Maximum a posteriori estimators as a limit of Bayes estimators
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Maximum a posteriori and Bayes estimators are two common methods of point estimation in Bayesian statistics. It is commonly accepted that maximum a posteriori estimators are a limiting case of Bayes estimators with 0–1 loss. In this paper, we provide a counterexample which shows that in general this claim is false. We then correct the claim that by providing a level-set condition for posterior densities such that the result holds. Since both estimators are defined in terms of optimization problems, the tools of variational analysis find a natural application to Bayesian point estimation.
Mathematics Subject Classification62C10 62F10 62F15 65K10
Both authors express their gratitude to Roger J.-B. Wets for his guidance and supervision. This paper is dedicated to him, in honor of his 80th birthday.
- 3.Chan, S.-O., Diakonikolas, I., Servedio, R.A., Sun, X.: Efficient density estimation via piecewise polynomial approximation. In: Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp. 604–613. ACM (2014)Google Scholar
- 6.Figueiredo, M.A.T.: Lecture Notes on Bayesian Estimation and Classification. Instituto de Telecomunicacoes-Instituto Superior Tecnico, Lisboa (2004)Google Scholar
- 10.Hoegele, W., Loeschel, R., Dobler, B., Koelbl, O., Zygmanski, P.: Bayesian estimation applied to stochastic localization with constraints due to interfaces and boundaries. Math. Probl. Eng. 2013, 5–6 (2013)Google Scholar
- 13.Knight, K: Epi-convergence in distribution and stochastic equi-semicontinuity. Unpublished manuscript, 37 (1999)Google Scholar
- 17.Pflug, G.C.: Asymptotic Dominance and Confidence for Solutions of Stochastic Programs. International Institute for Applied Systems Analysis, Laxenburg (1991)Google Scholar
- 20.Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Grundlehren der mathematischen Wissenschaften. Springer, Berlin (2009)Google Scholar
- 21.Royset, J.O., Wets, R.J.B.: Nonparametric density estimation via exponential epi-eplines: fusion of soft and hard information. Technical report (2013)Google Scholar