Optimal Remote Estimation of Discrete Random Variables Over the Collision Channel

Consider a system comprising sensors that communicate with a remote estimator by way of a so-called collision channel. Each sensor observes a discrete random variable and must decide whether to transmit it to the remote estimator or to remain silent. The variables are independent across sensors. There is no communication among the sensors, which precludes the use of coordinated transmission policies. The collision channel functions as an ideal link when a single sensor transmits. If there are two or more simultaneous transmissions, then a collision occurs and is detected at the remote estimator. The role of the remote estimator is to form estimates of all the observations at the sensors. Our goal is to design transmission policies that are globally optimal with respect to two criteria: the aggregate probability of error, which is a convex combination of the probabilities of error in estimating the individual observations; and the total probability of error. We show that, for the aggregate probability of error criterion, it suffices to sift through a structured finite set of candidate solutions to find a globally optimal one. In general, the cardinality of this set is exponential on the number of sensors, but we discuss important cases in which it becomes quadratic or even one. For the total probability of error criterion, we prove that the solution in which each sensor transmits when it observes all but a preselected most probable value is globally optimal. So, no search is needed in this case. Our results hold irrespective of the probability mass functions of the observed random variables, regardless the size of their support.