  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.
