From each ensemble i, we draw N i samples, labeled as {x in }, (n = 1, 2, . . . , N i ), and the whole sample set can be labeled as {x n }, n = (1, 2, . . . , N ), with N being the total sample size: N = K i=1 N i . MBAR theory states that the statistically optimal estimators (ĉ i ) for the partition functions c i of these K ensembles satisfy the following equation: