On Optimal Quantization Rules for Some Problems in Sequential Decentralized Detection

We consider the design of systems for sequential decentralized detection, a problem that entails several interdependent choices: the choice of a stopping rule (specifying the sample size), a global decision function (a choice between two competing hypotheses), and a set of quantization rules (the local decisions on the basis of which the global decision is made). This correspondence addresses an open problem of whether in the Bayesian formulation of sequential decentralized detection, optimal local decision functions can be found within the class of stationary rules. We develop an asymptotic approximation to the optimal cost of stationary quantization rules and exploit this approximation to show that stationary quantizers are not optimal in a broad class of settings. We also consider the class of blockwise-stationary quantizers, and show that asymptotically optimal quantizers are likelihood-based threshold rules.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  J. Wolfowitz,et al.  Optimum Character of the Sequential Probability Ratio Test , 1948 .

[3]  M. A. Girshick,et al.  Bayes and minimax solutions of sequential decision problems , 1949 .

[4]  Walter T. Federer,et al.  Sequential Design of Experiments , 1967 .

[5]  J. Andel.  Sequential Analysis , 2022, The SAGE Encyclopedia of Research Design.

[6]  G. Lorden.  On Excess Over the Boundary , 1970 .

[7]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal stopping rules , 1977 .

[8]  C.C. White,et al.  Dynamic programming and stochastic control , 1978, Proceedings of the IEEE.

[9]  Nils Sandell,et al.  Detection with Distributed Sensors , 1980, IEEE Transactions on Aerospace and Electronic Systems.

[10]  L. C. Thomas,et al.  Optimization over Time. Dynamic Programming and Stochastic Control. Volume 1 , 1983 .

[11]  J. Tsitsiklis.  On threshold rules in decentralized detection , 1986, 1986 25th IEEE Conference on Decision and Control.

[12]  H. Chernoff.  Sequential Analysis and Optimal Design , 1987 .

[13]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[14]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .

[15]  H. Vincent Poor,et al.  Decentralized Sequential Detection with a Fusion Center Performing the Sequential Test , 1992, 1992 American Control Conference.

[16]  J. Tsitsiklis.  Decentralized Detection' , 1993 .

[17]  John N. Tsitsiklis,et al.  Extremal properties of likelihood-ratio quantizers , 1993, IEEE Trans. Commun..

[18]  H. Vincent Poor,et al.  An Introduction to Signal Detection and Estimation , 1994, Springer Texts in Electrical Engineering.

[19]  Rick S. Blum,et al.  Distributed detection with multiple sensors I. Advanced topics , 1997, Proc. IEEE.

[20]  Pramod K. Varshney,et al.  Distributed detection with multiple sensors I. Fundamentals , 1997, Proc. IEEE.

[21]  Venugopal V. Veeravalli,et al.  Sequential decision fusion: theory and applications , 1999 .

[22]  T. Lai.  SEQUENTIAL ANALYSIS: SOME CLASSICAL PROBLEMS AND NEW CHALLENGES , 2001 .

[23]  Y. Mei.  Asymptotically optimal methods for sequential change-point detection , 2003 .

[24]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[25]  P.K. Varshney,et al.  Channel-aware distributed detection in wireless sensor networks , 2006, IEEE Signal Processing Magazine.

[26]  Peter Willett,et al.  On the optimality of the likelihood-ratio test for local sensor decision rules in the presence of nonideal channels , 2005, IEEE Transactions on Information Theory.

[27]  David Lindley,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 2010 .