We study sensor/agent data collection and collaboration policies for parameter estimation, accounting for resource constraints and correlation between observations collected by distinct sensors/agents. Specifically, we consider a group of sensors/agents each samples from different variables of a multivariate Gaussian distribution and has different estimation objectives, and we formulate a sensor/agent's data collection and collaboration policy design problem as a Fisher information maximization (or Cramer-Rao bound minimization) problem. When the knowledge of correlation between variables is available, we analytically identify two particular scenarios: (1) where the knowledge of the correlation between samples cannot be leveraged for collaborative estimation purposes and (2) where the optimal data collection policy involves investing scarce resources to collaboratively sample and transfer information that is not of immediate interest and whose statistics are already known, with the sole goal of increasing the confidence on the estimate of the parameter of interest. When the knowledge of certain correlation is unavailable but collaboration may still be worthwhile, we propose novel ways to apply multi-armed bandit algorithms to learn the optimal data collection and collaboration policy in our distributed parameter estimation problem and demonstrate that the proposed algorithms, DOUBLE-F, DOUBLE-Z, UCB-F, UCB-Z, are effective through simulations.

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