This ensures that free energy estimates directly satisfy the cycle consistency condition. Uncertainty estimates, derived from the standard deviation of the posterior distribution, reflect both the sampled configurations and the cycle consistency condition. Through two example applications, we demonstrate that CBayesMBAR significantly improves the accuracy of free energy estimates on perturbation graphs with cycles, outperforming both BayesMBAR and CCC. The computational cost of CBayesMBAR is comparable to that of multiple independent BayesMBAR calculations and is negligible compared to the cost of molecular dynamics simulations. For example, when computing relative protein-ligand binding free energies, CBayesMBAR takes about one minute to compute free energy differences and their uncertainties for all edges in the alchemical perturbation graph, using 2500 configurations from each state and running on a single RTX A5000 graphic processing unit.