Optimal quantization in the dependent Gaussian problem
Optimal quantization (for detection) rules are comparatively well-known for the case of independent observations. However, for even the simplest cases involving dependence little is beyond the anecdotal stage. A pair of prior papers examining what is perhaps the simplest dependent problem, that of detecting an additive signal is correlated Gaussian noise, have demonstrated a surprising complexity of behavior. Specifically, the existence of three `regions' (of signal/correlation values) was observed for the AND and OR fusion rules: in one region the quantization regions were provably simply-connected, and in another they were provably not simply-connected. In this paper we continue study of this problem. Our results are largely computational, but they reveal that little has been studied in dependent quantization for good reason. We show examples in which the XOR rule is optimal; in which multi-level quantization is optimal; and in which the optimal quantizer simply ignores one sensor.