Generalized Modes in Bayesian Inverse Problems

被引:11
作者
Clason, Christian [1 ]
Helin, Tapio [2 ]
Kretschmann, Remo [1 ]
Piiroinen, Petteri [3 ]
机构
[1] Univ Duisburg Essen, Fac Math, D-45127 Essen, Germany
[2] LUT Univ, Sch Engn Sci, FI-53851 Lappeenranta, Finland
[3] Univ Helsinki, Dept Math & Stat, FI-00014 Helsinki, Finland
基金
芬兰科学院;
关键词
Bayesian inverse problems; generalized MAP estimates; uniform priors; APPROXIMATIONS;
D O I
10.1137/18M1191804
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
Uncertainty quantification requires efficient summarization of high- or even infinite-dimensional (i.e., nonparametric) distributions based on, e.g., suitable point estimates (modes) for posterior distributions arising from model-specific prior distributions. In this work, we consider nonparametric modes and maximum a posteriori (MAP) estimates for priors that do not admit continuous densities, for which previous approaches based on small ball probabilities fail. We propose a novel definition of generalized modes based on the concept of approximating sequences, which reduce to the classical mode in certain situations that include Gaussian priors but also exist for a more general class of priors. The latter includes the case of priors that impose strict bounds on the admissible parameters and in particular of uniform priors. For uniform priors defined by random series with uniformly distributed coefficients, we show that generalized MAP estimates but not classical MAP estimates can be characterized as minimizers of a suitable functional that plays the role of a generalized Onsager-Machlup functional. This is then used to show consistency of nonlinear Bayesian inverse problems with uniform priors and Gaussian noise.
引用
收藏
页码:652 / 684
页数:33
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