  We investigate the problem of approximate Bayesian inference for a general
class of observation models by means of the expectation propagation (EP)
framework for large systems under some statistical assumptions. Our approach
tries to overcome the numerical bottleneck of EP caused by the inversion of
large matrices. Assuming that the measurement matrices are realizations of
specific types of ensembles we use the concept of freeness from random matrix
theory to show that the EP cavity variances exhibit an asymptotic
self-averaging property. They can be pre-computed using specific generating
functions, i.e. the R- and/or S-transforms in free probability, which do not
require matrix inversions. Our approach extends the framework of (generalized)
approximate message passing -- assumes zero-mean iid entries of the measurement
matrix -- to a general class of random matrix ensembles. The generalization is
via a simple formulation of the R- and/or S-transforms of the limiting
eigenvalue distribution of the Gramian of the measurement matrix. We
demonstrate the performance of our approach on a signal recovery problem of
nonlinear compressed sensing and compare it with that of EP.
