An Algebraic Framework for Hierarchical Probabilistic Abstraction

TL;DR

This work introduces a hierarchical probabilistic abstraction framework that generalizes measure-theoretic foundations to multi-layer settings, addressing the challenge of capturing complex probabilistic hierarchies. It defines three fundamental one-layer abstraction types—Direct, Divergent, and Convergent—and builds them into hierarchical constructs (HPAM-DAGs) with a boundary concept (HPoA) to preserve probabilistic integrity. The framework further combines these primitives into Sequential and Hybrid HPAM-DAGs, including illustrative Alzheimer’s disease scenarios, and provides propositions on HPoA existence, uniqueness, and intermediate states to support tractable, comprehensible reasoning across layers. By offering modular abstractions that link low-level perceptual data to high-level concepts, the approach aims to enable dual-level reasoning and scalable analyses across AI subfields, with future work identifying rigorous formalizations and broader domain applications.

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.