On the Complexity and Approximation of Binary Evidence in Lifted Inference

Lifted inference algorithms exploit symmetries in probabilistic models to speed up inference. They show impressive performance when calculating unconditional probabilities in relational models, but often resort to non-lifted inference when computing conditional probabilities, because the evidence breaks many of the model's symmetries. Recent theoretical results paint a grim picture, showing that conditioning on binary relations is #P-hard, and in the worst case, no lifting can be expected. In this paper, we identify Boolean rank of the evidence as a key parameter in the complexity of conditioning. We contrast the hardness result by showing that conditioning on binary evidence with bounded Boolean rank is efficient. This opens up the possibility of approximating evidence by a low-rank Boolean matrix factorization that maintains the model's symmetries and admits efficient lifted inference.