion of a Dirichlet process prior (Ferguson, 1973, Ann. Stat., 2, 209-230). The paper presents a generalization to the case of right censored observations of the rate result of an empirical Bayes nonparametric estimator of a distribution function of Korwar and Hollander ((1974, Tech. Rept. No. 288, Dept. of Statistics, Florida State University) in the uncensored case. The rate of asymptotic convergence to optimality is shown to be the best obtainable for the problem considered.

The portfolio choice problem: Comparison of certainty equivalence and optimal bayes portfolios

Abstract: For the portfolio problem with unknown parameter values, we compare the conventional certainty equivalence portfolio choice with the optimal Bayes portfolio. In the important single risky asset case a diffuse Bayes rule leads to portfolios that differ significantly from those suggested by a certainty equivalence rule which we show are inadmissible relative to a quadratic utility function for the range of parameters we consider. These results are invariant to arbitrary changes in the utility function parameters. We illustrate the results using a simple mutual fund example.

Generalized Bayes Estimators in Multivariate Problems

Abstract: Several problems involving multivariate generalized Bayes estimators are investigated. First, a characterization of admissible estimators as generalized Bayes estimators is developed for certain multivariate exponential families and quadratic loss. The problem of verifying whether or not an estimator is generalized Bayes is also considered. Next, an important class of estimators for a multivariate normal mean is considered. (The class includes many minimax, empirical Bayes, and ridge regression estimators of current interest.) Necessary conditions are developed for an estimator in this class to be "nearly" generalized Bayes, in the sense that if it were properly smoothed, it would be generalized Bayes. An application to adaptive ridge regression is given. The paper concludes with the development of an asymptotic approximation to generalized Bayes estimators for general losses and location vector densities. Using this approximation, weakened versions of the above results are obtained for general losses and densities.

Assessing Probability with Multiple Individuals: Group Interaction Versus Mathematical Aggregation.

Abstract: : The application of decision theory often involves assessing subjective probabilities and procedures for assessing them are quite well developed. But such procedures are based on assessments by a single person. Often multiple individuals are called on to provide the probabilistic judgments. Unanimity in judgments among the multiple individuals cannot be expected, thereby creating the problem of how to arrive at a single probability distribution that can be used in applying decision theory. Two general approaches to this problem exist. The individuals can interact as a group to reach a consensus, or the individual judgments can be mathematically aggregated to produce a single probability distribution. Each of these approaches has advantages and disadvantages. Group interaction allows the exchange of information, but may be susceptible to dominance by certain individuals or pressure for conformity. Mathematical aggregation is simple to use and ensures that a single distribution will result, but theoretical difficulties are encountered in specifying an appropriate aggregation model. Using several forms of group interaction and mathematical aggregation models, this research investigated the quality of probabilities produced by interaction versus mathematical models.

On the performance of some subset selection procedures

Abstract: In this paper, we restrict attention to the problem of subset selection of normal populations. The approaches and results of some previous comparison studies of subset selection procedures are discussed briefly. And then the result of a new Monte Carlo study comparing the performance of two classical procedures and the Bayes procedure is presented.

The Bhattacharyya distance and detection between Markov chains

Abstract: When the statistical structure under each of two hypotheses is time varying, the collection of infinitely many observations does not guarantee an error probability that approaches zero. A recursive formula for the Bhattacharyya distance between two Markov chains is derived, and it is used to derive necessary and sufficient conditions for asymptotically perfect detection (APD). It is shown that the use of incorrect prior probabilities in the Bayes detection rulee does not affect AID. The results are also extended to time-continuons finite-state Markov observations. An application is analyzed, in which the behavior of a message buffer is monitored for the purpose of detecting malfunctions in a computer communication network.

Bayes Linear Estimators

Abstract: A class of linear estimators, called Bayes linear estimators, is developed by finding, among all linear estimators, ones which have least average total mean squared error, averaged over parameter points. Ridge, generalized ridge, restricted least squares, subset least squares, least squares, best, and generalized inverse linear estimators are all either Bayes linear estimators or limits of Bayes linear estimators. Results on Bayes linear estimators are extended to affine estimators. “Bootstrapping” procedures, in which the data are recycled in the guise of prior information, are discussed.

