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The structure of optimal communication policies for remote estimation over the collision channel with private and common observations

We consider a remote estimation system formed by two sensors that access dependent observations and a remote estimator. Each sensor observes a private and a common random variable and must decide whether to attempt to transmit them to the estimator, which seeks to produce estimates of the private measurements of both sensors and the common observation. Information is transmitted from the sensors to the estimator over a collision channel that outputs a collision when both sensors attempt a transmission. The goal is to design decentralized transmission policies at the sensors that jointly minimize a mean squared error criterion. We show that there exists a team-optimal solution that, for each realization of the common random variable, is a threshold policy on the private random variable.