Characterizing Identifiability in Boolean Factor Models
Abstract
Boolean factor models, including prominent subfamilies such as Boolean matrix decompositions and cognitive diagnosis models, find broad applications ranging from social sciences to engineering. Despite their flexibility, a key challenge lies in establishing the identifiability of their graphical structures, which specify how latent variables influence observed variables. Existing identifiability conditions typically rely on the strong assumption of pure nodes, which may be unrealistic in many applications. We develop a novel approach leveraging the Hasse diagram to represent the distribution of observed variables and transform identifiability into a graph isomorphism challenge. Based on this, we establish {\it sufficient and necessary} graphical identifiability conditions that do not require pure nodes. We further derive equivalent algebraic conditions and develop an efficient Boolean satisfiability (SAT)-based verification algorithm. We extend the analysis to probabilistic Boolean factor models, establishing conditions for jointly identifying the graphical structure and additional model parameters without requiring pure nodes. Our results substantially broaden the class of identifiable and interpretable Boolean factor models by removing the pure-node requirement, yielding new theoretical insights, while also providing practitioners with a concrete and easily implementable tool to assess model identifiability.
Related
- Characterizing Identifiability and Generalization for Inverse Receding-Horizon Linear-Quadratic Regulator Problems
- Boolean Trusses as Rectangular Bands and Boolean Rings
- Canonical Directed Laplacian on F_2^4, Symmetry-Induced Two-Scale Coupling, and the Structural Isomorphism of a 64-Copy Boolean Graph Assembly
- The Boolean Prime Ideal Theorem and Vitali Sets
- Hyperstructures as complete atomic Boolean algebras
BibTeX
@misc{lin2026characterizing,
title = {{Characterizing Identifiability in Boolean Factor Models}},
author = {Mengqi Lin and Gongjun Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02682},
eprint = {2610.02682},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02682},
}
RIS
TY - UNPB
TI - Characterizing Identifiability in Boolean Factor Models
AU - Mengqi Lin
AU - Gongjun Xu
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02682
UR - https://arxiv.org/abs/2610.02682
AB - Boolean factor models, including prominent subfamilies such as Boolean matrix decompositions and cognitive diagnosis models, find broad applications ranging from social sciences to engineering. Despite their flexibility, a key challenge lies in establishing the identifiability of their graphical structures, which specify how latent variables influence observed variables. Existing identifiability conditions typically rely on the strong assumption of pure nodes, which may be unrealistic in many applications. We develop a novel approach leveraging the Hasse diagram to represent the distribution of observed variables and transform identifiability into a graph isomorphism challenge. Based on this, we establish {\it sufficient and necessary} graphical identifiability conditions that do not require pure nodes. We further derive equivalent algebraic conditions and develop an efficient Boolean satisfiability (SAT)-based verification algorithm. We extend the analysis to probabilistic Boolean factor models, establishing conditions for jointly identifying the graphical structure and additional model parameters without requiring pure nodes. Our results substantially broaden the class of identifiable and interpretable Boolean factor models by removing the pure-node requirement, yielding new theoretical insights, while also providing practitioners with a concrete and easily implementable tool to assess model identifiability.
ER -