AI 中文总结
研究针对现有布尔矩阵分解未利用相关数据集共享潜在结构的问题，提出JBBMF模型，通过共享潜在结构联合分解二元矩阵，在模拟和多发性骨髓瘤数据应用中提升了因子恢复与分析效果。
AI 中文摘要
布尔矩阵分解为高维数据中潜在二元模式的发现提供了可解释框架，但现有方法通常仅分析单个二元矩阵，或独立分解多个矩阵，未能利用相关数据集间的共享潜在结构。我们提出联合贝叶斯布尔矩阵分解（Joint Bayesian Boolean Matrix Factorization，JBBMF）模型，该模型通过一个共享的潜在布尔模式矩阵和数据集特定的载荷矩阵，同时分解两个相关的二元矩阵。为捕捉配对数据集间的依赖关系，我们引入了连接载荷矩阵的条件先验，使潜在因子在不同条件下可保持或改变，同时保留共同的可解释表示。该模型结合了布尔矩阵分解、伯努利观测模型及共轭先验，产生闭式全条件分布和用于后验推断的高效吉布斯采样器，可实现对潜在因子、重构矩阵及噪声参数的不确定性量化。模拟研究表明，与对每个数据集独立应用标准布尔矩阵分解相比，联合建模相关二元数据集可显著提升共享潜在因子的恢复效果，同时保持高重构精度。我们将JBBMF应用于从多发性骨髓瘤患者诊断期和复发期采集的配对染色体拷贝数变异谱，该分析识别出疾病分期间共享的复发性染色体变异特征，并量化了这些发现的不确定性。JBBMF为基因组学及其他应用领域中相关二元数据集的联合分析提供了灵活且可解释的贝叶斯模型。
英文摘要
Boolean matrix factorization provides an interpretable framework for discovering latent binary patterns in high-dimensional data, yet existing methods typically analyze a single binary matrix or factorize multiple matrices independently, failing to exploit shared latent structure across related datasets. We propose Joint Bayesian Boolean Matrix Factorization (JBBMF), a model that simultaneously factorizes two related binary matrices through a shared latent Boolean pattern matrix and dataset-specific loading matrices. To capture dependence between paired datasets, we introduce a conditional prior linking the loading matrices, allowing latent factors to persist or change across conditions while preserving a common interpretable representation. The model combines Boolean matrix factorization with a Bernoulli observation model and conjugate priors, yielding closed-form full conditional distributions and an efficient Gibbs sampler for posterior inference,uncertainty quantification for latent factors, reconstructed matrices, and noise parameters. Simulation studies demonstrate that jointly modeling related binary datasets substantially improves recovery of shared latent factors compared with independently applying standard Boolean matrix factorization to each dataset, while maintaining high reconstruction accuracy. We apply JBBMF to paired chromosomal copy number alteration profiles from multiple myeloma patients collected at diagnosis and relapse. The analysis identifies recurrent chromosomal alteration signatures shared between disease stages and quantifies the uncertainty of these findings. \texttt{JBBMF} offers a flexible and interpretable Bayesian model for the joint analysis of related binary datasets in genomics and other application domains.
