The replica method is a non-rigorous but widely-accepted technique from
statistical physics used in the asymptotic analysis of large, random, nonlinear
problems. This paper applies the replica method to non-Gaussian maximum a
posteriori (MAP) estimation. It is shown that with random linear measurements
and Gaussian noise, the asymptotic behavior of the MAP estimate of an
n-dimensional vector decouples as n scalar MAP estimators. The result is a
counterpart to Guo and Verdu's replica analysis of minimum mean-squared error
estimation.

The replica MAP analysis can be readily applied to many estimators used in
compressed sensing, including basis pursuit, lasso, linear estimation with
thresholding, and zero norm-regularized estimation. In the case of lasso
estimation the scalar estimator reduces to a soft-thresholding operator, and
for zero norm-regularized estimation it reduces to a hard threshold. Among
other benefits, the replica method provides a computationally-tractable method
for exactly computing various performance metrics including mean-squared error
and sparsity pattern recovery probability.