Lifted Inference uses the rules of first order predicate logic to improve the speed of the standard Markov Random Field algorithms applied to Markov Logic Networks. I wish I had been in Barcelona Spain in July last year for IJCAI11 because they had a cool tutorial on Lifted Inference. Here’s a quote

Much has been achieved in the field of AI, yet much remains to be done if we are to reach the goals we all imagine. One of the key challenges with moving ahead is closing the gap between logical and statistical AI. Recent years have seen an explosion of successes in combining probability and (subsets of) first-order logic respectively programming languages and databases in several subfields of AI: Reasoning, Learning, Knowledge Representation, Planning, Databases, NLP, Robotics, Vision, etc. Nowadays, we can learn probabilistic relational models automatically from millions of inter-related objects. We can generate optimal plans and learn to act optimally in uncertain environments involving millions of objects and relations among them. Exploiting shared factors can speed up message-passing algorithms for relational inference but also for classical propositional inference such as solving SAT problems. We can even perform exact lifted probabilistic inference avoiding explicit state enumeration by manipulating first-order state representations directly.

In the related paper “Lifted Inference Seen from the Other Side : The Tractable Features“, Jha, Gogate, Meliou, Suciu (2010) reverse this notion. Here’s the abstract:

Lifted Inference algorithms for representations that combine ﬁrst-order logic and graphical models have been the focus of much recent research. All lifted algorithms developed to date are based on the same underlying idea: take a standard probabilistic inference algorithm (e.g., variable elimination, belief propagation etc.) and improve its efﬁciency by exploiting repeated structure in the ﬁrst-order model. In this paper, we propose an approach from the other side in that we use techniques from logic for probabilistic inference. In particular, we deﬁne a set of rules that look only at the logical representation to identify models for which exact efﬁcient inference is possible. Our rules yield new tractable classes that could not be solved efﬁciently by any of the existing techniques.