Information Lattice Learning as Probabilistic Graphical Model Structure Learning
For researchers in probabilistic graphical models and interpretable machine learning, the paper provides a formal connection between ILL and constraint-based factor graphs, but the results are primarily theoretical and incremental.
The paper shows that information lattice learning (ILL) learns probabilistic rules that correspond to marginal constraints over interpretable abstractions, and that under Shannon-entropy lifting these constraints yield a log-linear factor graph. This provides a probabilistic graphical model interpretation for ILL, clarifying its relationship to maximum entropy models and suggesting new directions for inference and hybrid learning.
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.