Testing against independence with multiple decision centers

We consider a three-terminal distributed hypothesis testing against independence problem. Two terminals, which act as decision centers, are required to decide whether their observed sequences are probabilistically independent of the sequence observed at the third terminal. The third terminal communicates with the two decision centers over three rate-limited noise-free pipes (error free channels): A common pipe that is connected to both centers, and two private pipes that connect separately to each center. We characterize the optimal exponential decay of the type-II error probabilities at the two decision centers given that the type-I error probabilities vanish for increasing blocklengths. The optimal exponents are determined by a certain information-theoretic optimization problem that depends on the maximum rates allowed over the noise-free communication pipes.