  Computing the partition function and the marginals of a global probability
distribution are two important issues in any probabilistic inference problem.
In a previous work, we presented sub-tree based upper and lower bounds on the
partition function of a given probabilistic inference problem. Using the
entropies of the sub-trees we proved an inequality that compares the lower
bounds obtained from different sub-trees. In this paper we investigate the
properties of one specific lower bound, namely the lower bound computed by the
minimum entropy sub-tree. We also investigate the relationship between the
minimum entropy sub-tree and the sub-tree that gives the best lower bound.
