发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于哈斯图的方法，将布尔因子模型的图形可辨识性转化为图同构问题，建立无需纯节点的充分必要条件，并开发SAT验证算法，显著拓宽可辨识模型类别。
AI 中文摘要
布尔因子模型，包括布尔矩阵分解和认知诊断模型等显著子族，在从社会科学到工程学的广泛领域中得到应用。尽管其灵活性，一个关键挑战在于确立其图形结构的可辨识性，该结构指定了潜在变量如何影响观测变量。现有的可辨识性条件通常依赖于纯节点的强假设，这在许多应用中可能不切实际。我们开发了一种新方法，利用哈斯图来表示观测变量的分布，并将可辨识性转化为图同构挑战。基于此，我们建立了不要求纯节点的充分且必要的图形可辨识性条件。我们进一步推导出等价的代数条件，并开发了一种高效的基于布尔可满足性（SAT）的验证算法。我们将分析扩展到概率布尔因子模型，建立了在无需纯节点的情况下联合识别图形结构和额外模型参数的条件。我们的结果通过移除纯节点要求，显著拓宽了可辨识且可解释的布尔因子模型的类别，产生了新的理论见解，同时也为实践者提供了具体且易于实现的工具来评估模型可辨识性。
英文摘要
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.
