Optimal state estimation for distributed algorithm with noise adding mechanism

Networked systems often relies on distributed algorithms to achieve a global computation/statistic goal with iterative local information exchanges between neighbor nodes. Due to the concerns of privacy, one node usually adds a random noise to its original data for information exchange at each iteration to preserve the privacy. But a neighbor node can still infer/estimate the node's state based on the information it received, no matter what type of noises is used. However, how to obtain the optimal state estimation is a critical and open issue. Therefore, in this paper, we investigate how to obtain the optimal estimation of each node's state based on the information outputs of the node and its neighbors. Firstly, we introduce two novel definitions on the estimation, named eaccurate estimation and optimal state estimation, to depict the optimal state estimation problem. Then, we obtain the optimal state estimation and its closed-form of expression, followed with some important properties considering different available information sets for the estimation.