Let's say we want to know the value of some $x \in \mathbb{R}$ and we have $n$ "noisy votes (estimates)" $v_1, v_2, ... v_n$ drawn from $V_1, V_2,..V_n$. For each of these votes, we also have an approximate mean squared error of the voter $E[(V_i - x)^2]$ and all pairwise covariances $Cov(V_i, V_j)$, with $ i, j \in [1...n].$ What is the best way to ensemble the votes to predict $x$?
My thoughts: Intuitively I feel like if any voters have an MSE of 0, their votes should be averaged and all others neglected. Generalizing, it seems that the weight assigned to each vote should be inversely related to the error. I am not sure how to prove the optimal weighting. Also, is a weighted combination of votes the best we can do, or could a more complex scheme achieve lower error?