We propose an approach to lifted approximate inference for first-order probabilistic models, such as Markov logic networks. It is based on performing exact lifted inference in a simplified first-order model, which is found by relaxing first-order constraints, and then compensating for the relaxation. These simplified models can be incrementally improved by carefully recovering constraints that have been relaxed, also at the first-order level. This leads to a spectrum of approximations, with lifted belief propagation on one end, and exact lifted inference on the other. We discuss how relaxation, compensation, and recovery can be performed, all at the firstorder level, and show empirically that our approach substantially improves on the approximations of both propositional solvers and lifted belief propagation.
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