IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 39, NO. 2, MARCH 1993 433 Decentralized Sequential Detection with a Fusion Center Performing the Sequential Test Venugopal V. Veeravalli, Student Member, IEEE, Tamer BaSar, Fellow, IEEE, and H. Vincent Poor, Fellow, IEEE Abstract-A decentralized sequential detection problem is con- sidered in which each one of a set of sensors receives a sequence of observations about the hypothesis. Each sensor sends a sequence of summary messages to the fusion center where a sequential test is carried out to determine the true hypothesis. A Bayesian framework for this problem is introduced, and for the case when the information structure in the system is quasi-classical, it is shown that the problem is tractable. A detailed analysis of this case is presented along with some numerical results. Index Terms-Decentralized detection, sequential analysis, dy- namic programming. . I. INTRODUCTION W ITH THE INCREASING INTEREST in decentralized detection problems in recent years, extensions of vari- ous centralized detection problems to decentralized cases have been formulated and studied [l]. In particular, there has been considerable interest in the solution to decentralized detection problems of a sequential nature [2]-[6]. In decentralized sequential hypotheses testing, each one of a set of sensors receives a sequence of observations about the hypothesis. Two distinct formulations are possible. In one case, first each sensor performs a sequential test on its observations and arrives at a final local decision; subsequently the local decisions are used for a common purpose at a site possibly remote to all the sensors. In the other case, each sensor sends a sequence of summary messages to the fusion center, where a sequential test is carried out to determine the true hypothesis. In this paper, we study the latter case. More formally, let there be N sensors 5’1,. . . , 5’~ in the system. At time IcE{l, 2,. . .}, sensor 5’1 observes a random variable XL, and forms a summary message U: of the information available for Manuscript received November 1, 1991; revised June 15, 1992. This work was supported in part by the Joint Services Electronics Program under Grant N00014-90-J-1270, through the University of Illinois, and in part by the Office of Naval Research under Grant N00014-89-J-1321, through Princeton University. This work was presented in part at the 1992 American Control Conference, Chicago, IL, June 24-26, 1992. V.V. Veeravalli was with the Department of Electrical and Computer Engineering, and the Coordinated Science Laboratory, University of Illinois, Urbana, IL. He is now with the Division of Applied Sciences, Harvard University, Cambridge, MA 02139. T. Ba$ar is with the Department of Electrical and Computer Engineering, and the Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801. H. V. Poor is with the Electrical Engineering Department, Princeton Uni- versity, Princeton, NJ 08544. IEEE Log Number 9203865. . . Fig. 1. General setting for decentralized sequential detection with a fusion center performing the sequential test. decision at time k. In a general setting, we allow a two-way communication between the sensors and the fusion center as shown in Fig. 1. In particular, the fusion center could relay past decisions from the other sensors. This means that at time Ic, each sensor has access to all its observations up to time k and all the decisions of all the other sensors up to time k - 1. We now introduce a Bayesian framework for this sequential hypothesis testing problem. The two hypotheses HO and HI are assumed to have known prior probabilities. Also, the conditional joint distributions of the sensor observations under each hypothesis are assumed to be known. A positive cost c is associated with each time step taken for decision making. The fusion center stops receiving additional information at a stopping time r and makes a final decision S based on the observations up to time 7. Decision errors are penalized through a decision cost function W(S; H). The Bayesian optimization problem then is the minimization of E(c7 + W(S; H)} over all admissible decision policies at the fusion 001%9448/93$03.00 0 1993 IEEE