The Bayesian framework used in BayesMBAR enables the coupling of multiple BayesMBAR calculations on a perturbation graph with cycles. For example, consider three thermodynamic states, A, B, and C. To compute their free energy differences, we construct a perturbation graph connecting these states (Fig. ). In this graph, red, orange, and blue circles represent the three end states, and arrows indicate the paths connecting them. Small black circles along the paths represent intermediate states between the end states. Configurations are sampled from both the end states and the intermediate states along each path. We use X h to denote the configurations sampled from states on the h-th path, where h ranges from 1 to 3, and Y h to denote the state indices of X h . In other words, X h and Y h correspond to X and Y in Eq. 2 and 3. We use F h to denote the free energy of states on the h-th path, expressed as