A Variational Inference Framework for Soft-In-Soft-Out Detection in   Multiple Access Channels
  We propose a unified framework for deriving and studying soft-in-soft-out (SISO) detection in interference channels using the concept of variational inference. The proposed framework may be used in multiple-access interference (MAI), inter-symbol interference (ISI), and multiple-input multiple-outpu (MIMO) channels. Without loss of generality, we will focus our attention on turbo multiuser detection, to facilitate a more concrete discussion. It is shown that, with some loss of optimality, variational inference avoids the exponential complexity of a posteriori probability (APP) detection by optimizing a closely-related, but much more manageable, objective function called variational free energy. In addition to its systematic appeal, there are several other advantages to this viewpoint. First of all, it provides unified and rigorous justifications for numerous detectors that were proposed on radically different grounds, and facilitates convenient joint detection and decoding (utilizing the turbo principle) when error-control codes are incorporated. Secondly, efficient joint parameter estimation and data detection is possible via the variational expectation maximization (EM) algorithm, such that the detrimental effect of inaccurate channel knowledge at the receiver may be dealt with systematically. We are also able to extend BPSK-based SISO detection schemes to arbitrary square QAM constellations in a rigorous manner using a variational argument. 
D. D. Lin and T. J. Lim