  We consider the problem of distributed estimation under the Bayesian
criterion and explore the design of optimal quantizers in such a system. We
show that, for a conditionally unbiased and efficient estimator at the fusion
center and when local observations have identical distributions, it is optimal
to partition the local sensors into groups, with all sensors within a group
using the same quantization rule. When all the sensors use identical number of
decision regions, use of identical quantizers at the sensors is optimal. When
the network is constrained by the capacity of the wireless multiple access
channel over which the sensors transmit their quantized observations, we show
that binary quantizers at the local sensors are optimal under certain
conditions. Based on these observations, we address the location parameter
estimation problem and present our optimal quantizer design approach. We also
derive the performance limit for distributed location parameter estimation
under the Bayesian criterion and find the conditions when the widely used
threshold quantizer achieves this limit. We corroborate this result using
simulations. We then relax the assumption of conditionally independent
observations and derive the optimality conditions of quantizers for
conditionally dependent observations. Using counter-examples, we also show that
the previous results do not hold in this setting of dependent observations and,
therefore, identical quantizers are not optimal.
