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

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

We derive Onsager-Machlup functionals for countable product measures on weighted $\ell^p$ subspaces of the sequence space $\mathbb{R}^{\mathbb{N}}$. Each measure in the product is a shifted and scaled copy of a reference probability measure on $\mathbb{R}$ that admits a sufficiently regular Lebesgue density. We study the equicoercivity and $Γ$-convergence of sequences of Onsager-Machlup functionals associated to convergent sequences of measures within this class. We use these results to establish analogous results for probability measures on separable Banach or Hilbert spaces, including Gaussi…

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。

贝叶斯逆问题中的广义众数

Generalized modes in Bayesian inverse problems

本文针对无连续密度先验下非参数 MAP 估计难题，提出基于逼近序列的广义众数定义，并刻画 uniform 先验的广义 Onsager–Machlup 泛函，证明非线性贝叶斯逆问题的一致性。

对角高斯先验下 $\ell^p$ 中最大后验估计的良好定义性

Maximum a posteriori estimators in $\ell^p$ are well-defined for diagonal Gaussian priors

本文证明了对角高斯先验下 $\ell^p$ 空间中最大后验估计的良好定义性，并给出 $p=2$ 情形的简化证明及与 Onsager-Machlup 泛函的联系，同时提出处理 $1 \leq p < \infty$ 的新凸化结果。