Before considering the approximate inference methods, let's solve the exact inference problem using the concepts that we have so far developed in this chapter. In the previous sections, we saw that maximizing the energy function is equivalent to minimizing the relative entropy between Q and . So now, if we restrict ourselves to calibrated cluster trees, we can further simplify the objective function. Restricting ourselves to calibrated cluster trees allows us to rewrite the energy function in a factored form as a sum of terms, each depending directly on only one of the beliefs in Q. This form also reveals structure in the distribution, and is therefore a much better starting point for further analysis.
Given a cluster tree T with a set of beliefs Q and an assignment , which maps factors in to clusters in T, we define the factored energy function as follows:
Here, is the initial potential assigned to :
Here, represents the expectation on the value given...