Deriving the Optimal Scoring Matrix

Our next goal is to construct a scoring matrix that is optimal to estimate distances. An optimal estimator (scoring function) has the smallest variance. Let d^* = S_E^-1(w/n) be our estimate of the distance using w. To compute the expected value of d^* and its variance, we use the Taylor expansion of S_E^-1(w/n) around S_E(d). These computations were done with a computer algebra system (Maple), they are too tedious and error prone to be done by hand.