  We construct team-optimal estimation algorithms over distributed networks for
state estimation in the finite-horizon mean-square error (MSE) sense. Here, we
have a distributed collection of agents with processing and cooperation
capabilities. These agents observe noisy samples of a desired state through a
linear model and seek to learn this state by interacting with each other.
Although this problem has attracted significant attention and been studied
extensively in fields including machine learning and signal processing, all the
well-known strategies do not achieve team-optimal learning performance in the
finite-horizon MSE sense. To this end, we formulate the finite-horizon
distributed minimum MSE (MMSE) when there is no restriction on the size of the
disclosed information, i.e., oracle performance, over an arbitrary network
topology. Subsequently, we show that exchange of local estimates is sufficient
to achieve the oracle performance only over certain network topologies. By
inspecting these network structures, we propose recursive algorithms achieving
the oracle performance through the disclosure of local estimates. For practical
implementations we also provide approaches to reduce the complexity of the
algorithms through the time-windowing of the observations. Finally, in the
numerical examples, we demonstrate the superior performance of the introduced
algorithms in the finite-horizon MSE sense due to optimal estimation.
