During the optimization, to calculate properties at states we have not visited, we must calculate the reduced internal energy u = β l (U l (X k,t ) + P l V k,t ) for each sample, where k is the state sampled from, l is the state to be reweighted at, and t is the label of the sample from the ensemble at k. This must be calculated at each iteration for property estimation corresponding to the force field parameters (i. e. corresponding to changing l) suggested by the optimization algorithm.