Abstracting Probabilistic Models: A Logical Perspective
This work addresses the lack of a fully understood framework for abstracting probabilistic models, which is incremental in extending abstraction concepts from deterministic to probabilistic systems.
The paper tackles the problem of abstracting probabilistic models by providing a semantical framework from first principles, proving properties at both parameter and structural levels, and concluding with observations on automatic derivation.
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.