1. Assume we have a calibrated clique tree for a distribution PΦ. Show how we can use the clique tree to generate samples that are sampled exactly from PΦ.
2. Prove proposition 12.2.
Let ξ be a sample generated by algorithm 12.2 and w be its weight. Then the distribution over ξ is as defined by the network BZ=z, and
The proof is not di‑cult and is left as an exercise. It is important to note, however, that the algorithm does not require the explicit construction of the mutilated network. It simply traverses the original network.