The Bayesian solution to a statistical inverse problem can be summarised by a mode of the posterior distribution, i.e. a MAP estimator. The MAP estimator essentially coincides with the (regularised) variational solution to the inverse problem, seen as minimisation of the Onsager--Machlup functional of the posterior measure. An open problem in the stability analysis of inverse problems is to establish a relationship between the convergence properties of solutions obtained by the variational approach and by the Bayesian approach. To address this problem, we propose a general convergence theory for modes that is based on the $\Gamma$-convergence of Onsager--Machlup functionals, and apply this theory to Bayesian inverse problems with Gaussian and edge-preserving Besov priors. Part II of this paper considers more general prior distributions.

翻译：对统计逆向问题的巴伊西亚解决办法可以用一种后向分布模式,即MAP 估计符来概括。MAP 估计符基本上与反向问题的(常规)变式解决办法相吻合,被视为后向计量的Onsager-Machlup功能的最小化。反向问题稳定分析的一个公开问题是,在变向办法和巴伊西亚办法获得的解决办法的趋同特性之间建立一种关系。为了解决这一问题,我们为基于Onsager-Machlup功能的 $\Gamma$-converggence的模型提出了一个一般趋同理论,并将这一理论应用于巴伊西亚与Gaussian和边缘保留Besov 先前的反向问题。本文第二部分比较概括了先前的分布。