Multi-Hop Network with Multiple Decision Centers under Expected-Rate Constraints

TL;DR

This paper characterizes the fundamental limits of type-II error exponents in a multi-hop distributed hypothesis testing system with multiple decision centers under expected rate constraints, revealing a boosted exponents region compared to maximum-rate constraints.

Contribution

It introduces a new multiplexing and rate-sharing strategy for achieving the exponents region and provides novel converse proofs using change of measure arguments and asymptotic Markov chains.

Findings

Exponents region is larger under expected-rate constraints than maximum-rate constraints.

When all DCs have equal type-I error probabilities, the exponents region is rectangular.

Special cases for K=2 and K=3 are explicitly characterized.

Abstract

We consider a multi-hop distributed hypothesis testing problem with multiple decision centers (DCs) for testing against independence and where the observations obey some Markov chain. For this system, we characterize the fundamental type-II error exponents region, i.e., the type-II error exponents that the various DCs can achieve simultaneously, under expected rate-constraints. Our results show that this fundamental exponents region is boosted compared to the region under maximum-rate constraints, and that it depends on the permissible type-I error probabilities. When all DCs have equal permissible type-I error probabilities, the exponents region is rectangular and all DCs can simultaneously achieve their optimal type-II error exponents. When the DCs have different permissible type-I error probabilities, a tradeoff between the type-II error exponents at the different DCs arises. New achievability and converse proofs are presented. For the achievability, a new multiplexing and rate-sharing strategy is proposed. The converse proof is based on applying different change of measure arguments in parallel and on proving asymptotic Markov chains. For the special cases $K = 2$ and $K = 3$, we provide simplified expressions for the exponents region; a similar simplification is conjectured for arbitrary $K\geq 2$.