论文标题
Onsager-Machlup功能强大的后验模式的最小化器是最小化的吗？
Are minimizers of the Onsager-Machlup functional strong posterior modes?
论文作者
论文摘要
在这项工作中，我们连接了两个概念：概率度量的非参数模式的概念（由渐近小球概率定义，以及Onsager-Machlup功能的概率，也通过渐近小球概率定义了广义密度。我们表明，在可分离的希尔伯特空间环境中，在轻度条件下，基于高斯之前存在的贝叶斯后验分布的模式存在，并同意其Onsager-Machlup功能的最小化，因此也与后验模式较弱。我们将此结果应用于逆问题，并在前向映射的情况下得出条件，后者模式的这种变异表征所保持。我们的结果严格地表明，在无限维数据被加性高斯或拉普拉斯噪声损坏的极限情况下，非参数最大值后验估计等于Tikhonov-Phillips的正则化。与Dashti，Law，Stuart和Voss（2013）的工作相比，对可能性的假设是放松的，因此它们尤其涵盖了白色高斯工艺噪声的重要情况。我们通过将其应用于拉普拉斯噪声的严重不良线性问题来说明我们的结果，在该问题上，我们可以通过分析表达最大的后验估计量，并研究其在小噪声极限中的收敛速度。
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.
