Rate distortion codes for the collective estimation from independent noisy observations

We present a collective behavior in the optimal aggregation of noisy observations of a source. The quality of estimation of the source state involves a difficult tradeoff between sensing quality which increases by increasing the number of sensors, and aggregation quality which decreases if the number of sensors is too large. We analytically study the optimal strategy for large scale aggregation, and obtain an explicit and exact result by introducing a basic model. We show that larger scale aggregation always outperforms smaller scale aggregation at higher noise levels, while below a critical value of noise, there exist moderate scale aggregation levels at which optimal estimation is realized. We also examine the practical tradeoff between the above two aggregation strategies by applying an iterative encoding to linear codes.