Projectivity revisited
This work addresses the limitation of projectivity in handling structured data for researchers in statistical relational learning, though it is incremental as it builds on prior formalizations.
The paper extends the concept of projectivity from statistical relational representations based solely on domain size to include structured input from databases, enabling its application to a wider range of problems. It transfers existing results, proves a correspondence with infinite domains, and introduces a strengthened version called σ-projectivity for consistent representation across modes.
The behaviour of statistical relational representations across differently sized domains has become a focal area of research from both a modelling and a complexity viewpoint.Recently, projectivity of a family of distributions emerged as a key property, ensuring that marginal probabilities are independent of the domain size. However, the formalisation used currently assumes that the domain is characterised only by its size. This contribution extends the notion of projectivity from families of distributions indexed by domain size to functors taking extensional data from a database. This makes projectivity available for the large range of applications taking structured input. We transfer key known results on projective families of distributions to the new setting. This includes a characterisation of projective fragments in different statistical relational formalisms as well as a general representation theorem for projective families of distributions. Furthermore, we prove a correspondence between projectivity and distributions on countably infinite domains, which we use to unify and generalise earlier work on statistical relational representations in infinite domains. Finally, we use the extended notion of projectivity to define a further strengthening, which we call $σ$-projectivity, and which allows the use of the same representation in different modes while retaining projectivity.