We compute the alchemical free energy differences among the ligands using BayesMBAR, CCC, and CBayesMBAR with configurations sampled over simulation times ranging from 20 ps to 5000 ps. To ensure statistically meaningful comparisons, we repeat the calculations, including the molecular dynamics simulations, 10 times. The performance of the three methods were evaluated using the same metrics as in the harmonic oscillator example. Since the exact alchemical free energy differences are unknown, the reference free energy differences are calculated using all configurations sampled from all 10 repeats of the molecular dynamics simulations. The mean RMSEs of the free energy differences between end states of all paths are shown in Table . For both the water phase and the protein phase, CBayesMBAR has significantly smaller RMSEs than BayesMBAR, with p-values less than 0.05 for all cases (Fig. and). Compared to CCC, the RMSE of CBayesMBAR is either significantly (p-value ≤ 0.05) smaller or statistically indistinguishable from that of CCC (Fig. and). We also compare the three methods using the MAEs of their free energy estimates on individual paths. The results are shown in Table and S3 for the water phase and Table and S4 for the protein phase. Although CBayesMBAR and CCC have reduced RMSEs aggregated over all paths compared to BayesMBAR (Table ), their improvement over BayesMBAR on individual paths is not as consistent as in the harmonic oscillator example. CBayesMBAR and CCC have smaller MAEs than BayesMBAR for some paths, while for other paths, they have larger MAEs. This indicates that incorporating the cycle consistency condition does not uniformly enhance the accuracy of free energy estimates on all paths, although it does improve the overall accuracy.