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This paper presents a new approach to soft demodulation for MIMO channels. The proposed method is an approximation to the exact a posteriori probability-per-bit computer. The main idea is to marginalize the posterior density for the received data exactly over the subset of the transmitted bits that are received with the lower signal-to-noise-ratio (SNR), and marginalize this density approximately over the remaining bits. Unlike the exact demodulator, whose complexity is huge due to the need for enumerating all possible combinations of transmitted constellation points, the proposed method has very low complexity. The algorithm has a fully parallel structure, suitable for implementation in parallel hardware. Additionally, its complexity is fixed, which makes it suitable for pipelined implementation. We also show how the method can be extended to the situation when the receiver has only partial channel state information, and how it can be modified to take soft-input into account. Numerical examples illustrate its performance on slowly fading 4 times 4 and 6 times 6 complex MIMO channels.