Asymptotic design of quantizers for decentralized MMSE estimation

Conceptual and practical encoding/decoding, aimed at accurately reproducing remotely collected observations, has been heavily investigated since the pioneering works by Shannon about source coding. However, when the goal is not to reproduce the observables, but making inference about an embedded parameter and the scenario consists of many unconnected remote nodes, the landscape is less certain. We consider a multiterminal system designed for efficiently estimating a random parameter according to the minimum mean square error (MMSE) criterion. The analysis is limited to scalar quantizers followed by a joint entropy encoder, and it is performed in the high-resolution regime where the problem can be more easily mathematically tackled. Focus is made on the peculiarities deriving from the estimation task, as opposed to that of reconstruction, as well as on the multiterminal, as opposite to centralized, character of the inference. The general form of the optimal nonuniform quantizer is derived and examples are given.