Priorconditioned CGLS-Based Quasi-MAP Estimate, Statistical Stopping Rule, and Ranking of Priors
Priorconditioned CGLS-Based Quasi-MAP Estimate, Statistical Stopping Rule, and Ranking of Priors
复制标题
基于先验 CGLS 的准 MAP 估计、统计停止规则和先验排序
DOI：
10.1137/16m108272x
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
B. Vantaggi
中科院分区：
文献类型：
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作者：
D. Calvetti;F. Pitolli;J. Prezioso;E. Somersalo;B. Vantaggi
我们考虑贝叶斯框架内的线性离散不适定问题，假设高斯加性噪声模型和高斯先验，其协方差矩阵可能是已知的模乘缩放因子。在这种情况下，我们提出了一种新的后验密度点估计器，即基于先验 CGLS 的准 MAP (qMAP)，作为经典最大后验 (MAP) 估计的计算上有吸引力的近似，特别是当矩阵 ${\mathsf A}$ 的有效秩远小于未知维度时。利用贝叶斯范式以及标准正态随机变量与其欧几里德范数平方的 $\chi^2$ 分布之间的联系，我们提出了一种新的停止规则，即 Max$\chi^2$ 规则，以终止 qMAP 估计计算中的 CGLS 迭代。此外，我们表明所提出的停止规则可用于估计噪声协方差的可能未知的缩放因子，这相当于估计 t...
We consider linear discrete ill-posed problems within the Bayesian framework, assuming a Gaussian additive noise model and a Gaussian prior whose covariance matrices may be known modulo multiplicative scaling factors. In that context, we propose a new pointwise estimator for the posterior density, the priorconditioned CGLS-based quasi-MAP (qMAP) as a computationally attractive approximation of the classical maximum a posteriori (MAP) estimate, in particular when the effective rank of the matrix ${\mathsf A}$ is much smaller than the dimension of the unknown. Exploiting the Bayesian paradigm and the connection between standard normal random variables and the $\chi^2$ distribution of their squared Euclidean norms, we propose a new stopping rule, the Max$\chi^2$ rule, to terminate the CGLS iterations in the computation of the qMAP estimate. Moreover, we show that the proposed stopping rule can be used to estimate the possibly unknown scaling factor of the noise covariance, which is tantamount to estimating t...