  We study computational aspects of relational marginal polytopes which are
statistical relational learning counterparts of marginal polytopes, well-known
from probabilistic graphical models. Here, given some first-order logic
formula, we can define its relational marginal statistic to be the fraction of
groundings that make this formula true in a given possible world. For a list of
first-order logic formulas, the relational marginal polytope is the set of all
points that correspond to the expected values of the relational marginal
statistics that are realizable. In this paper, we study the following two
problems: (i) Do domain-liftability results for the partition functions of
Markov logic networks (MLNs) carry over to the problem of relational marginal
polytope construction? (ii) Is the relational marginal polytope containment
problem hard under some plausible complexity-theoretic assumptions? Our
positive results have consequences for lifted weight learning of MLNs. In
particular, we show that weight learning of MLNs is domain-liftable whenever
the computation of the partition function of the respective MLNs is
domain-liftable (this result has not been rigorously proven before).
