Projectivity revisited

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 inference is independent of the domain size and that under certain additional requirements parameter estimation from random samples is statistically consistent. However, the currently used formalisation 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 the known results on projective families of distributions to the new setting. This includes statistical consistency of learning, 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 $\sigma$-projectivity, and which allows the use of the same representation in different modes while retaining projectivity.