Lifted Model Construction under Approximate Commutativity

Lifted inference algorithms enable scalable probabilistic inference even for large object domains by leveraging the indistinguishability of objects in a probability distribution. An essential prerequisite for constructing a lifted representation is to identify commutative factors, i.e., functions whose output values are invariant under permutations of a subset of their input values, in a potential-based factorisation. In practice, however, parameters learned from data inevitably deviate even if associated objects are indistinguishable, causing their corresponding factors to be only approximately commutative instead of being exactly commutative. We address this problem by introducing the concept of ε-commutativity, a relaxation of commutativity where output values are only approximately invariant under permutations of input values. Specifically, we show how ε-commutativity can be exploited for lifted model construction, downstream probabilistic inference, and prove strict bounds on the induced approximation error, thereby ensuring the practical applicability of lifted model construction while maintaining highly accurate query results. These theoretical guarantees are confirmed empirically, demonstrating comparable query accuracy at lower runtime.