Probabilistic Modelling and Reasoning Solutions for Tutorial 2 Spring 2018

Solution. The Markov blanket of a node in a undirected graphical model equals the set of its neighbours: MB(x4) = ne(x4) = ne4 = {x1, x5}. This implies, for example, that x4 ⊥ x2, x3 | x1, x5. (e) On which minimal set of variables A do we need to condition to have x1 ⊥ x5 | A? Solution. We first identify all trails from x1 to x5. There are three such trails: (x1, x2, x5), (x1, x3, x2, x5), and (x1, x4, x5). Conditioning on x2 blocks the first two trails, conditioning on x4 blocks the last. We thus have: x1 ⊥ x5 | x2, x4, so that A = {x2, x4}.