Optimization of detection networks. II. Tree structures

A distributed binary detection problem with multimessage (⩾1 bit) communications is considered, wherein the nodes (sensors, decision-makers (DMs)) of the system are organized in the form of a tree with multiple root nodes. A numerical algorithm is developed for determining the optimal decision rule at each node assuming monotone cost functions imposed only on the root nodes. It is assumed that the observations of each node are conditionally independent of those of the other nodes. It is shown that the problem is equivalent to solving a nonlinear optimal control problem, and the necessary conditions of optimality using Bayes´ risk as the optimization criterion are derived. The optimal control approach provides an interpretation of certain functions of the co-state variables in terms of thresholds, and leads to a computationally efficient min-H algorithm to solve for the optimal decision rule at each node. The numerical algorithm provides a tool to investigate the organizational issues of adaptation, structure, and robustness