Suppose you have a relational probabilistic model for movie prediction, which represents P(likes(P, M) | age(P), genre(M))

where age(P) and genre(M) are a priori independent

(a) What is the treewidth (page 417) of the ground belief network (after pruning irrelevant variables) for querying age(Sam) given age and genre are not observed and given the observations for likes in Figure 17.14.

(b) For the same probabilistic model, for m movies, n people, and r ratings, what is the worst-case treewidth of the corresponding graph (after pruning irrelevant variables), where only ratings are observed? [Hint: The treewidth depends on the structure of the observations; think about how the observations can be structured to maximize the treewidth.]

(c) For the same probabilistic model, for m movies, n people, and r ratings, what is the worst-case treewidth of the corresponding graph, where only some of the ratings but all of the genres are observed?