Information Lattice Learning as Probabilistic Graphical Model Structure Learning

2026-06-10

AI Summary

This work investigates how to learn interpretable abstract rules from probabilistic signals and establishes their connection to probabilistic graphical models. It proposes a learning framework grounded in information lattices, where rules are interpreted as marginal constraints on quotient variables by alternately projecting signals onto partition lattices and lifting the resulting rules back to the original domain. This approach explicitly links information lattice learning to constraint-based factor graph learning for the first time, revealing an intrinsic relationship with maximum entropy models and introducing a novel interpretable modeling paradigm centered on quotient variables. The contributions include deriving the corresponding log-linear factor graph representation, clarifying the fundamental distinction between information lattices and Bayesian networks, and opening new avenues for hybrid symbolic–probabilistic learning.

Abstract

Information lattice learning (ILL) learns interpretable rules of a signal by alternately projecting the signal onto a partition lattice that encodes a hierarchy of abstractions and lifting selected rules back to the signal domain. When the signal is a probability mass function, we show the probabilistic rules learned by ILL admit a natural probabilistic graphical model (PGM) interpretation and develop this interpretation in detail. A partition in ILL induces a deterministic quotient variable, and a rule is the marginal law of that quotient variable. A rule set is therefore a collection of marginal constraints over interpretable abstractions. General lifting is the feasible family of all joint distributions satisfying those constraints, while special lifting chooses a maximum-ignorance reconstruction, implemented in ILL by an L2 uniformity principle closely related to maximum entropy. Under a Shannon-entropy lifting, the same constraints yield a log-linear factor graph whose factors are indexed by learned abstractions. The information lattice itself, however, is not a Bayesian network: its edges encode refinement and coarsening of abstractions, not conditional dependence. Thus ILL is best viewed as structure learning for interpretable constraint-based factor graphs over quotient variables. This view clarifies how ILL relates to graphical models and maximum entropy models, while suggesting new directions for inference, identifiability, and hybrid symbolic-probabilistic learning.

Problem

• Information Lattice Learning
• Probabilistic Graphical Model
• Structure Learning
• Interpretable Rules
• Quotient Variables

Innovation

• Information Lattice Learning
• Probabilistic Graphical Models
• Quotient Variables
• Structure Learning
• Maximum Entropy