Free energy calculations play an essential role in computational physics. As often observed, based on the same set of simulated data, various free energy estimators, such as thermodynamic integration, overlap histogramming, and free energy perturbation, usually have different convergence behavior. In the present work, a lambda-WHAM approach is developed to forge such missing link by refining the energy derivative histograms with the maximal likelihood treatment. Using these lambda-WHAM generated energy derivative histograms, free energy estimation efficiency and accuracy are optimally improved; more interestingly, with this additional lambda-WHAM component, these seemingly different theoretical approaches become identical. Published by Elsevier B.V.