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818Chapter 18.Integral Equations and Inverse TheoryCITED REFERENCES AND FURTHER READING:Backus, G.E., and Gilbert, F. 1968, Geophysical Journal of the Royal Astronomical Society,vol. 16, pp. 169–205. [1]Backus, G.E., and Gilbert, F. 1970, Philosophical Transactions of the Royal Society of LondonA, vol. 266, pp.
123–192. [2]Parker, R.L. 1977, Annual Review of Earth and Planetary Science, vol. 5, pp. 35–64. [3]Loredo, T.J., and Epstein, R.I. 1989, Astrophysical Journal, vol. 336, pp. 896–919. [4]18.7 Maximum Entropy Image RestorationAbove, we commented that the association of certain inversion methodswith Bayesian arguments is more historical accident than intellectual imperative.Maximum entropy methods, so-called, are notorious in this regard; to summarizethese methods without some, at least introductory, Bayesian invocations would beto serve a steak without the sizzle, or a sundae without the cherry. We shouldSample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).(Don’t let this notation mislead you into inverting the full matrix W(x) + λS. Youonly need to solve for some y the linear system (W(x) + λS) · y = R, and thensubstitute y into both the numerators and denominators of 18.6.12 or 18.6.13.)Equations (18.6.12) and (18.6.13) have a completely different character fromthe linearly regularized solutions to (18.5.7) and (18.5.8). The vectors and matrices in(18.6.12) all have size N , the number of measurements. There is no discretization ofthe underlying variable x, so M does not come into play at all. One solves a differentN × N set of linear equations for each desired value of x.
By contrast, in (18.5.8),one solves an M × M linear set, but only once. In general, the computational burdenof repeatedly solving linear systems makes the Backus-Gilbert method unsuitablefor other than one-dimensional problems.How does one choose λ within the Backus-Gilbert scheme? As alreadymentioned, you can (in some cases should) make the choice before you see anyactual data. For a given trial value of λ, and for a sequence of x’s, use equation(18.6.12) to calculate q(x); then use equation (18.6.6) to plot the resolution functionsb x0 ) as a function of x0 . These plots will exhibit the amplitude with whichδ(x,different underlying values x0 contributepto the point ub(x) of your estimate. For theu(x)] using equation (18.6.8). (Yousame value of λ, also plot the function Var[bneed an estimate of your measurement covariance matrix for this.)As you change λ you will see very explicitly the trade-off between resolutionand stability.
Pick the value that meets your needs. You can even choose λ to be afunction of x, λ = λ(x), in equations (18.6.12) and (18.6.13), should you desire todo so. (This is one benefit of solving a separate set of equations for each x.) Forthe chosen value or values of λ, you now have a quantitative understanding of yourinverse solution procedure. This can prove invaluable if — once you are processingreal data — you need to judge whether a particular feature, a spike or jump forexample, is genuine, and/or is actually resolved. The Backus-Gilbert method hasfound particular success among geophysicists, who use it to obtain information aboutthe structure of the Earth (e.g., density run with depth) from seismic travel time data.18.7 Maximum Entropy Image Restoration819Prob(A|B) = Prob(A)Prob(B|A)Prob(B)(18.7.1)Here Prob(A|B) is the probability of A given that B has occurred, and similarly forProb(B|A), while Prob(A) and Prob(B) are unconditional probabilities.“Bayesians” (so-called) adopt a broader interpretation of probabilities than doso-called “frequentists.” To a Bayesian, P (A|B) is a measure of the degree ofplausibility of A (given B) on a scale ranging from zero to one.
In this broader view,A and B need not be repeatable events; they can be propositions or hypotheses.The equations of probability theory then become a set of consistent rules forconducting inference [1,2] . Since plausibility is itself always conditioned on some,perhaps unarticulated, set of assumptions, all Bayesian probabilities are viewed asconditional on some collective background information I.Suppose H is some hypothesis.
