Problem
Suppose we are given a set of factors Φ = {φ1, . . . , φK} over X = {X1, . . . , Xn}. Our aim is to convert these factors into a pairwise Markov network by introducing new auxiliary variables Y = {Y1, . . . , Yk} so that Yj denotes a joint assignment to Scope[φj]. Show how to construct a set of factors Φ' that is a pairwise Markov network over X ∪ Y such that PΦ' (x) = PΦ(x) for each assignment to X.