Faithfulness and learning hypergraphs from discrete distributions

TL;DR

This paper introduces the concept of strong-faithfulness for hypergraphs in discrete distributions, providing bounds and methods to ensure consistent learning of hierarchical log-linear models.

Contribution

It extends faithfulness concepts to hypergraphs, enabling consistent hypergraph learning and quantifying the prevalence of non-strong-faithful distributions.

Findings

Bounds for distributions not satisfying strong-faithfulness are derived.

Hypergraph models better capture association structures in discrete data.

Methods for assessing and ensuring strong-faithfulness are proposed.

Abstract

The concepts of faithfulness and strong-faithfulness are important for statistical learning of graphical models. Graphs are not sufficient for describing the association structure of a discrete distribution. Hypergraphs representing hierarchical log-linear models are considered instead, and the concept of parametric (strong-) faithfulness with respect to a hypergraph is introduced. Strong-faithfulness ensures the existence of uniformly consistent parameter estimators and enables building uniformly consistent procedures for a hypergraph search. The strength of association in a discrete distribution can be quantified with various measures, leading to different concepts of strong-faithfulness. Lower and upper bounds for the proportions of distributions that do not satisfy strong-faithfulness are computed for different parameterizations and measures of association.