I have two measurements with known errors but unknown correlation. How can I combine them to extract the best value?
Unfortunately this is not possible. If you write the measurements as x1 ± e1 and x2 ± e2 with correlation coefficient r (so the error matrix for x1 and x2 has diagonal elements e12 and e22, and off-diagonal elements r×e1×e2), then the result and its error are very sensitive to the value of r. For r = 0, we get the standard result for uncorrelated measurements xbest = w1× x1 + w2×x2, where w1 = 1/e12 / (1/e12 + 1/e22) and the error e is given by 1/e2 = 1/e12 + 1/e22. But, for example, for r = e1/e2 (where e1 is the smaller error), the x2 measurement is completely ignored, and xbest = x1 ± e1. For even larger r, the error on the best value decreases, and reaches zero when r=+1. Meanwhile the best value is no longer between x1 and x2, but goes further and further beyond x1. It is very sensitive to the precise value of r. For negative r, the best value is between x1 and x2, but the error again goes to zero as r approaches -1. All this makes the problem sound impossible to answer. In real l
