On Performance Limit for Bandwidth-Constrained Distributed Parameter Estimation Systems

We consider a bandwidth-constrained distributed parameter estimation problem, where each sensor makes a noisy observation of an unknown random source
$\theta$
. Each sensor is unaware of
$\theta$
's prior distribution and the actual dynamic range of its observation, and simply assumes that its observation is limited to a finite interval [
$\tau_{k}, \tau_{k}$
]. Each sensor quantizes its observation using a multi-bit uniform quantizer, where the quantization step size is chosen according to
$\tau_{k}$
. Sensors send their quantized observations to a fusion center (FC), that is tasked with estimating
$\theta$
based on the received data from the sensors. We derive the Bayesian Fisher information, which is the inverse of the Bayesian Cramer-Rao lower bound, for two types of random
$\theta$
, namely Gaussian and Laplacian
$\theta$
. To quantify the amount of information loss on
$\theta$
when the FC uses the quantized observation for estimating
$\theta$
, due to both limited dynamic ranges at the sensors and uniform quantization, we examine the derived Fisher information at the asymptotic regimes, when the quantization rate and
$\tau_{k}$
go to infinity. We also provide two accurate approximations of the Fisher information for two cases of (i) binary and (ii) high rate multi-bit quantizers. Through simulations we explore the conditions under which the information loss on
$\theta$
is negligible and demonstrate the accuracy of the provided approximations.