The key to the second type of method lies in effectively coupling the calculations of free energies along the edges of a cycle using the cycle consistency condition. Intuitively, this coupling should not significantly alter the free energy estimates on edges where the estimates are already highly precise. Instead, it should leverage these high-precision edges and the cycle consistency condition to refine the estimates on edges with lower precision. Here, we propose a probabilistic method that accomplishes this coupling, which we call the coupled Bayesian multistate Bennett acceptance ratio (CBayesMBAR) method. We applied CBayesMBAR to compute free energy differences among four harmonic oscillators and relative protein-ligand binding free energies. In both examples, CBayesMBAR provides more accurate results than those obtained without considering the cycle consistency condition. Furthermore, CBayesM-BAR outperformed the widely used cycle closure correction method that also uses the cycle consistency condition. We have implemented CBayesMBAR as part of the BayesMBAR package, which is freely available at