# Information Lattice Learning as Probabilistic Graphical Model Structure Learning

arXiv: https://arxiv.org/abs/2606.19366 · HTML: https://mlsanity.com/paper/2606.19366/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 23 references that are in the map.
- First posted on arXiv 11 Jun 2026.

## Builds on (most cited first, 3 of 3)

- [When Does LeJEPA Learn a World Model?](https://mlsanity.com/paper/2605.26379/) (2026)
- [Learning from One and Only One Shot](https://mlsanity.com/paper/2201.08815/) (2022)
- [Learning Gaussian Graphical Models with Observed or Latent FVSs](https://mlsanity.com/paper/1311.2241/) (2013)

## Abstract (arXiv)

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.