  We generalise the distribution semantics underpinning probabilistic logic
programming by distilling its essential concept, the separation of a free
random component and a deterministic part. This abstracts the core ideas beyond
logic programming as such to encompass frameworks from probabilistic databases,
probabilistic finite model theory and discrete lifted Bayesian networks. To
demonstrate the usefulness of such a general approach, we completely
characterise the projective families of distributions representable in the
generalised distribution semantics and we demonstrate both that large classes
of interesting projective families cannot be represented in a generalised
distribution semantics and that already a very limited fragment of logic
programming (acyclic determinate logic programs) in the determinsitic part
suffices to represent all those projective families that are representable in
the generalised distribution semantics at all.
