Concept lattices as interpretable models

The interpretability literature has spent a decade looking for a class of models whose decisions are legible by construction. Decision trees, sparse linear models, generalised additive models, prototype-based networks — each candidate trades some expressive capacity for some guarantee of human readability. Concept lattices, in the sense of Formal Concept Analysis, belong on that list, and they have a property that none of the others quite share: their structure is determined, canonically, by the data alone. Once a context is fixed, the lattice is fixed. There are no architectural choices, no regularisation hyperparameters, no training noise. The model is a theorem about the data.

That sounds, on first hearing, like an extreme form of interpretability — and in some ways it is. Every node of a concept lattice is a precise statement of the form “the set of objects with exactly this combination of attributes is exactly this set.” Every edge is an inclusion. The full implication theory of the context can be read off the lattice. There is nothing inside the model that is not also visible from outside it.

The strain shows up when one tries to use such a lattice as a predictive model rather than a descriptive one. Generalisation, in the classical FCA setting, is a delicate matter: the lattice describes the data given, and saying anything about data not given requires an additional commitment — a probabilistic semantics, an external implication closure, a notion of attribute exploration. This essay, forthcoming, will develop that distinction carefully, with examples from interpretability benchmarks and a contrast with rule-based post-hoc explainers.