About the non-asymptotic behaviour of Bayes estimators. (English) Zbl 1394.62024
Summary: This paper investigates the nonasymptotic properties of Bayes procedures for estimating an unknown distribution from \(n\) i.i.d. observations. We assume that the prior is supported by a model \((\mathcal{S}, h)\) (where \(h\) denotes the Hellinger distance) with suitable metric properties involving the number of small balls that are needed to cover larger ones. We also require that the prior put enough probability on small balls.
We consider two different situations. The simplest case is the one of a parametric model containing the target density for which we show that the posterior concentrates around the true distribution at rate \(1 / \sqrt{n}\). In the general situation, we relax the parametric assumption and take into account a possible misspecification of the model. Provided that the Kullback-Leibler Information between the true distribution and the model \(\mathcal{S}\) is finite, we establish risk bounds for the Bayes estimators.
We consider two different situations. The simplest case is the one of a parametric model containing the target density for which we show that the posterior concentrates around the true distribution at rate \(1 / \sqrt{n}\). In the general situation, we relax the parametric assumption and take into account a possible misspecification of the model. Provided that the Kullback-Leibler Information between the true distribution and the model \(\mathcal{S}\) is finite, we establish risk bounds for the Bayes estimators.
References:
| [2] | Barron, A. R., Complexity regularization with applications to artificial neural networks, (Roussas, G., Nonparametric Functional Estimation (1991), Kluwer: Kluwer Dordrecht), 561-576 · Zbl 0739.62001 |
| [3] | Birgé, L., Approximation dans les espaces métriques et théorie de l’estimation, Z. Wahrscheinlichkeitstheorie Verw. Geb., 65, 181-237 (1983) · Zbl 0506.62026 |
| [4] | Birgé, L., Model selection via testing : an alternative to (penalized) maximum likelihood estimators, Ann. Inst. Henri Poincaré, Probab. et Statist., 42, 273-325 (2006) · Zbl 1333.62094 |
| [5] | Ghosal, S.; Ghosh, J. K.; van der Vaart, A. W., Convergence rates of posterior distributions, Ann. Statist., 28, 500-531 (2000) · Zbl 1105.62315 |
| [6] | Ghosal, S.; Lember; van der Vaart, A. W., On Bayesian adaptation, Acta Appl. Math., 79, 165-175 (2003) · Zbl 1030.62030 |
| [7] | Ibragimov, I. A.; Has’minskii, R. Z., Statistical Estimation: Asymptotic Theory (1981), Springer-Verlag: Springer-Verlag New York · Zbl 0389.62024 |
| [8] | Kolmogorov, A. N.; Tikhomirov, V. M., \( \varepsilon \)-entropy and \(\varepsilon \)-capacity of sets in function spaces, Amer. Math. Soc. Transl. Ser. 2, 17, 277-364 (1961) · Zbl 0133.06703 |
| [9] | Le Cam, L. M., Convergence of estimates under dimensionality restrictions, Ann. Statist., 1, 38-53 (1973) · Zbl 0255.62006 |
| [10] | Le Cam, L. M., On the risk of Bayes estimates, (Gupta, S. S.; Berger, J. O., Statistical Decision Theory and Related Topics III, Vol. 2 (1982), Academic Press: Academic Press New York), 121-137 · Zbl 0573.62029 |
| [11] | Le Cam, L. M., Asymptotic Methods in Statistical Decision Theory (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0605.62002 |
| [12] | Lorentz, G. G., Approximation of Functions (1966), Holt, Rinehart, Winston: Holt, Rinehart, Winston New York · Zbl 0153.38901 |
| [13] | Massart, P., Concentration inequalities and model selection, (Picard, J., Lecture on Probability Theory and Statistics, Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003. Lecture on Probability Theory and Statistics, Ecole d’Eté de Probabilités de Saint-Flour XXXIII - 2003, Lecture Note in Mathematics (2007), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1170.60006 |
| [14] | van der Vaart, A. W., Convergence rates of nonparametric posteriors, (Green, P. J.; Hjort, N.; Richardson, S., Highly Structured Stochastic Systems (2003), Oxford University Press: Oxford University Press Oxford) |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
