Lifted inference, handling whole sets of indistinguishable objects together, is critical to the effective application of probabilistic relational models to realistic real world tasks. Recently, lifted belief propagation (LBP) has been proposed as an efficient approximate solution of this inference problem. It runs a modified BP on a lifted network where nodes have been grouped together if they have -- roughly speaking -- identical computation trees, the tree-structured unrolling of the underlying graph rooted at the nodes. In many situations, this purely syntactic criterion is too pessimistic: message errors decay along paths. Intuitively, for a long chain graph with weak edge potentials, distant nodes will send and receive identical messages yet their computation trees are quite different. To overcome this, we propose iLBP, a novel, easy-to-implement, informed LBP approach that interleaves lifting and modified BP iterations. In turn, we can efficiently monitor the true BP messages se...