标题： 贝叶斯MAP估计中的超参数估计：参数化与一致性 显示英文标题

标题： Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

摘要： 贝叶斯反问题公式具有三个主要优点：它提供了一个清晰的建模框架；提供了不确定性量化的方法；并且允许对超参数进行合理的学习。 后验分布可以通过采样方法进行探索，但对于许多问题来说，这样做在计算上是不可行的。 在这种情况下，通常会寻求最大后验（MAP）估计量。 尽管这些估计量计算相对便宜，并且具有吸引人的变分公式，但一个关键缺点是它们在参数化变化下缺乏不变性。 当使用层次先验来学习超参数时，这是一个特别重要的问题。 在本文中，我们研究了在采用条件高斯层次先验分布时，参数化选择对MAP估计量的影响。 具体而言，我们考虑了中心参数化，即直接求解未知状态的自然参数化，以及非中心参数化，其中与一个白化高斯分布一起处理未知状态变量，并且在考虑维度鲁棒的MCMC算法时出现；只有在非参数设置中，非中心参数化下的MAP估计才是定义良好的。 然而，我们证明基于非中心参数化的MAP估计作为超参数的估计量并不一致；相反，我们证明随着维度趋于无穷大，有限维中心MAP估计的极限是一致的。 我们还考虑了经验贝叶斯超参数估计，证明了这些估计的一致性，并表明它们在噪声方面比中心MAP估计更稳健。 贯穿全文的一个基本概念是，超参数只能恢复到测度等价的程度，这是在Ornstein-Uhlenbeck过程背景下已知的现象。 显示英文摘要

摘要： The Bayesian formulation of inverse problems is attractive for three primary reasons: it provides a clear modelling framework; means for uncertainty quantification; and it allows for principled learning of hyperparameters. The posterior distribution may be explored by sampling methods, but for many problems it is computationally infeasible to do so. In this situation maximum a posteriori (MAP) estimators are often sought. Whilst these are relatively cheap to compute, and have an attractive variational formulation, a key drawback is their lack of invariance under change of parameterization. This is a particularly significant issue when hierarchical priors are employed to learn hyperparameters. In this paper we study the effect of the choice of parameterization on MAP estimators when a conditionally Gaussian hierarchical prior distribution is employed. Specifically we consider the centred parameterization, the natural parameterization in which the unknown state is solved for directly, and the noncentred parameterization, which works with a whitened Gaussian as the unknown state variable, and arises when considering dimension-robust MCMC algorithms; MAP estimation is well-defined in the nonparametric setting only for the noncentred parameterization. However, we show that MAP estimates based on the noncentred parameterization are not consistent as estimators of hyperparameters; conversely, we show that limits of finite-dimensional centred MAP estimators are consistent as the dimension tends to infinity. We also consider empirical Bayesian hyperparameter estimation, show consistency of these estimates, and demonstrate that they are more robust with respect to noise than centred MAP estimates. An underpinning concept throughout is that hyperparameters may only be recovered up to measure equivalence, a well-known phenomenon in the context of the Ornstein-Uhlenbeck process.