Marginal Probability undirected graph G=(V,E) finite integer g≥2 Gibbs distribution uG over all configurations in [g] V possible boundary condition o E [g]A specified on an arbitrary subset ACD marginal distribution at vertexy: VxE [q]:u()=Pr [X=XA=0] XoHG a）=4o)）=Π-1(a) ZG i=1Marginal Probability Gibbs distribution µG over all configurations in [q]V undirected graph G = (V, E) finite integer q ≥ 2 specified on an arbitrary subset Λ⊂V ∀ possible boundary condition σ ∈ [q]Λ marginal distribution at vertex µ v ∈ V : ￾ v 8x 2 [q] : µ￾ v (x) = Pr X⇠µG [Xv = x | X⇤ = ￾] w(￾) ZG = µ(￾) = Y n i=1 µ￾1,...,￾i￾1 vi (￾i)