  Abstraction is a powerful idea widely used in science, to model, reason and
explain the behavior of systems in a more tractable search space, by omitting
irrelevant details. While notions of abstraction have matured for deterministic
systems, the case for abstracting probabilistic models is not yet fully
understood.
  In this paper, we provide a semantical framework for analyzing such
abstractions from first principles. We develop the framework in a general way,
allowing for expressive languages, including logic-based ones that admit
relational and hierarchical constructs with stochastic primitives. We motivate
a definition of consistency between a high-level model and its low-level
counterpart, but also treat the case when the high-level model is missing
critical information present in the low-level model. We prove properties of
abstractions, both at the level of the parameter as well as the structure of
the models. We conclude with some observations about how abstractions can be
derived automatically.
