Maximum Entropy And Bayesian Spectral Analysis And Estimation Problems

This volume has its origin in the third ·Workshop on Maximum-Entropy and Bayesian Methods in Applied Statistics,· held at the University of Wyoming, August 1 to 4, 1983. It was anticipated that the proceedings of this workshop could not be prepared in a timely fashion, so most of the papers were not collected until a year or so ago. Because most of the papers are in the nature of advancing theory or solving specific problems, as opposed to status reports, it is believed that the contents of this volume will be of lasting interest to the Bayesian community. The workshop was organized to bring together researchers from differ ent fields to examine critically maximum-entropy and Bayesian methods in science, engineering, medicine, economics, and other disciplines. Some of the papers were chosen specifically to kindle interest in new areas that may offer new tools or insight to the reader or to stimulate work on pressing problems that appear to be ideally suited to the maximum-entropy or Bayes ian method.

This volume contains the text of the twenty-five papers presented at two workshops entitled Maximum-Entropy and Bayesian Methods in Applied Statistics, which were held at the University of Wyoming from June 8 to 10, 1981, and from August 9 to 11, 1982. The workshops were organized to bring together researchers from different fields to critically examine maxi mum-entropy and Bayesian methods in science, engineering, medicine, oceanography, economics, and other disciplines. An effort was made to maintain an informal environment where ideas could be easily ~xchanged. That the workshops were at least partially successful is borne out by the fact that there have been two succeeding workshops, and the upcoming Fifth Workshop promises to be the largest of all. These workshops and their proceedings could not have been brought to their final form without the substantial help of a number of people. The support of David Hofmann, the past chairman, and Glen Rebka, Jr. , the present chairman of the Physics Department of the University of Wyoming, has been strong and essential. Glen has taken a special interest in seeing that the proceedings have received the support required for their comple tion. The financial support of the Office of University Research Funds, University of Wyoming, is gratefully acknowledged. The secretarial staff, in particular Evelyn Haskell, Janice Gasaway, and Marce Mitchum, of the University of Wyoming Physics Department has contributed a great number of hours in helping C. Ray Smith organize and direct the workshops.

The 10th International Workshop on Maximum Entropy and Bayesian Methods, MaxEnt 90, was held in Laramie, Wyoming from 30 July to 3 August 1990. This volume contains the scientific presentations given at that meeting. This series of workshops originated in Laramie in 1981, where the first three of what were to become annual workshops were held. The fourth meeting was held in Calgary. the fifth in Laramie, the sixth and seventh in Seattle, the eighth in Cambridge, England, and the ninth at Hanover, New Hampshire. It is most appropriate that the tenth workshop, occurring in the centennial year of Wyoming's statehood, was once again held in Laramie. The original purpose of these workshops was twofold. The first was to bring together workers from diverse fields of scientific research who individually had been using either some form of the maximum entropy method for treating ill-posed problems or the more general Bayesian analysis, but who, because of the narrow focus that intra-disciplinary work tends to impose upon most of us, might be unaware of progress being made by others using these same techniques in other areas. The second was to introduce to those who were somewhat aware of maximum entropy and Bayesian analysis and wanted to learn more, the foundations, the gestalt, and the power of these analyses. To further the first of these ends, presenters at these workshops have included workers from area. s as varied as astronomy, economics, environmenta.

Forty years ago, in 1957, the Principle of Maximum Entropy was first intro duced by Jaynes into the field of statistical mechanics. Since that seminal publication, this principle has been adopted in many areas of science and technology beyond its initial application. It is now found in spectral analysis, image restoration and a number of branches ofmathematics and physics, and has become better known as the Maximum Entropy Method (MEM). Today MEM is a powerful means to deal with ill-posed problems, and much research work is devoted to it. My own research in the area ofMEM started in 1980, when I was a grad uate student in the Department of Electrical Engineering at the University of Sydney, Australia. This research work was the basis of my Ph.D. the sis, The Maximum Entropy Method and Its Application in Radio Astronomy, completed in 1985. As well as continuing my research in MEM after graduation, I taught a course of the same name at the Graduate School, Chinese Academy of Sciences, Beijingfrom 1987to 1990. Delivering the course was theimpetus for developing a structured approach to the understanding of MEM and writing hundreds of pages of lecture notes.

The 10th International Workshop on Maximum Entropy and Bayesian Methods, MaxEnt 90, was held in Laramie, Wyoming from 30 July to 3 August 1990. This volume contains the scientific presentations given at that meeting. This series of workshops originated in Laramie in 1981, where the first three of what were to become annual workshops were held. The fourth meeting was held in Calgary. the fifth in Laramie, the sixth and seventh in Seattle, the eighth in Cambridge, England, and the ninth at Hanover, New Hampshire. It is most appropriate that the tenth workshop, occurring in the centennial year of Wyoming's statehood, was once again held in Laramie. The original purpose of these workshops was twofold. The first was to bring together workers from diverse fields of scientific research who individually had been using either some form of the maximum entropy method for treating ill-posed problems or the more general Bayesian analysis, but who, because of the narrow focus that intra-disciplinary work tends to impose upon most of us, might be unaware of progress being made by others using these same techniques in other areas. The second was to introduce to those who were somewhat aware of maximum entropy and Bayesian analysis and wanted to learn more, the foundations, the gestalt, and the power of these analyses. To further the first of these ends, presenters at these workshops have included workers from area. s as varied as astronomy, economics, environmenta.

This volume has its origin in the Fifth, Sixth and Seventh Workshops on and Bayesian Methods in Applied Statistics", held at "Maximum-Entropy the University of Wyoming, August 5-8, 1985, and at Seattle University, August 5-8, 1986, and August 4-7, 1987. It was anticipated that the proceedings of these workshops would be combined, so most of the papers were not collected until after the seventh workshop. Because all of the papers in this volume are on foundations, it is believed that the con tents of this volume will be of lasting interest to the Bayesian community. The workshop was organized to bring together researchers from different fields to critically examine maximum-entropy and Bayesian methods in science and engineering as well as other disciplines. Some of the papers were chosen specifically to kindle interest in new areas that may offer new tools or insight to the reader or to stimulate work on pressing problems that appear to be ideally suited to the maximum-entropy or Bayesian method. A few papers presented at the workshops are not included in these proceedings, but a number of additional papers not presented at the workshop are included. In particular, we are delighted to make available Professor E. T. Jaynes' unpublished Stanford University Microwave Laboratory Report No. 421 "How Does the Brain Do Plausible Reasoning?" (dated August 1957). This is a beautiful, detailed tutorial on the Cox-Polya-Jaynes approach to Bayesian probability theory and the maximum-entropy principle.

This work is essentially an extensive revision of my Ph.D. dissertation, [1J. It 1S primarily a research document on the application of probability theory to the parameter estimation problem. The people who will be interested in this material are physicists, economists, and engineers who have to deal with data on a daily basis; consequently, we have included a great deal of introductory and tutorial material. Any person with the equivalent of the mathematics background required for the graduate level study of physics should be able to follow the material contained in this book, though not without eIfort. From the time the dissertation was written until now (approximately one year) our understanding of the parameter estimation problem has changed extensively. We have tried to incorporate what we have learned into this book. I am indebted to a number of people who have aided me in preparing this docu ment: Dr. C. Ray Smith, Steve Finney, Juana Sunchez, Matthew Self, and Dr. Pat Gibbons who acted as readers and editors. In addition, I must extend my deepest thanks to Dr. Joseph Ackerman for his support during the time this manuscript was being prepared.