Posterior Pdf Research Articles

Comments on "A general method of minimum cross-entropy spectral estimation"

In a recent correspondence Tzannes et al. [1] introduced a new form of minimum cross-entropy (MCE) spectral analysis based on the observation that normalized spectra and symmetric probability density functions (pdf's) are axiomatically indistinguishable. Thus, the form of the minimum cross-entropy posterior pdf can be used as a model for spectral estimation. In this correspondence we derive the same result without using a Lagrange multiplier methodology. We also establish the connection of this estimate to the cepstral representation of the signal and propose an efficient algorithm not requiring explicit solution of nonlinear equations.

Bayesian estimation of hypocenter with origin time eliminated.

In this study a new approach to hypocenter location is formulated from a Bayesian point of view. Observational data for hypocenter location are the arrival times of phases at seismic stations. Unknown parameters are the spatial coordinates and origin time of a quake focus. In general, we have some prior information about the spatial coordinates, but not about the origin time. Then, according to Bayes' theorem, the probability density function (pdf) of hypocenter parameters posterior to observed data is proportional to a product of the likelihood and the prior pdf of the spatial coordinates. Integrating the posterior pdf over the whole range of origin time, we can eliminate the origin time from the location problem, and obtain the marginal pdf of only the spatial coordinates. The best estimate of a hypocenter is given by a set of the spatial coordinates which maximizes the marginal pdf. Supposing Gaussian errors in both observational and prior data, we obtain a simple and robust algorithm to invert arrival time data for a hypocenter. Estimation errors of parameters are evaluated by an asymptotic covariance matrix, which gives a good approximation to exact covariance when the estimate is linearly close to a true hypocenter. The advantages of this approach are as follows : i) the use of prior data resolves unknown parameters completely within finite errors, and ii) the elimination of origin time suppresses the instability of estimates.

A minimum Bayes risk approach to optimal portfolio choice

This paper presents a Bayes decision approach to optimal portfolio choice under uncertainty. The approach ties together the estimation and decision problems. It duly recognizes the uncertainty of estimation of unknown parameters of asset return distributions by postulating a posterior PDF. It differentiates between the true-optimal solution and a quasi-optimal solution and defines the difference between the realized utilities under these two different solutions as a loss. Then it treats the expected loss with respect to the posterior PDF as the Bayes risk. The optimal portfolio choice is formulated as a constrained optimization (nonlinear programming) problem. The Bayes solution is derived in general and is solved for diffuse and conjugate priors in particular. An illustrative example is provided. The numerical results are reported and compared to those obtained according to the traditional method that ignores estimation risk. To observe how estimation risk affects optimal portfolio decision, the Bayes ex...

A Bayesian Note on Reliability Growth During a Development Testing Program

The problem of estimating the reliability of a system which is undergoing development testing is considered from a Bayesian standpoint. Formally, m sets of binomial trials are performed under conditions which lead to an ordering, θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> < θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2m</sub> , of the binomial parameters. The parameter of interest is θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> , the final underlying reliability of the system. The marginal posterior pdf for θ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> is easily obtained when uniform prior pdf's are assumed. The method is illustrated.

A Bayesian test of the product cycle hypothesis applied to Japanese crude steel production

Abstract It is postulated that the product cycle of crude steel production may be observed as a gradual shift in the time trend parameter in the family of the CES production functions. Quarterly data from 1957 to 1973 are used for the empirical test. Posterior pdf's for the join point as well as for the adjustment parameter are derived, and they indicate that Japanese crude steel production passed from the new to standardized product stage between 1968 and 1972. With the arrival of the standardized state it is noted that the industry has become less competitive.

Bayesian Limits for the Reliability of Pass/Fail Parallel Units

This paper employs the Mellin transform to derive the exact posterior pdf and Cdf of the reliability of a system consisting of s-independent units in parallel, based on pass-fail data obtained from testing the units. From a tabulation of the posterior Cdf, arbitrary Bayesian limits can be obtained for the reliability of the parallel system. A computer program for determining the requisite posterior pdf and Cdf is operational.

