  This paper considers the problem of distributed estimation in wireless sensor
networks (WSN), which is anticipated to support a wide range of applications
such as the environmental monitoring, weather forecasting, and location
estimation. To this end, we propose a joint model and data driven distributed
estimation method by designing the optimal quantizers and fusion center (FC)
based on the Bayesian and minimum mean square error (MMSE) criterions. First,
universal mean square error (MSE) lower bound for the quantization-based
distributed estimation is derived and adopted as the design metric for the
quantizers. Then, the optimality of the mean-fusion operation for the FC with
MMSE criterion is proved. Next, by exploiting different levels of the statistic
information of the desired parameter and observation noise, a joint model and
data driven method is proposed to train parts of the quantizer and FC modules
as deep neural networks (DNNs), and two loss functions derived from the MMSE
criterion are adopted for the sequential training scheme. Furthermore, we
extend the above results to the case with multi-bit quantizers, considering
both the parallel and one-hot quantization schemes. Finally, simulation results
reveal that the proposed method outperforms the state-of-the-art schemes in
typical scenarios.
