Priorconditioned CGLS-Based Quasi-MAP Estimate, Statistical Stopping Rule, and Ranking of Priors

Abstract

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 the noise level. It is shown that the Max$\chi^2$ stopping rule and the proposed noise level estimation are affected by the correlation structure of priorconditioner, i.e., prior-related right preconditioner, but not by its scaling, which in the classical regularization framework would correspond to the Tikhonov regularization parameter. We will also show how the relation between priorconditioners and whitening transformations of the unknown can be used to rank different priorconditioners according to their agreement with the data.