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# Lifted MAP Inference for Markov Logic Networks

*Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics*, PMLR 33:859-867, 2014.

#### Abstract

In this paper, we present a new approach for lifted MAP inference in Markov Logic Networks (MLNs). Our approach is based on the following key result that we prove in the paper: if an MLN has no shared terms then MAP inference over it can be reduced to MAP inference over a Markov network having the following properties: (i) the number of random variables in the Markov network is equal to the number of first-order atoms in the MLN; and (ii) the domain size of each variable in the Markov network is equal to the number of groundings of the corresponding first-order atom. We show that inference over this Markov network is exponentially more efficient than ground inference, namely inference over the Markov network obtained by grounding all first-order atoms in the MLN. We improve this result further by showing that if non-shared MLNs contain no self joins, namely every atom appears at most once in each of its formulas, then all variables in the corresponding Markov network need only be bi-valued. Our approach is quite general and can be easily applied to an arbitrary MLN by simply grounding all of its shared terms. The key feature of our approach is that because we reduce lifted inference to propositional inference, we can use any propositional MAP inference algorithm for performing lifted MAP inference. Within our approach, we experimented with two propositional MAP inference algorithms: Gurobi and MaxWalkSAT. Our experiments on several benchmark MLNs clearly demonstrate our approach is superior to ground MAP inference in terms of scalability and solution quality.