Relational Marginal Problems: Theory and Estimation
This work addresses parameter estimation in relational models for settings with varying data sizes, offering theoretical insights and practical adjustments, but it is incremental as it extends propositional concepts to the relational domain.
The paper tackles the relational marginal problem by comparing relational marginals, establishing a duality with maximum likelihood estimation, and developing a statistically sound parameter learning method for relational models when training and test data have different numbers of constants, including error bounds and adjustments for feature groundings.
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.