Belief propagation (BP) based multiple-input multiple-output (MIMO) detection algorithm can achieve near optimal performance, but the technical analysis remains unsolved. In this paper, we give theoretical analysis about the relationship between BP detection and joint maximum likelihood (JML) detection via an optimization framework of Bethe approximation to free energy and constrained JML detection. The proposed approach can explain why BP can achieve high performance in some extent and simulation result verifies the conclusion.
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