Mean estimated uncertainties of free energy estimates on individual paths are presented in Table and S6 for the water and protein phases, respectively. Similar to the harmonic oscillator example, CBayesMBAR demonstrates smaller uncertainties than BayesMBAR, underscoring improved precision in free energy estimates by leveraging the cycle consistency condition. Comparing the uncertainties and the MAEs of the free energy estimates on individual paths, we observe that whether the cycle consistency condition improves the accuracy of free energy estimates on a path depends on the initial precision of the estimate on the path relative to that of other paths in the cycle. For paths with low precision, CBayesMBAR and CCC significantly improve the accuracy of the estimates, while for paths with high precision, CBayesMBAR and CCC could perform worse than BayesMBAR. This phenomenon occurs because coupling allows noise from low-precision paths to pass into higher precision paths, thereby slightly reducing the accuracy of the estimates on the Notably, for paths where CBayesMBAR and CCC perform worse than BayesMBAR, CBayesMBAR tends to have smaller MAEs than CCC. This suggests that the cycle consistency condition is more effectively utilized in CBayesMBAR than in CCC to improve the accuracy of free energy estimates on individual paths. In this work, we introduce CBayesMBAR, a new method for computing free energy differences on perturbation graphs with cycles. As a Bayesian approach, CBayesMBAR integrates the cycle consistency condition to couple free energy estimates across all edges of a perturbation graph in a principled manner. It incorporates the cycle consistency condition into the prior distribution of free energies and combines this with sampled configurations to form the posterior distribution, which is then used to estimate the free energies and their uncertainties.