Entropic Projections and Dominating Points

Christian Léonard

ESAIM: Probability and Statistics (2010)

  • Volume: 14, page 343-381
  • ISSN: 1292-8100

Abstract

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Entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for the existence of entropic projections, generalized entropic projections and dominating points. Representations of the generalized entropic projections are obtained. It is shown that they are the “measure component" of the solutions to some extended entropy minimization problem. This approach leads to new results and offers a unifying point of view. It also permits to extend previous results on the subject by removing unnecessary topological restrictions. As a by-product, new proofs of already known results are provided.

How to cite

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Léonard, Christian. "Entropic Projections and Dominating Points." ESAIM: Probability and Statistics 14 (2010): 343-381. <http://eudml.org/doc/250830>.

@article{Léonard2010,
abstract = { Entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for the existence of entropic projections, generalized entropic projections and dominating points. Representations of the generalized entropic projections are obtained. It is shown that they are the “measure component" of the solutions to some extended entropy minimization problem. This approach leads to new results and offers a unifying point of view. It also permits to extend previous results on the subject by removing unnecessary topological restrictions. As a by-product, new proofs of already known results are provided. },
author = {Léonard, Christian},
journal = {ESAIM: Probability and Statistics},
keywords = {Conditional laws of large numbers; random measures; large deviations; entropy; convex optimization; entropic projections; dominating points; Orlicz spaces; laws of large numbers; large deviations},
language = {eng},
month = {12},
pages = {343-381},
publisher = {EDP Sciences},
title = {Entropic Projections and Dominating Points},
url = {http://eudml.org/doc/250830},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Léonard, Christian
TI - Entropic Projections and Dominating Points
JO - ESAIM: Probability and Statistics
DA - 2010/12//
PB - EDP Sciences
VL - 14
SP - 343
EP - 381
AB - Entropic projections and dominating points are solutions to convex minimization problems related to conditional laws of large numbers. They appear in many areas of applied mathematics such as statistical physics, information theory, mathematical statistics, ill-posed inverse problems or large deviation theory. By means of convex conjugate duality and functional analysis, criteria are derived for the existence of entropic projections, generalized entropic projections and dominating points. Representations of the generalized entropic projections are obtained. It is shown that they are the “measure component" of the solutions to some extended entropy minimization problem. This approach leads to new results and offers a unifying point of view. It also permits to extend previous results on the subject by removing unnecessary topological restrictions. As a by-product, new proofs of already known results are provided.
LA - eng
KW - Conditional laws of large numbers; random measures; large deviations; entropy; convex optimization; entropic projections; dominating points; Orlicz spaces; laws of large numbers; large deviations
UR - http://eudml.org/doc/250830
ER -

References

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  1. R. Azencott, Grandes déviations et applications, in École d'Eté de Probabilités de Saint-Flour VIII (1978).  
  2. J.M. Borwein and A.S. Lewis, Strong rotundity and optimization. SIAM J. Optim.1 (1994) 146–158.  
  3. C. Boucher, R.S. Ellis and B. Turkington, Spatializing random measures: doubly indexed processes and the large deviation principle. Ann. Probab.27 (1999) 297–324.  
  4. I. Csiszár, I-divergence geometry of probability distributions and minimization problems. Ann. Probab.3 (1975) 146–158.  
  5. I. Csiszár, Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab.12 (1984) 768–793.  
  6. I. Csiszár, Generalized projections for non-negative functions. Acta Math. Hungar.68 (1995) 161–185.  
  7. I. Csiszár, F. Gamboa and E. Gassiat, MEM pixel correlated solutions for generalized moment and interpolation problems. IEEE Trans. Inform. Theory45 (1999) 2253–2270.  
  8. D. Dacunha-Castelle and F. Gamboa, Maximum d'entropie et problème des moments. Ann. Inst. H. Poincaré. Probab. Statist.26 (1990) 567–596.  
  9. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Second edition. Appl. Math. 38. Springer-Verlag (1998).  
  10. U. Einmahl and J. Kuelbs, Dominating points and large deviations for random vectors. Probab. Theory Relat. Fields105 (1996) 529–543.  
  11. R.S. Ellis, J. Gough and J.V. Puli, The large deviations principle for measures with random weights. Rev. Math. Phys.5 (1993) 659–692.  
  12. F. Gamboa and E. Gassiat, Bayesian methods and maximum entropy for ill-posed inverse problems. Ann. Statist.25 (1997) 328–350.  
  13. H. Gzyl, The Method of Maximum Entropy. World Scientific (1994).  
  14. J. Kuelbs, Large deviation probabilities and dominating points for open convex sets: nonlogarithmic behavior. Ann. Probab.28 (2000) 1259–1279.  
  15. C. Léonard, Large deviations for Poisson random measures and processes with independent increments. Stoch. Proc. Appl.85 (2000) 93–121.  
  16. C. Léonard, Convex minimization problems with weak constraint qualifications. J. Convex Anal.17 (2010) 321–348.  
  17. C. Léonard, Minimization of energy functionals applied to some inverse problems. J. Appl. Math. Optim.44 (2001) 273–297.  
  18. C. Léonard, Minimizers of energy functionals under not very integrable constraints. J. Convex Anal.10 (2003) 63–88.  
  19. C. Léonard, Minimization of entropy functionals. J. Math. Anal. Appl.346 (2008) 183–204.  
  20. C. Léonard and J. Najim, An extension of Sanov's theorem: application to the Gibbs conditioning principle. Bernoulli8 (2002) 721–743.  
  21. J. Najim, A Cramér type theorem for weighted random variables. Electron. J. Probab.7 (2002) 1–32.  
  22. P. Ney, Dominating points and the asymptotics of large deviations for random walks on Rd. Ann. Probab.11 (1983) 158–167.  
  23. P. Ney, Convexity and large deviations. Ann. Probab.12 (1984) 903–906.  
  24. M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, Pure Appl. Math. 146. Marcel Dekker, Inc. (1991).  
  25. R.T. Rockafellar, Integrals which are convex functionals. Pacific J. Math.24 (1968) 525–539.  
  26. R.T. Rockafellar, Conjugate Duality and Optimization, volume 16 of Regional Conf. Series in Applied Mathematics. SIAM, Philadelphia (1974).  
  27. R.T. Rockafellar and R. Wets, Variational Analysis, in Grundlehren der Mathematischen Wissenschaften, volume 317. Springer (1998).  
  28. A. Schied, Cramér's condition and Sanov's theorem. Statist. Probab. Lett.39 (1998) 55–60.  
  29. C.R. Smith, G.J. Erickson and P.O. Neudorfer (Eds.). Maximum Entropy and Bayesian Methods, Proc. of 11th Int. Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, Seattle, 1991. Kluwer.  
  30. D.W. Stroock and O. Zeitouni, Microcanonical distributions, Gibbs states and the equivalence of ensembles, in Festchrift in Honour of F. Spitzer, edited by R. Durrett and H. Kesten. Birkhaüser (1991) 399–424.  

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