Inverse Problems as Statistics

  • P. B. Stark


What mathematicians, scientists, engineers, and statisticians mean by “inverse problem” differs. For a statistician, an inverse problem is an inference or estimation problem. The data are finite in number and contain errors, as they do in classical estimation or inference problems, and the unknown typically is infinite-dimensional, as it is in nonparametric regression. The additional complication in an inverse problem is that the data are only indirectly related to the unknown. Standard statistical concepts, questions, and considerations such as bias, variance, mean-squared error, identifiability, consistency, efficiency, and various forms of optimality apply to inverse problems. This article discusses inverse problems as statistical estimation and inference problems, and points to the literature for a variety of techniques and results.


Inverse Problem Separable Banach Space Prior Probability Distribution Wavelet Shrinkage Maximum Risk 
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  1. 1.
    N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337–404, 1950.MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    G. Backus. Inference from inadequate and inaccurate data, I. Proc. Natl. Acad. Sci., 65:1–7, 1970.CrossRefGoogle Scholar
  3. 3.
    G. Backus. Inference from inadequate and inaccurate data, II. Proc. Natl. Acad. Sci., 65:281–287, 1970.CrossRefMathSciNetGoogle Scholar
  4. 4.
    G. Backus. Inference from inadequate and inaccurate data, III. Proc. Natl. Acad. Sci., 67:282–289, 1970.CrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Backus and F. Gilbert. The resolving power of gross Earth data. Geophys. J. R. astron. Soc., 16:169 205, 1968.Google Scholar
  6. 6.
    G.E. Backus. Isotropic probability measures in infinite-dimensional spaces. Proc. Natl. Acad. Sci., 84:8755–8757, 1987.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    G.E. Backus. Bayesian inference in geomagnetism. Geophys. J., 92:125–142, 1988.MATHCrossRefGoogle Scholar
  8. 8.
    G.E. Backus. Comparing hard and soft prior bounds in geophysical inverse problems. Geophys. J., 94:249–261, 1988.MATHCrossRefGoogle Scholar
  9. 9.
    G.E. Backus. Confidence set inference with a prior quadratic bound. Geophys. J., 97:119–150, 1989.MATHCrossRefGoogle Scholar
  10. 10.
    G.E. Backus. Trimming and procrastination as inversion techniques. Phys. Earth Planet. Inter., 98:101–142, 1996.CrossRefGoogle Scholar
  11. 11.
    R.F. Bass. Probabilistic Techniques in Analysis. Springer-Verlag, New York, 1995.MATHGoogle Scholar
  12. 12.
    S.C. Constable, R.L. Parker, and C.G. Constable. Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics, 52:289–300, 1987.CrossRefGoogle Scholar
  13. 13.
    P.W. Diaconis and D. Freedman. Consistency of Bayes estimates for nonparametric regression: normal theory. Bernoulli, 4:411–444, 1998.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    D.L. Donoho. One-sided inference about functionals of a density. Ann. Stat., 16:1390–1420, 1988.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    D.L. Donoho. Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Technical report, Dept. of Statistics, Stanford Univ., 1991.Google Scholar
  16. 16.
    D.L. Donoho. Exact asymptotic minimax risk for sup norm loss via optimal recovery. Probab. Theory and Rel. Fields, 99:145–170, 1994.MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    D.L. Donoho. Statistical estimation and optimal recovery. Ann. Stat., 22:238–270, 1994.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    D.L. Donoho and I.M. Johnstone. Wavelets and optimal nonlinear function estimates. Technical Report 281, Dept. Statistics, Univ. of Calif., Berkeley, 1990.Google Scholar
  19. 19.
    D.L. Donoho and I.M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81:425–455, 1994.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    D.L. Donoho and I.M. Johnstone. Minimax estimation via wavelet shrinkage. Ann. Stat., 26:879–921, 1998.MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    D.L. Donoho, I.M. Johnstone, G. Kerkyacharian, and D. Picard. Wavelet shrinkage: asymptopia? (with discussion). J. Roy. Stat. Soc, Ser. B, 57:301–369, 1995.MATHMathSciNetGoogle Scholar
