Converses for distributed estimation via strong data processing inequalities

Abstract: We consider the problem of distributed estimation, where local processors observe independent samples conditioned on a common random parameter of interest, map the observations to a finite number of bits, and send these bits to a remote estimator over independent noisy channels. We derive converse results for this problem, such as lower bounds on Bayes risk. The main technical tools include a lower bound on the Bayes risk via mutual information and small ball probability, as well as strong data processing inequalities for the relative entropy. Our results can recover and improve some existing results on distributed estimation with noiseless channels, and also capture the effect of noisy channels on the estimation performance.