Distributed Hypothesis Testing with Collaborative Detection

A detection system with a single sensor and two detectors is considered, where each of the terminals observes a memoryless source sequence, the sensor sends a message to both detectors and the first detector sends a message to the second detector. Communication of these messages is assumed to be error-free but rate-limited. The joint probability mass function (pmf) of the source sequences observed at the three terminals depends on an M-ary hypothesis ( \mathrm{M}\ge 2), and the goal of the communication is that each detector can guess the underlying hypothesis. Detector k, k = 1,2, aims to maximize the error exponent under hypothesis {i}-{k}, i-{k}\, \in\{ 1,\ldots ,\mathrm{M}\}, while ensuring a small probability of error under all other hypotheses. We study this problem in the case in which the detectors aim to maximize their error exponents under the same hypothesis (i.e., i-{1}\,= \quad i-{2}) and in the case in which they aim to maximize their error exponents under distinct hypotheses (i.e., i-{1}\, 6 = \quad i-{2}). For the setting in which i-{1}\,= \quad i-{2}, we present an achievable exponents region for the case of positive communication rates, and show that it is optimal for a specific case of testing against independence. We also characterize the optimal exponents region in the case of zero communication rates. For the setting in which i-{1}\, 6 = \quad i-{2}, we characterize the optimal exponents region in the case of zero communication rates.