Information Lattice Learning Interpreted as Probabilistic Graphical Model Structure

Information Lattice Learning (ILL) is a technique designed to uncover interpretable rules within a signal by iteratively projecting the signal onto a partition lattice, which represents a hierarchy of abstractions, and then lifting selected rules back to the signal domain. This research specifically examines the scenario where the signal is a probability mass function. The paper demonstrates that when applied to probability mass functions, the probabilistic rules learned by ILL can be naturally interpreted within the framework of Probabilistic Graphical Models (PGMs). In this context, a partition in ILL generates a deterministic quotient variable, and a learned rule corresponds to the marginal probability distribution of that quotient variable. Consequently, a set of rules forms a collection of marginal constraints over these interpretable abstractions. The concept of "general lifting" in ILL is shown to correspond to the feasible family of all joint distributions that satisfy these constraints. "Special lifting," which aims for a maximum-ignorance reconstruction, is implemented in ILL through an L2 uniformity principle that is closely related to the principle of maximum entropy. Under a Shannon-entropy lifting, these constraints result in a log-linear factor graph where factors are indexed by the learned abstractions. The information lattice itself, however, is distinct from a Bayesian network; its edges denote refinement and coarsening of abstractions, not conditional dependencies. This perspective positions ILL primarily as a method for structure learning in interpretable constraint-based factor graphs over quotient variables, clarifying its connections to graphical models and maximum entropy models while suggesting new avenues for research in inference, identifiability, and hybrid symbolic-probabilistic learning.