In this exercise, we consider the use of tree-structured local models in the undirected setting.

a. Show how we can use a structure similar to tree-CPDs to represent a factor in a Markov network. What do the values at the leaves of such a tree represent?

b. Given a context U = u, define a simple algorithm that takes a tree factor (Y ) and returns the reduced factor [U = u](Y U) (see definition 4.5).

c. The preceding expression takes Y U to be the scope of the reduced factor. In some cases it turns out that we can further reduce the scope. Give an example and specify a general rule for when a variable in Y U can be eliminated from the scope of the reduced tree-factor.