  This paper studies the asymptotic performance of maximum-a-posteriori
estimation in the presence of prior information. The problem arises in several
applications such as recovery of signals with non-uniform sparsity pattern from
underdetermined measurements. With prior information, the maximum-a-posteriori
estimator might have asymmetric penalty. We consider a generic form of this
estimator and study its performance via the replica method. Our analyses
demonstrate an asymmetric form of the decoupling property in the large-system
limit. Employing our results, we further investigate the performance of
weighted zero-norm minimization for recovery of a non-uniform sparse signal.
Our investigations illustrate that for a given distortion, the minimum number
of required measurements can be significantly reduced by choosing weighting
coefficients optimally.
