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.
Mots-clés : conditional laws of large numbers, random measures, large deviations, entropy, convex optimization, entropic projections, dominating points, Orlicz spaces
@article{PS_2010__14__343_0,
author = {L\'eonard, Christian},
title = {Entropic projections and dominating points},
journal = {ESAIM: Probability and Statistics},
pages = {343--381},
publisher = {EDP-Sciences},
volume = {14},
year = {2010},
doi = {10.1051/ps/2009003},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2009003/}
}
Léonard, Christian. Entropic projections and dominating points. ESAIM: Probability and Statistics, Tome 14 (2010), pp. 343-381. doi : 10.1051/ps/2009003. http://www.numdam.org/articles/10.1051/ps/2009003/
[1] , Grandes déviations et applications, in École d'Eté de Probabilités de Saint-Flour VIII (1978). | Zbl
[2] and , Strong rotundity and optimization. SIAM J. Optim. 1 (1994) 146-158. | Zbl
[3] , and , Spatializing random measures: doubly indexed processes and the large deviation principle. Ann. Probab. 27 (1999) 297-324. | Zbl
[4] , I-divergence geometry of probability distributions and minimization problems. Ann. Probab. 3 (1975) 146-158. | Zbl
[5] , Sanov property, generalized I-projection and a conditional limit theorem. Ann. Probab. 12 (1984) 768-793. | Zbl
[6] , Generalized projections for non-negative functions. Acta Math. Hungar. 68 (1995) 161-185. | Zbl
[7] , and , MEM pixel correlated solutions for generalized moment and interpolation problems. IEEE Trans. Inform. Theory 45 (1999) 2253-2270. | Zbl
[8] and , Maximum d'entropie et problème des moments. Ann. Inst. H. Poincaré. Probab. Statist. 26 (1990) 567-596. | Numdam | Zbl
[9] and , Large Deviations Techniques and Applications. Second edition. Appl. Math. 38. Springer-Verlag (1998). | Zbl
[10] and , Dominating points and large deviations for random vectors. Probab. Theory Relat. Fields 105 (1996) 529-543. | Zbl
[11] , and , The large deviations principle for measures with random weights. Rev. Math. Phys. 5 (1993) 659-692. | Zbl
[12] and , Bayesian methods and maximum entropy for ill-posed inverse problems. Ann. Statist. 25 (1997) 328-350. | Zbl
[13] , The Method of Maximum Entropy. World Scientific (1994). | Zbl
[14] , Large deviation probabilities and dominating points for open convex sets: nonlogarithmic behavior. Ann. Probab. 28 (2000) 1259-1279. | Zbl
[15] , Large deviations for Poisson random measures and processes with independent increments. Stoch. Proc. Appl. 85 (2000) 93-121. | Zbl
[16] , Convex minimization problems with weak constraint qualifications. J. Convex Anal. 17 (2010) 321-348. | Zbl
[17] , Minimization of energy functionals applied to some inverse problems. J. Appl. Math. Optim. 44 (2001) 273-297. | Zbl
[18] , Minimizers of energy functionals under not very integrable constraints. J. Convex Anal. 10 (2003) 63-88. | Zbl
[19] , Minimization of entropy functionals. J. Math. Anal. Appl. 346 (2008) 183-204. | Zbl
[20] and , An extension of Sanov's theorem: application to the Gibbs conditioning principle. Bernoulli 8 (2002) 721-743. | Zbl
[21] , A Cramér type theorem for weighted random variables. Electron. J. Probab. 7 (2002) 1-32. | Zbl
[22] , Dominating points and the asymptotics of large deviations for random walks on Rd. Ann. Probab. 11 (1983) 158-167. | Zbl
[23] , Convexity and large deviations. Ann. Probab. 12 (1984) 903-906. | Zbl
[24] and , Theory of Orlicz spaces, Pure Appl. Math. 146. Marcel Dekker, Inc. (1991). | Zbl
[25] , Integrals which are convex functionals. Pacific J. Math. 24 (1968) 525-539. | Zbl
[26] , Conjugate Duality and Optimization, volume 16 of Regional Conf. Series in Applied Mathematics. SIAM, Philadelphia (1974). | Zbl
[27] and , Variational Analysis, in Grundlehren der Mathematischen Wissenschaften, volume 317. Springer (1998). | Zbl
[28] , Cramér's condition and Sanov's theorem. Statist. Probab. Lett. 39 (1998) 55-60. | Zbl
[29] , and (Eds.). Maximum Entropy and Bayesian Methods, Proc. of 11th Int. Workshop on Maximum Entropy and Bayesian Methods of Statistical Analysis, Seattle, 1991. Kluwer. | Zbl
[30] and , 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. | Zbl
Cité par Sources :
