Distributed hypothesis testing with a fusion center: The conditionally dependent case

The paper deals with decentralized Bayesian detection with M hypotheses, and N sensors making conditionally correlated measurements regarding these hypotheses. Each sensor sends to a fusion center an integer from {0, 1, D - 1}, and the fusion center makes a decision on the actual hypothesis based on the messages it receives from the sensors so as to minimize the average probability of error. Such conditionally dependent scenarios arise in several applications of decentralized detection such as sensor networks and network security. Conditional dependence leads to a non-standard distributed decision problem where threshold based policies (on likelihood ratios) are no longer optimal, which results in a challenging distributed optimization/decision making problem. We show that, in this case, the minimum average probability of error cannot be expressed as a function of the marginal distributions of the sensor messages. Instead, we characterize this probability based on the joint distributions of these messages. We also provide some numerical results for the case where the sensors' measurements follow bivariate normal distributions.