ACA Bayes with Wi-PHY apps - Asymptotically correct approximate Bayes with wireless PHY applications

In the areas of communications and sensing, the demand for effective approximate Bayesian estimation techniques is paramount. Sparse channel modeling extends traditional model selection, enabling optimized models based on available training data. Compressed sensing techniques extend Linear Minimum Mean Squared Error (LMMSE) estimation by a hierarchical Bayesian formulation. In multi-user detection or (semi-)blind channel estimation, going beyond LMMSE and Gaussian models represents a leap. A straightforward approximate Bayesian estimation method is Variational Bayes (VB). VB can be seen as an extension of the Expectation-Maximization (EM) technique to scenarios involving random parameters, thereby yielding not only point estimates but also approximate posterior distributions. Notably, while VB yields accurate means in Gaussian problems, it tends to underestimate variances significantly. An even more refined technique for approximate Bayes is Expectation Propagation (EP). Both VB and EP share the underlying concept of minimizing the Kullback-Leibler Divergence (KLD), albeit with different trial posteriors. In the case of EP, the KLD leads to the so-called Bethe Free Energy (BFE). Exact alternating constrained minimization of the BFE leads to Belief Propagation (BP), whereas EP restricts approximating pdfs to be in an exponential family. Taking a fresh look at alternating minimization of a KLD, the Central Limit Theorem leads to Gaussianity of the extrinsics in the marginal posteriors in moderate asymptotic settings. This in turn leads to what we call Gaussian Extrinsic Propagation, which sheds new light on characterizing performance beyond the loose Bayesian Cramer-Rao bound. Focusing on the Generalized Linear Model, assuming a n.i.i.d. (sign) statistical model for the measurement matrix allows asymptotically to find the correct variances without matrix inversions. This leads to Approximate Message Passing (AMP) in which the extrinsics are related to Componentwise Conditionally Unbiased MMSE estimation. Reformulating AMP to correspond to alternating minimization of an asymptotic version of the BFE leads to a provably convergent algorithm. Alternating minimization becomes tricky in the presence of constraints and we shed some light on the desirable behavior of the Alternating Directions Method of Multipliers (ADMM) approach. As an application we handle the pilot contamination problem in cellular or cell-free (MIMO) systems by semi-blind channel estimation. We review various techniques and bounds but in particular we consider EP approaches for approximate MMSE channel estimation.