Showing papers on "Bayesian inference published in 1971"

The Bernstein-Von Mises Theorem for Markov Processes

Abstract: Since the appearance of P. Billingsley's monograph [2] on the large sample inference in Markov processes in which the weak consistency and asymptotic normality of the maximum likelihood estimate was investigated, there has been considerable interest in the further development of the theory along other directions. Billingsley's work was mainly concerned with extending the results of H. Cramer ([4] page 500). Among more recent developments one might mention the proof of the almost sure consistency of the maximum likelihood estimator following the ideas of A. Wald by G. Roussas [7], and the results on asymptotic Bayes estimates obtained by Lorraine Schwartz [9]. In the present paper we extend to Markov processes one of the fundamental results in the asymptotic theory of inference, viz., the approach of the posterior density (in a sense to be made precise later) to the normal. When the observed chance variables are independent and identically distributed, this result was obtained by L. LeCam in [5] (page 309). The same author offers another derivation of this result in [6]. Special cases of the theorem were first given by S. Bernstein and R. von Mises (for reference see [5]). The regularity conditions satisfied by the transition probability density of the Markov process are given in Section 1. We prove in Theorem 2.4 of Section 2 those properties of the maximum likelihood estimator that are needed for the proof of the main result of the paper given in Section 3 (Theorem 3.1). Theorem 3.1 is stated in a form which is general enough to include the Bernstein-von Mises theorem as well as the somewhat sharper versions that are available when it is known that the prior probability distribution has a finite absolute moment of order $m$. Theorem 3.2 deduces these results as a consequence of Theorem 3.1. The latter result also enables us to prove a theorem on the asymptotic behavior of regular Bayes estimates. This is done in Theorem 4.1 of Section 4.

An introduction to probability, decision, and inference

Abstract: (1971) An Introduction to Probability, Decision, and Inference Technometrics: Vol 13, No 2, pp 450-450

Bayesian inference and the classical test theory model: Reliability and true scores

Abstract: A general one-way analysis of variance components with unequal replication numbers is used to provide unbiased estimates of the true and error score variance of classical test theory. The inadequacy of the ANOVA theory is noted and the foundations for a Bayesian approach are detailed. The choice of prior distribution is discussed and a justification for the Tiao-Tan prior is found in the particular context of the “n-split” technique. The posterior distributions of reliability, error score variance, observed score variance and true score variance are presented with some extensions of the original work of Tiao and Tan. Special attention is given to simple approximations that are available in important cases and also to the problems that arise when the ANOVA estimate of true score variance is negative. Bayesian methods derived by Box and Tiao and by Lindley are studied numerically in relation to the problem of estimating true score. Each is found to be useful and the advantages and disadvantages of each are discussed and related to the classical test-theoretic methods. Finally, some general relationships between Bayesian inference and classical test theory are discussed.

A study of an ordinary and empirical Bayes approach to reliability estimation in the gamma life testing model

Abstract: Service life testing and reliability estimation, using ordinary and empirical Bayes approach in failure model with gamma probability distribution

Statistics: Probability Inference and Decision, Volume 1

Abstract: (1971). Statistics: Probability Inference and Decision, Volume 1. Technometrics: Vol. 13, No. 4, pp. 920-920.

Bayesian Techniques for Test Selection.

Abstract: ing amount of attention has been given to the potential of the techniques included under the rubric of Bayesian statistics (Meyer, 1966). While the basic Bayesian theorem dates back to 1763, an interest in applications of the theorem has come into prominence only in the past few years. A perusal of this Bayesian literature indicates that while there can be little question as to the validity of the theorem, its applications have generated a great deal of controversy (Binder, 1964). The purpose of this article is not to present a detailed analysis of the rationale behind Bayesian inferential techniques. Several references are available which provide such information (Meyer, 1966; Binder, 1964; Edwards, Lindman, and Savage, 1963), and the reader desiring to explore in depth is referred to these sources. Rather, this article will present an example of Bayesian techniques for a specific problem, that of test selection. Horst (1966) indicated a need for new applications of decision theory to the area of psychological measurement and prediction, and this article is addressed to this need. The format of the article will be to present a hypothetical example of the application of Bayesian procedures in test selection and to discuss some of the ramifications of the method

A Geometrical Version of Bayes' Theorem

Abstract: (1971). A Geometrical Version of Bayes' Theorem. The American Statistician: Vol. 25, No. 5, pp. 45-46.