  As an important piece of the multi-tier computing architecture for future
wireless networks, over-the-air computation (OAC) enables efficient function
computation in multiple-access edge computing, where a fusion center aims to
compute a function of the data distributed at edge devices. Existing OAC relies
exclusively on the maximum likelihood (ML) estimation at the fusion center to
recover the arithmetic sum of the transmitted signals from different devices.
ML estimation, however, is much susceptible to noise. In particular, in the
misaligned OAC where there are channel misalignments among received signals, ML
estimation suffers from severe error propagation and noise enhancement. To
address these challenges, this paper puts forth a Bayesian approach by letting
each edge device transmit two pieces of statistical information to the fusion
center such that Bayesian estimators can be devised to tackle the
misalignments. Numerical and simulation results verify that, 1) For the aligned
and synchronous OAC, our linear minimum mean squared error (LMMSE) estimator
significantly outperforms the ML estimator. In the low signal-to-noise ratio
(SNR) regime, the LMMSE estimator reduces the mean squared error (MSE) by at
least 6 dB; in the high SNR regime, the LMMSE estimator lowers the error floor
of MSE by 86.4%; 2) For the asynchronous OAC, our LMMSE and sum-product maximum
a posteriori (SP-MAP) estimators are on an equal footing in terms of the MSE
performance, and are significantly better than the ML estimator. Moreover, the
SP-MAP estimator is computationally efficient, the complexity of which grows
linearly with the packet length.
