 MCFactorization tells us that once we know the conditional probability distribution of each variable given its parents, \(\Pr(X_i \mid \bPA(X_i))\), we can compute the complete joint distribution over all of the variables. It is relatively easy to see that MCFactorization follows from MCScreening_off. Since \(\bG\) is acyclic, we may re-label the subscripts on the variables so that they are ordered from ‘earlier’ to ‘later’, with only earlier variables being ancestors of later ones. It follows from the probability calculus that   \[\Pr(X_1, X_2 , \ldots ,X_n)  = \Pr(X_1) \times \Pr(X_2 \mid X_1) \times \ldots \times \Pr(X_n \mid X_1, X_2 , \ldots ,X_{n-1})\]    (this is a version of the theorem of total probability). For each term \(\Pr(X_i \mid X_1, X_2 , \ldots ,X_{i-1})\), our ordering ensures that all of the parents of \(X_i\) will be included on the right hand side, and none of its descendants will. MCScreening_off then tells us that we can eliminate all of the terms from the right hand side except for the parents of \(X_i\).