On optimal quantization rules for some sequential decision problems
We consider the problem of 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). In this paper we resolve an open problem concerning whether optimal local decision functions for the Bayesian formulation of sequential decentralized detection can be found within the class of stationary rules. We develop an asymptotic approximation to the optimal cost of stationary quantization rules and show how this approximation yields a negative answer to the stationarity question. We also consider the class of blockwise stationary quantizers and show that asymptotically optimal quantizers are likelihood-based threshold rules.