A pattern deformational model and Bayes error- correcting recognition system

Abstract: A pattern deformational model is proposed in this paper. Pattern deformations are categorized into two types: local deformation and structural deformation. A structure-preserving local deformation can be decomposed into a syntactic deformation followed by a semantic deformation, the former being induced on primitive structures and the latter on primitive properties. Bayes error-correcting parsing algorithms are proposed accordingly which not only can perform normal syntax analysis but also can make statistical decisions. An optimum Bayes error-correcting recognition system is then formulated for pattern classification. The system can be considered as a hybrid pattern classifier which uses both syntactic and statistical pattern recognition techniques.

A new approach to filtering and adaptive control

Abstract: A simple distribution-free method is proposed for directly estimating and updating a criterion function without recourse to prior state space specification, updated state probabilities, and Bayes' rule. Optimality properties and efficiency advantages of the method are illustrated in terms of a two-armed bandit problem. The relationship between direct criterion function estimation and Kalman-Bucy filtering is clarified.

State estimation under uncertain observations with unknown statistics

Abstract: The asymptotic behavior of a Bayes optimal adaptive estimation scheme for a linear, discrete-time system with interrupted observations is investigated. The interrupted observation mechanism is expressed in terms of a stationary two-state Markov chain. The transition probability matrix is unknown and can take values only from a finite set.

Hypothesis Generation and Plausibility Assessment.

Abstract: : A hypothesis generation model is described which consists of two sub-processes. Hypotheses are retrieved from memory using several data as retrieval cues in the hypothesis retrieval sub-process. These hypotheses are then evaluated by a plausibility assessment sub-process. Two experiments are described. A memory retrieval experiment examined hypothesis retrieval from memory using multiple data. A memory-tagging model is described which predicts the probability of multi-data hypothesis retrieval. Performance in this task was poor; subjects rarely generated an adequate hypothesis set. A second plausibility assessment experiment was performed where subjects estimated the plausibility of specified hypotheses using varying amounts of data. Plausibility assessments for specified hypotheses were usually extreme in comparison to the posterior odds calculated by Bayes' theorem. This result was also attributed to deficiencies in hypothesis retrieval from memory. (Author)

The Bayes Sequential Procedure for Estimating the Arrival Rate of a Poisson Process

Abstract: Let Wn, n = 0, 1, …, be the time until the nth arrival of a Poisson process with rate θ. Using loss L (θ, ) = θ−2(θ − )2 and sampling costs involving cost per arrival and cost per unit time, the Bayes sequential procedure is derived. The large-sample properties of the procedure are then studied in the classical framework, and the optimal stopping time N* is shown to be asymptotically equivalent to n*, that function of θ which minimizes the expected total cost given θ. Asymptotic normality of the optimal sequential estimator is also shown. Finally, the asymptotic saving in expected total cost from using the Bayes sequential procedure instead of the Bayes fixed sample size procedure is computed.

Universal Bayes Estimators

Abstract: Let $x_1, \cdots, x_n$ be i.i.d. random variables with a distribution depending on the real parameter. Under what conditions is a generalized Bayes estimator independent of the choice of the even loss function? The known answer to this question is that this independence holds if the posterior density is symmetric and unimodal. The description of distributions and corresponding generalized prior densities on the real line, for which the posterior density is symmetric and unimodal, is presented. These families form an important subclass of all exponential laws with two-dimensional sufficient statistics.

A Note on Decision Theoretic Coefficients for Tests.

Abstract: Recently it was suggested that the Bayes risk might be used to characterize tests. To conform to common practices about indexes, a rescaling of the Bayes risk was proposed. The motivation for this new coefficient, d, was to provide an index that has a large value when the Bayes risk is small and that has a value in the closed interval [0, 1]. However, since d might have a value outside this range, a modification of d is described which yields an index that always has a value between zero and one.