On optimal quantization rules for sequential decision problems

We consider the problem of sequential decentralized detection, a problem that entails 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). The main result of this paper is to resolve an open problem posed by Veeravalli et al. (1993) concerning whether optimal local decision functions for the Bayesian formulation of sequential decentralized detection can be found within the class of stationary rules. We provide a negative answer to this question by exploiting an asymptotic approximation to the optimal cost of stationary quantization rules, and the asymmetry of the Kullback-Leibler divergences. In addition, we show that asymptotically optimal quantizers, when restricted to the space of blockwise stationary quantizers, are likelihood-based threshold rules