As exact inference for first-order probabilistic graphical models at the propositional level can be formidably expensive, there is an ongoing effort to design efficient lifted inference algorithms for such models. This paper discusses directed first-order models that require an aggregation operator when a parent random variable is parameterized by logical variables that are not present in a child random variable. We introduce a new data structure, aggregation parfactors, to describe aggregation in directed first-order models. We show how to extend Milch et al.'s C-FOVE algorithm to perform lifted inference in the presence of aggregation parfactors. We also show that there are cases where the polynomial time complexity (in domain size of logical variables) of the C-FOVE algorithm can be reduced to logarithmic time complexity using aggregation parfactors.

Jacek Kisyński, Dawid Poole