On Threshold Rules in Decentralized Detection 1

We consider a decentralized detection problem in which a number of identical sensors transmit a binary function of their observations to a fusion center which then decides which one of two alternative hypotheses is true. We show that, when the number of sensors grows to infinity, optimality is not lost (in terms of the probability of error) if we constrain the sensors to use the same decision rule in deciding what to transmit. This results in considerable simplification of the problem. We also discuss the case where the messages may take more than two values and the case of M-ary (M > 2) hypotheses. Next we consider two variants of a decentralized sequential detection problem. For one variant we show that each sensor should decide what to transmit based on a likelihood ratio test; for the other, we demonstrate that such a result fails to hold and that more complicated decision rules are required. The static decentralized detection problem is defined as follows. There are two hypotheses Ho and H 1 , with given prior probabilities and N sensors.th sensor, be a random variable taking values in a set Y. We assume that the yi's are conditionally independent, given either hypothesis, with a known conditional distribution P(yilHj), j = 1,2. Each sensor i evaluates a binary message ui E {0, 1} as a function of its own observation; that is ui = 7i(Yi), where the function S7i: Y {-* (0, 1} is the decision rule of sensor i. The messages ul, ... ,UN are all transmitted to a fusion center which declares hypothesis Ho or H 1 to be true, based on a decision rule 70o: {, 1)N {-(0, 1}. That is, the final decision u 0 of the fusion center is given by uo = y0o(u 1 ,...,UN). To any set of decision rules rN = {-(7,1,- ..-,7N} we associate a cost JN (rN) which is equal to the probability that the fusion center declares true the wrong hypothesis. The problem consists of finding a set of decision rules rN which minimzes the cost J(rN). The above defined problem and its variants have been the subject of a fair amount of recent research [TeSa, Ek, TsAt, LaSa]. It is known that any optimal set of decision rules has the following structure. Each one of the sensors evaluates its message ui using a likelihood ratio test with an appropriate threshold …