Relational Marginal Problems: Theory and Estimation
  In the propositional setting, the marginal problem is to find a
(maximum-entropy) distribution that has some given marginals. We study this
problem in a relational setting and make the following contributions. First, we
compare two different notions of relational marginals. Second, we show a
duality between the resulting relational marginal problems and the maximum
likelihood estimation of the parameters of relational models, which generalizes
a well-known duality from the propositional setting. Third, by exploiting the
relational marginal formulation, we present a statistically sound method to
learn the parameters of relational models that will be applied in settings
where the number of constants differs between the training and test data.
Furthermore, based on a relational generalization of marginal polytopes, we
characterize cases where the standard estimators based on feature's number of
true groundings needs to be adjusted and we quantitatively characterize the
consequences of these adjustments. Fourth, we prove bounds on expected errors
of the estimated parameters, which allows us to lower-bound, among other
things, the effective sample size of relational training data.
