1. Let A, B, S and T be non-empty subsets of vertices of a DAG G. If A and B are d-separated by S and A and S are dseparated by T, are A and B d-separated by T? (prove the claim, or provide a counter-example) 2. Let G = (V, E) be a DAG and let GM 2 (V, EM) be an undirected graph such that we have an edge 1' j in EM when i, j are both parents of the same node in G, or when there is a directed edge 1' > 3' or 3' > 2' in E (GM is known as the moral graph of G). Let AB, 5' be three sets of disjoint vertices. Show that if A and B are separated by S in G M, then they are d-separated by S in G. 3. Consider a v-structure network X } Z ( Y, where the variables X, Y, Z are binary, with values in {01 1}. Assume that the conditional probability of Z is given by the following \"noisy-or" model: 10(2 = 0lay) = (1 W1 MEG VI\". where A, ,u, V E (U, 1) and A is a so-called \"leak\" probability. Intuitively, this encodes the fact that Z is less likely to be 0 when either X or Y are equal to 1, hence making this a \"noisy\" OR operation. Show that this network must statisfy the explaining away property: p(:t=1|z=l) 2p(m=l|y=l,z= 1)