A Joint Bayesian Boolean Matrix Factorization with Application to Chromosomal Copy Number Alterations in Multiple Myeloma

Abstract

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.