Zusammenfassung
We derive lower bounds on the Bayes risk in decentralized estimation, where
the estimator does not have direct access to the random samples generated
conditionally on the random parameter of interest, but only to the data
received from local processors that observe the samples. The received data are
subject to communication constraints, due to quantization and the noise in the
communication channels from the processors to the estimator. We first derive
general lower bounds on the Bayes risk using information-theoretic quantities,
such as mutual information, information density, small ball probability, and
differential entropy. We then apply these lower bounds to the decentralized
case, using strong data processing inequalities to quantify the contraction of
information due to communication constraints. We treat the cases of a single
processor and of multiple processors, where the samples observed by different
processors may be conditionally dependent given the parameter, for
noninteractive and interactive communication protocols. Our results recover and
improve recent lower bounds on the Bayes risk and the minimax risk for certain
decentralized estimation problems, where previously only conditionally
independent sample sets and noiseless channels have been considered. Moreover,
our results provide a general way to quantify the degradation of estimation
performance caused by distributing resources to multiple processors, which is
only discussed for specific examples in existing works.
Nutzer