A generative probabilistic model for relational data consists of a family of\nprobability distributions for relational structures over domains of different\nsizes. In most existing statistical relational learning (SRL) frameworks, these\nmodels are not projective in the sense that the marginal of the distribution\nfor size-$n$ structures on induced sub-structures of size $k<n$ is equal to the\ngiven distribution for size-$k$ structures. Projectivity is very beneficial in\nthat it directly enables lifted inference and statistically consistent learning\nfrom sub-sampled relational structures. In earlier work some simple fragments\nof SRL languages have been identified that represent projective models.\nHowever, no complete characterization of, and representation framework for\nprojective models has been given. In this paper we fill this gap: exploiting\nrepresentation theorems for infinite exchangeable arrays we introduce a class\nof directed graphical latent variable models that precisely correspond to the\nclass of projective relational models. As a by-product we also obtain a\ncharacterization for when a given distribution over size-$k$ structures is the\nstatistical frequency distribution of size-$k$ sub-structures in much larger\nsize-$n$ structures. These results shed new light onto the old open problem of\nhow to apply Halpern et al.'s "random worlds approach" for probabilistic\ninference to general relational signatures.\n