Extrinsics and linearized CWCU MMSE estimation as in GAMP

Generalized Approximate Message Passing (GAMP) algorithms have demonstrated significant efficacy in signal recovery. GAMP has been derived by applying asymptotic approximations to Expectation Propagation (EP). EP algorithms start from a factored approximate posterior in an exponential family. They update a factor by fitting an exponential family pdf to a approximate posterior which is obtained by replacing one approximate factor by the original (prior) factor. The remaining factors form the approximate extrinsic. Hence extrinsics are obtained by marginalizing the product of all pdf factors except for the prior. A marginal posterior is then obtained by combining the extrinsic with the prior. Low complexity algorithms like GAMP in turn obtain the extrinsic from the posterior. In the Gaussian case, we reveal the intimate relation of extrinsics to ComponentWise Conditionally Unbiased Minimum Mean Squared Error (CWCU MMSE) estimation, whereas the posterior allows MMSE estimation. In the Gaussian case, MMSE estimation means Linear MMSE estimation, non-Gaussianity leads to nonlinear estimators. We rederive the revisited GVAMP algorithm as asymptotic alternating minimization of a Kullback-Leibler Divergence. We then explore the extrinsics in GAMP by asymptotic perturbations relating posterior beliefs and extrinsics.