Even before there exist any explicit data,a Bayesian can assign to H some degree of plausibility Prob(H|I), called the“Bayesian prior.” Now, when some data D1 comes along, Bayes theorem tells howto reassess the plausibility of H,Prob(H|D1 I) = Prob(H|I)Prob(D1 |HI)Prob(D1 |I)(18.7.2)The factor in the numerator on the right of equation (18.7.2) is calculable as theprobability of a data set given the hypothesis (compare with “likelihood” in §15.1).The denominator, called the “prior predictive probability” of the data, is in this casemerely a normalization constant which can be calculated by the requirement thatthe probability of all hypotheses should sum to unity.
(In other Bayesian contexts,the prior predictive probabilities of two qualitatively different models can be usedto assess their relative plausibility.)If some additional data D2 comes along tomorrow, we can further refine ourestimate of H’s probability, asProb(H|D2 D1 I) = Prob(H|D1 I)Prob(D2 |HD1 I)Prob(D1 |D1 I)(18.7.3)Using the product rule for probabilities, Prob(AB|C) = Prob(A|C)Prob(B|AC),we find that equations (18.7.2) and (18.7.3) implyProb(H|D2 D1 I) = Prob(H|I)Prob(D2 D1 |HI)Prob(D2 D1 |I)(18.7.4)which shows that we would have gotten the same answer if all the data D1 D2had been taken together.Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5)Copyright (C) 1988-1992 by Cambridge University Press.Programs Copyright (C) 1988-1992 by Numerical Recipes Software.Permission is granted for internet users to make one paper copy for their own personal use.
Further reproduction, or any copying of machinereadable files (including this one) to any servercomputer, is strictly prohibited. To order Numerical Recipes books,diskettes, or CDROMsvisit website http://www.nr.com or call 1-800-872-7423 (North America only),or send email to trade@cup.cam.ac.uk (outside North America).also comment in passing that the connection between maximum entropy inversionmethods, considered here, and maximum entropy spectral estimation, discussed in§13.7, is rather abstract. For practical purposes the two techniques, though bothnamed maximum entropy method or MEM, are unrelated.Bayes’ Theorem, which follows from the standard axioms of probability, relatesthe conditional probabilities of two events, say A and B:820Chapter 18.Integral Equations and Inverse TheoryFrom a Bayesian perspective, inverse problems are inference problems [3,4].The underlying parameter set u is a hypothesis whose probability, given the measureddata values c, and the Bayesian prior Prob(u|I) can be calculated.
We might wantto report a single “best” inverse u, the one that maximizesProb(u|I)Prob(c|I)(18.7.5)over all possible choices of u. Bayesian analysis also admits the possibility ofreporting additional information that characterizes the region of possible u’s withhigh relative probability, the so-called “posterior bubble” in u.The calculation of the probability of the data c, given the hypothesis u proceedsexactly as in the maximum likelihood method. For Gaussian errors, e.g., it is given by1Prob(c|uI) = exp(− χ2 )∆u1 ∆u2 · · · ∆uM2(18.7.6)where χ2 is calculated from u and c using equation (18.4.9), and the ∆uµ’s areconstant, small ranges of the components of u whose actual magnitude is irrelevant,because they do not depend on u (compare equations 15.1.3 and 15.1.4).In maximum likelihood estimation we, in effect, chose the prior Prob(u|I) tobe constant.
That was a luxury that we could afford when estimating a small numberof parameters from a large amount of data. Here, the number of “parameters”(components of u) is comparable to or larger than the number of measured values(components of c); we need to have a nontrivial prior, Prob(u|I), to resolve thedegeneracy of the solution.In maximum entropy image restoration, that is where entropy comes in.
Theentropy of a physical system in some macroscopic state, usually denoted S, is thelogarithm of the number of microscopically distinct configurations that all havethe same macroscopic observables (i.e., consistent with the observed macroscopicstate). Actually, we will find it useful to denote the negative of the entropy, alsocalled the negentropy, by H ≡ −S (a notation that goes back to Boltzmann). Insituations where there is reason to believe that the a priori probabilities of themicroscopic configurations are all the same (these situations are called ergodic), thenthe Bayesian prior Prob(u|I) for a macroscopic state with entropy S is proportionalto exp(S) or exp(−H).MEM uses this concept to assign a prior probability to any given underlyingfunction u.
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