A Bayes tour of inversion: A tutorial

It is unclear whether one can (or should) write a tutorial about Bayes. It is a little like writing a tutorial about the sense of humor. However, this tutorial is about the Bayesian approach to the solution of the ubiquitous inverse problem. Inasmuch as it is a tutorial, it has its own special ingredients. The first is that it is an overview; details are omitted for the sake of the grand picture. In fractal language, it is the progenitor of the complex pattern. As such, it is a vision of the whole. The second is that it does, of necessity, assume some ill‐defined knowledge on the part of the reader. Finally, this tutorial presents our view. It may not appeal to, let alone be agreed to, by all.

[1]  L. M. M.-T..  Theory of Probability , 1929, Nature.

[2]  Tadeusz J. Ulrych,et al.  Signal and noise separation: Art and science , 1999 .

[3]  M. H. Bretherton,et al.  Statistics in Theory and Practice , 1966 .

[4]  Rodney W. Johnson,et al.  Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy , 1980, IEEE Trans. Inf. Theory.

[5]  G. Kitagawa,et al.  Akaike Information Criterion Statistics , 1988 .

[6]  E. Jaynes.  On the rationale of maximum-entropy methods , 1982, Proceedings of the IEEE.

[7]  Marvin H. J. Guber.  Bayesian Spectrum Analysis and Parameter Estimation , 1988 .

[8]  F. J. Jacobs,et al.  Spiking band-limited traces with a relative-entropy algorithm , 1991 .

[9]  B. Roy Frieden,et al.  Probability, Statistical Optics, And Data Testing , 1982 .

[10]  Tad J. Ulrych,et al.  Information theory measures with application to model identification , 1986, IEEE Trans. Acoust. Speech Signal Process..

[11]  D. Oldenburg,et al.  NON-LINEAR INVERSION USING GENERAL MEASURES OF DATA MISFIT AND MODEL STRUCTURE , 1998 .

[12]  T. Fearn,et al.  Bayesian statistics : principles, models, and applications , 1990 .

[13]  A. Duijndam.  BAYESIAN ESTIMATION IN SEISMIC INVERSION. PART I: PRINCIPLES1 , 1988 .

[14]  Anatolii V. Andreev.  Probability, statistical optics, and data testing , 1992 .

[15]  Michel Roussignol,et al.  Bayesian statistics of non-linear inverse problems: example of the magnetotelluric 1-D inverse problem , 1994 .

[16]  T. Ulrych,et al.  Minimum relative entropy and probabilistic inversion in groundwater hydrology , 1998 .

[17]  A. Duijndam.  BAYESIAN ESTIMATION IN SEISMIC INVERSION. PART II: UNCERTAINTY ANALYSIS1 , 1988 .

[18]  F. Guess.  Bayesian Statistics: Principles, Models, and Applications , 1990 .

[19]  石黒 真木夫,et al.  Akaike information criterion statistics , 1986 .

[20]  Mauricio D. Sacchi,et al.  High‐resolution velocity gathers and offset space reconstruction , 1995 .

[21]  Mauricio D. Sacchi,et al.  Interpolation and extrapolation using a high-resolution discrete Fourier transform , 1998, IEEE Trans. Signal Process..

[22]  Luis Tenorio,et al.  Prior information and uncertainty in inverse problems , 2001 .

[23]  Amin Bassrei,et al.  Minimum relative entropy inversion of 1D data with applications , 1990 .

[24]  B. Efron,et al.  Stein's Estimation Rule and Its Competitors- An Empirical Bayes Approach , 1973 .