  22. 22.
    D.L. Donoho and M. Nussbaum. Minimax quadratic estimation of a quadratic functional. J. Complexity, 6:290–323, 1990.MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    N. Dunford and J.T. Schwartz. Linear Operators. John Wiley and Sons, New York, 1988.Google Scholar
  24. 24.
    S.N. Evans and P.B. Stark. Shrinkage estimators, Skorokhod’s problem, and stochastic integration by parts. Ann. Stat., 24:809–815, 1996.MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    D.A. Freedman. Invariants under mixing which generalize de Finetti’s Theorem: continuous time parameter. Ann. Math. Stat., 34:1194–1216, 1963.MATHCrossRefGoogle Scholar
  26. 26.
    D.A. Freedman. Browman Motion and Diffusion. Springer-Verlag, New York, 1983.Google Scholar
  27. 27.
    A. Gelman, J. Carlin, H. Stern, and D.B. Rubin. Bayesian Data Analysis. Chapman & Hall, London, 1995.Google Scholar
  28. 28.
    C.R. Genovese and P.B. Stark. Data reduction and statistical consistency of ℓ p misfit norms in linear inverse problems. Phys. Earth Planet. Inter., 98:143–162, 1996.CrossRefGoogle Scholar
  29. 29.
    I.J. Good. The Estimation of Probabilities: An Essay on Modern Bayesian Methods. MIT Press, Cambridge, MA, 1965.MATHGoogle Scholar
  30. 30.
    J.A. Hartigan. Bayes Theory. Springer-Verlag, New York, 1983.MATHGoogle Scholar
  31. 31.
    N.W. Hengartner and P.B. Stark. Confidence bounds on the probability density of aftershocks. Technical Report 352, Dept. Statistics, Univ. of Calif., Berkeley, 1992.Google Scholar
  32. 32.
    N.W. Hengartner and P.B. Stark. Finite-sample confidence envelopes for shape-restricted densities. Ann. Stat., 23:525–550, 1995.MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    W. James and C. Stein. Estimation with quadratic loss. In Proc. Fourth Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 361–380, Berkeley, 1961. Univ. California Press.Google Scholar
  34. 34.
    A.N. Kolmogorov. Foundations of the Theory of Probability. Chelsea Publishing Co., New York, 2nd edition, 1956.MATHGoogle Scholar
  35. 35.
    L. Le Cam. Asymptotic Methods in Statistical Decision Theory. Springer-Verlag, New York, 1986.MATHCrossRefGoogle Scholar
  36. 36.
    L. Le Cam. Maximum likelihood: an introduction. Intl. Stat. Rev., 58:153–171, 1990.MATHCrossRefGoogle Scholar
  37. 37.
    E.L. Lehmann and G. Casella. Theory of Point Estimation. Springer-Verlag, New York, 2nd edition, 1998.MATHGoogle Scholar
  38. 38.
    F. O’Sullivan. A statistical perspective on ill-posed inverse problems. Statistical Science, 1:502–518, 1986.MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    X. Shen. On methods of sieves and penalization. Ann. Stat., 25:2555–2591, 1997.MATHCrossRefGoogle Scholar
  40. 40.
    P.B. Stark. Inference in infinite-dimensional inverse problems: Discretization and duality. J. Geophys. Res., 97:14 055–14 082, 1992.CrossRefGoogle Scholar
  41. 41.
    P.B. Stark. Minimax confidence intervals in geomagnetism. Geophys. J. Intl., 108:329–338, 1992.CrossRefGoogle Scholar
  42. 42.
    C. Stein. Inadmissibility of the usual estimator of the mean of a multivariate normal distribution. In Proc. Third Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 197–206, Berkeley, 1956. Univ. California Press.Google Scholar
  43. 43.
    A. Tarantola. Inverse Problem Theory: methods for data fitting and model parameter estimation. Elsevier Science Publishing Co., Amsterdam, 1987.MATHGoogle Scholar
  44. 44.
    L. Tenorio, P.B. Stark, and C.H. Lineweaver. Bigger uncertainties and the Big Bang. Inverse Problems, 15:329–341, 1999.MATHCrossRefGoogle Scholar
  45. 45.
    G. Wahba. Spline Models for Observational Data. Soc. for Industrial and Appl. Math., Philadelphia, PA, 1990.Google Scholar
  46. 46.
    A. Wald. Statistical Decision Functions. John Wiley and Sons, New York, 1950.MATHGoogle Scholar
  47. 47.
    L. Zadeh. Fuzzy sets. Inf. Control, 8:338–353, 1965.MATHCrossRefMathSciNetGoogle Scholar

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© Springer-Verlag/Wien 2000

Authors and Affiliations

  • P. B. Stark
    • 1
  1. 1.Department of StatisticsUniversity of CaliforniaBerkeleyUSA

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