An Algebraic Framework for Hierarchical Probabilistic Abstraction

TL;DR

This paper introduces a hierarchical probabilistic abstraction framework based on measure theory, enabling modular, layered analysis of complex stochastic systems to improve interpretability and system understanding.

Contribution

It extends measure-theoretic foundations to hierarchical probabilistic abstraction, allowing layered, modular analysis of stochastic models.

Findings

Facilitates detailed and system-wide analysis through layered mappings.

Enhances interpretability of probabilistic models across AI subfields.

Supports alignment of high-level and low-level system understanding.

Abstract

Abstraction is essential for reducing the complexity of systems across diverse fields, yet designing effective abstraction methodology for probabilistic models is inherently challenging due to stochastic behaviors and uncertainties. Current approaches often distill detailed probabilistic data into higher-level summaries to support tractable and interpretable analyses, though they typically struggle to fully represent the relational and probabilistic hierarchies through single-layered abstractions. We introduce a hierarchical probabilistic abstraction framework aimed at addressing these challenges by extending a measure-theoretic foundation for hierarchical abstraction. The framework enables modular problem-solving via layered mappings, facilitating both detailed layer-specific analysis and a cohesive system-wide understanding. This approach bridges high-level conceptualization with low-level perceptual data, enhancing interpretability and allowing layered analysis. Our framework provides a robust foundation for abstraction analysis across AI subfields, particularly in aligning System 1 and System 2 thinking, thereby supporting the development of diverse abstraction methodologies.