Both BayesMBAR and CBayesMBAR quantify uncertainties in free energy estimates using the standard deviation of the posterior distribution. Table presents the mean estimated uncertainties for free energy differences between the end states of individual paths. Consistently, CBayesMBAR yields smaller uncertainties compared to BayesMBAR, highlighting how coupling free energy estimates via cycle consistency conditions enhances estimation precision. The performance improvement of CBayesMBAR over BayesMBAR in estimating free energy differences along individual paths (Table ) correlates closely with the uncertainties associated with these estimates (Table ). For instance, paths with smaller uncertainties (e.g., A → B) demonstrate modest improvements in MAE, whereas paths with larger uncertainties (e.g., B → D) have more substantial improvements. This observation aligns with the theoretical expectation that cycle-based coupling primarily benefits estimates with lower precision by leveraging high-precision paths and the cycle consistency condition. Next, we apply CBayesMBAR to compute the relative binding free energies of multiple ligands to a given protein. This type of calculation typically employs an alchemical per-turbation graph connecting the ligands. Free energy differences along the graph's edges are computed with alchemical methods. In our study, we apply CBayesMBAR to calculate the relative binding free energies of 6 ligands to the tyrosine kinase 2 (Tyk2) protein (Protein Data Bank ID: 4GIH ), a system widely used as a benchmark in the field. The 6 ligands share a common scaffold structure, with the R group representing the chemical group that differs among them (Fig. ). We construct an alchemical perturbation graph with multiple cycles to connect the ligands (Fig. ).