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.