×

zbMATH — the first resource for mathematics

Bayesian methods and maximum entropy for ill-posed inverse problems. (English) Zbl 0871.62010
Summary: We study linear inverse problems where some generalized moments of an unknown positive measure are observed. We introduce a new construction, called the maximum entropy on the mean method (MEM), which relies on a suitable sequence of finite-dimensional discretized inverse problems. Its advantage is threefold: It allows us to interpret all usual deterministic methods as Bayesian methods; it gives a very convenient way of taking into account prior information; it also leads to new criteria for the existence question concerning the linear inverse problem which will be a starting point for the investigation of superresolution phenomena. The key tool in this work is the large deviations property of some discrete random measure connected with the reconstruction procedure.

MSC:
62B10 Statistical aspects of information-theoretic topics
62A01 Foundations and philosophical topics in statistics
62F15 Bayesian inference
60A99 Foundations of probability theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] BALDI, P. 1988. Large deviations and stochastic homogenization. Ann. Mat. Pura Appl. 11 161 177. Z. · Zbl 0654.60024
[2] BARRY, D. 1986. Nonparametric Bayesian regression. Ann. Statist. 14 934 953. Z. · Zbl 0608.62052
[3] BORWEIN, J. M. and LEWIS, A. S. 1993. Partially-finite programming in L and the existence of 1 maximum entropy estimates. SIAM J. Optim. 3 248 267. Z. · Zbl 0780.49015
[4] CATTIAUX, P. and LEONARD, C. 1994. Minimization of Kullback information for diffusion proćesses. Ann. Inst. H. Poincare Probab. Statist. 30 83 132. Ź.
[5] CATTIAUX, P. and GAMBOA, F. 1995. Large deviations and variational theorems for marginal problems. Preprint, Orsay. Z. · Zbl 0922.60029
[6] CSISZAR, I. 1984. Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab. 12 768 793. Z. · Zbl 0544.60011
[7] CSISZAR, I. 1985. An extended maximum entropy principle and a Bayesian justification theorem. In Bayesian Statistics 2 83 98. North-Holland, Amsterdam. Z. · Zbl 0672.62004
[8] CSISZAR, I. 1991. Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Statist. 19 2032 2066. Z. DACUNHA-CASTELLE, D. and DUFLO, M. 1986. Probability and Statistics. Springer, New York. Z. DACUNHA-CASTELLE, D. and GAMBOA, F. 1990. Maximum d’entropie et probleme des moments. Ann. Inst. H. Poincare Probab. Statist. 26 567 596. Ź. · Zbl 0753.62003
[9] DIACONIS, P. and FREEDMAN, D. 1986. On the consistency of Bay es estimates. Ann. Statist. 14 1 26. Z. · Zbl 0595.62022
[10] DONOHO, D., JOHNSTONE, I., HOCH, J. and STERN, A. 1992. Maximum entropy and the nearly black object. J. Roy. Statist. Soc. Ser. B 54 41 82. Z. JSTOR: · Zbl 0788.62103
[11] DONOHO, D. 1993. Superresolution via sparsity constraints. SIAM J. Math. Anal. 23 1309 1331. Z. · Zbl 0769.42007
[12] DONOHO, D. and GASSIAT, E. 1992. Superresolution via positivity constraints. Unpublished manuscript. Z.
[13] DOUKHAN, P. and GAMBOA, F. 1996. Superresolution rates in Prokhorov metric. Canad. J. Math. 48 316 329.Z. · Zbl 0861.60005
[14] EKELAND, I. and TEMAM, R. 1976. Convex Analy sis and Variational Problems. North-Holland, Amsterdam. Z.
[15] FERGUSON, T. S. 1974. Prior distributions on spaces of probability. Ann. Statist. 2 615 629. Z. · Zbl 0286.62008
[16] FRIEDEN, B. R. 1985. Dice, entropy and likelihood. Proc. IEEE 73 1764 1770. Z.
[17] GAMBOA, F. 1989. Methode du maximum d’entropie sur la moy enne et applications. These Órsay. Z.
[18] GAMBOA, F. 1994. New Bayesian methods for ill posed problems. Preprint, Orsay. Z. · Zbl 0952.62027
[19] GAMBOA, F. and GASSIAT, E. 1991. Maximum d’entropie et probleme des moments: cas multidi mensionnel. Probab. Math. Statist. 12 67 83. Z. · Zbl 0766.60003
[20] GAMBOA, F. and GASSIAT, E. 1994. The maximum entropy method on the mean: applications to linear programming and superresolution. Math. Programming Ser. A 66 103 122. Z. · Zbl 0811.90112
[21] GAMBOA, F. and GASSIAT, E. 1996a. Sets of superresolution and the maximum entropy method on the mean. SIAM J. Math. Anal. 27 1129 1152. · Zbl 0936.62005
[22] GAMBOA, F. and GASSIAT, E. 1996b. Blind deconvolution of discrete linear sy stems. Ann. Statist. 24 1964 1981. Z. · Zbl 0867.62073
[23] GASSIAT, E. 1991. Probleme des moments et concentration de mesures C. R. Acad. Sci. Paris Ser. I Math. 310 41 44. Ź.
[24] KREIN, M. G. and NUDEL’MAN, A. A. 1977. The Markov moment problem and extremal problems. Translations of Mathematical Monographs 50. Amer. Math. Soc., Providence, RI. Z. · Zbl 0361.42014
[25] LIVESEY, A. K. and SKILLING, J. 1985. Maximum entropy theory. Acta Cry st. Sect. A 41 113 122. Z.
[26] MCLAUGHLIN, D. W. 1984. Inverse Problems. Amer. Math. Soc., Providence, RI. Z. · Zbl 0551.34014
[27] ROBERT, C. 1990a. An entropy concentration theorem. J. Appl. Probab. 27 303 313. Z. JSTOR: · Zbl 0705.60031
[28] ROBERT, R. 1990b. Etats d’equilibre statistique pour l’ecoulement bidimensionnel d’un fluide \' ṕarfait. C. R. Acad. Sci. Paris Ser. I Math. 311 575 578. Ź. · Zbl 0707.76002
[29] ROCKAFELLAR, R. T. 1970. Convex Analy sis. Princeton Univ. Press. Z.
[30] ROCKAFELLAR, R. T. 1971. Integrals which are convex functionals II. Pacific J. Math. 39 439 469. Z. · Zbl 0236.46031
[31] VAN CAMPENHOUT, J. M. and COVER, T. M. 1981. Maximum entropy and conditional probability. IEEE Trans. Inform. Theory 27 483 489. Z. · Zbl 0459.94009
[32] VARADHAN, S. R. S. 1984. Large Deviations and Applications. SIAM, Philadelphia, PA. Z. · Zbl 0549.60023
[33] WAHBA, G. 1990. Spline Models for Observational Data. SIAM, Philadelphia, PA. · Zbl 0813.62001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.