Sufficient condition on hypothesis statistics for suboptimality of the identical-quantizer parallel distributed detection system

A challenge in designing an optimal parallel distributed detection system is its exponentially increasing number of possible combinations of local quantization rules with respect to n local sensors. Instead, one may focus on a much easier task of finding the best identical-quantizer system (IQS). This raises an important query about when the simple best IQS is globally optimal in Bayes detection error, particularly under a finite number of sensors. We consider this issue by asking the different but relevant question of when the best IQS is only suboptimal. By numerical experiments over the simplest system setting with ternary local observations and binary local quantizers, we find and prove that for every even n ≥ 2, if the output statistics of a local quantizer are symmetric with respective to the two hypotheses, the corresponding IQS design can never be optimal in the sense of minimizing the Bayes detection error subject to equal prior probabilities. Our analysis also indicates that asymptotically, one can almost halve the detection error of an IQS by replacing one of the identical local quantizers with a different one when the two hypothesis distributions place most of their probability masses, respectively, on two different local observation outcomes.