Abstracting Probabilistic Models: A Logical Perspective

TL;DR

This paper develops a semantic framework for understanding and analyzing the abstraction of probabilistic models, extending concepts from deterministic systems to more expressive, logic-based probabilistic languages.

Contribution

It introduces a first-principles semantic framework for probabilistic model abstraction, including definitions of consistency and properties of abstractions.

Findings

Framework accommodates expressive, logic-based probabilistic languages

Defines and analyzes consistency between high-level and low-level models

Discusses automatic derivation of abstractions

Abstract

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.