In this work we connect two notions: That of the nonparametric mode of a probability measure, defined by asymptotic small ball probabilities, and that of the Onsager-Machlup functional, a generalized density also defined via asymptotic small ball probabilities. We show that in a separable Hilbert space setting and under mild conditions on the likelihood, modes of a Bayesian posterior distribution based upon a Gaussian prior exist and agree with the minimizers of its Onsager-Machlup functional and thus also with weak posterior modes. We apply this result to inverse problems and derive conditions on the forward mapping under which this variational characterization of posterior modes holds. Our results show rigorously that in the limit case of infinite-dimensional data corrupted by additive Gaussian or Laplacian noise, nonparametric maximum a posteriori estimation is equivalent to Tikhonov-Phillips regularization. In comparison with the work of Dashti, Law, Stuart, and Voss (2013), the assumptions on the likelihood are relaxed so that they cover in particular the important case of white Gaussian process noise. We illustrate our results by applying them to a severely ill-posed linear problem with Laplacian noise, where we express the maximum a posteriori estimator analytically and study its rate of convergence in the small noise limit.

翻译：本文连接了两个概念：通过渐近小球概率定义的概率测度的非参数众数，以及同样通过渐近小球概率定义的广义密度——Onsager-Machlup泛函。我们证明，在可分离希尔伯特空间框架下，且似然函数满足温和条件时，基于高斯先验的贝叶斯后验分布的众数存在，且与Onsager-Machlup泛函的最小化子一致，因此也与弱后验众数一致。我们将此结果应用于反问题，并推导了在该变分表征成立条件下前向映射所需满足的条件。我们的结果严格表明：在数据被加性高斯噪声或拉普拉斯噪声污染的无限维极限情形中，非参数最大后验估计等价于Tikhonov-Phillips正则化。与Dashti、Law、Stuart和Voss（2013）的研究相比，本文放宽了对似然函数的假设，使其特别覆盖了高斯白过程噪声这一重要情形。我们通过将其应用于一个带有拉普拉斯噪声的严重不适定线性问题来展示结果，在该问题中我们解析地表达了最大后验估计量，并研究了其在噪声趋零极限下的收敛速率。