Γ-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces

Γ-convergence of Onsager-Machlup functionals. Part I: With applications to maximum a posteriori estimation in Bayesian inverse problems

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 fo…

Onsager-Machlup 泛函的极小元是强后验众数吗？

Are minimizers of the Onsager-Machlup functional strong posterior modes?

本文证明可分 Hilbert 空间中 Gaussian 先验 Bayesian 后验的非参数众数等于 Onsager-Machlup 泛函极小元，并将其用于反问题，建立 MAP 估计与 Tikhonov-Phillips 正则化的等价性。

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

We consider the inverse problem of recovering an unknown functional parameter $u$ in a separable Banach space, from a noisy observation $y$ of its image through a known possibly non-linear ill-posed map ${\mathcal G}$. The data $y$ is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community. Our key result is to show that in this non-param…

发展过程的最可能路径

Most probable paths for developed processes

本文提出了一种通用方法，通过测量Onsager-Machlup函数在反发展上的值，定义和识别最可能路径，适用于多种流形值过程，如Lie群、同质空间和形状分析中的地标空间。