Asymptotic Normality of the Posterior Distribution for Exponential Models

Let $f(x)$ be a $\operatorname{pdf}$ of exponential form with respect to the measure $\mu$. Suppose a prior $\operatorname{pdf}$ $\pi$ has been placed on the natural parameter space $\Omega$, where $\pi$ is a density (with respect to $m$-dimensional Lebesgue measure) which is both positive and continuous at $\tau^\ast$, the true but unknown parameter value. Using basic properties of exponential families and certain associated convex functions, it is shown that the posterior pdf tends to the multivariate normal.

Bayesian Confidence Limits for the Reliability of Mixed Cascade and Parallel Independent Exponential Subsystems

This paper deals with the theoretical problem of derving Bayesian confidence intervals for the reliability of a system consisting of both cascade and parallel subsystems where each subsystem is independent and has an exponential failure probability density function (pdf). This approach is applicable when test data are available for each individual subsystem and not for the enfire system. The Mellin integral transform is used to analyze the system in a step-by-step procedure until the posterior pdf of the system reliability is obtained. The posterior cumulative distribution function is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the accuracy of the derived pdf. A computer program has been written in FORTRAN IV to evaluate the confidence limits. An example is presented which uses the computer program.

Use of Prior Information in the Analysis and Estimation of Cobb-Douglas Production Function Models

IN ANALYSES OF PRODUCTION-FUNCTION MODELS employing aggregate time-series data, it is often difficult to obtain precise estimates of parameters because input variables are usually highly intercorrelated. For example, in the case of a CobbDouglas (CD) production function with neutral technical change, the labor, capital, and time-trend variables are frequently highly intercorrelated, a fact that may lead to imprecise parameter estimates and, as will be seen below with U.S. data, highly implausible point estimates for certain parameters. It is generally recognized that introduction of prior information is one possible way of dealing with the above problems-sometimes referred to as multicollinearity problems. Another course of action in dealing with these problems is to extend the data base-say, by combining cross-section data with the available timeseries data. In both approaches, or in a combination of them, additional information is added to the information in the aggregate time-series sample in an attempt to improve the quality of inferences. In the present paper, we review a sampling-theory analysis of the CD production function put forward by Morishima and Saito [3], in which nondata-based prior information was added to aggregate time-series information in order to deal with what appears to be a multicollinearity problem. Then we turn to an analysis of the Morishima-Saito (MS) production function problem, using the. Bayesian approach in which nondata-based prior information is introduced by use of prior probability density functions (pdf's) for the parameters. By the use of prior pdf's in the Bayesian approach, it will be seen that prior information can be introduced in a rather flexible manner, and posterior pdf's for parameters of interest, that reflect both sample and prior information, can be readily computed. In particular, we wish to assess how sensitive inferences about the technical-change parameter and other parameters are to the form and manner in which prior information is introduced. The plan of the paper is as follows. In Section 2, we state the problem and review some sampling-theory results. We then present Bayesian analyses of the problem in Section 3 and provide a summary of results and some concluding remarks in Section 4.

A Bayes Analysis of Availability for a System Consisting of Several Independent Subsystems

A Bayes analysis of system availability A is carried out on the basis of snapshot data obtained on each of the N component subsystems. Application of Bayes formula, using a natural conjugate prior probability density function (pdf), yields the posterior pdf of subsystem availability. Determination of the posterior pdf of system availability requires the derivation of the pdf of the product of N independent random variables, which is accomplished through the use of the Mellin integral transform. From the knowledge of the posterior pdf of A, confidence limits on system availability can be obtained.

Bayesian Confidence Limits for the Reliability of Mixed Exponential and Distribution-Free Cascade Subsystems

The problem treated here is the theoretical one of deriving exact Bayesian confidence intervals for the reliability of a system consisting of some independent cascade subsystems with exponential failure probability density functions (pdf) mixed with other independent cascade subsystems whose failure pdf's are unknown. The Mellin integral transform is used to derive the posterior pdf of the system reliability. The posterior cumulative distribution function (cdf) is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the derived pdf. A computer program written in Fortran IV is operational. It utilizes multiprecision to obtain the posterior pdf to any desired degree of accuracy in both functional and tabular form. The posterior cdf is tabulated at any desired increments to any required degree of accuracy.

The Posterior $t$ Distribution

The Posterior $t$ Distribution