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

Technical Summary: Joint Bayesian Boolean Matrix Factorization (JBBMF)

1. Problem Statement

Boolean Matrix Factorization (BooMF) offers an interpretable framework for discovering latent binary patterns in high-dimensional discrete data, such as genomic copy number alterations (CNAs), user-item interactions, and network relations. Unlike real-valued methods like PCA or NMF, BooMF preserves the binary nature of the data, yielding factors with direct semantic meaning. However, existing BooMF methods typically analyze a single matrix or factorize multiple matrices independently. This approach fails to exploit shared latent structures across related datasets (e.g., paired samples collected at different time points, such as diagnosis and relapse in cancer). Analyzing such data independently risks learning incomparable latent factors, discarding weak but consistent patterns, and ignoring the shared biological or structural mechanisms connecting the datasets. While previous attempts at joint factorization exist (e.g., JSBMF), they rely on deterministic, greedy optimization, lack uncertainty quantification, and do not explicitly model noise, limiting their robustness in high-noise or sparse regimes.

2. Methodology

The authors propose Joint Bayesian Boolean Matrix Factorization (JBBMF), a fully probabilistic generative model designed to simultaneously factorize two related binary matrices, $\mathbf{X}^{(1)}$ and $\mathbf{X}^{(2)}$.

• Generative Model: The model assumes both datasets share a common latent Boolean pattern matrix $\mathbf{H} \in \{0, 1\}^{R \times G}$, where $R$ is the rank and $G$ is the number of features. Each dataset $i \in \{1, 2\}$ has a dataset-specific loading matrix $\mathbf{W}^{(i)} \in \{0, 1\}^{K \times R}$, where $K$ is the number of observations. The observed matrix is approximated by the Boolean product $\mathbf{Z}^{(i)} = \mathbf{W}^{(i)} \circ \mathbf{H}$, where $\circ$ denotes Boolean matrix multiplication (logical AND/OR).
• Noise Model: To handle real-world imperfections, the model employs a Bernoulli observation model. Given the latent signal $Z^{(i)}_{kg}$, the observation $X^{(i)}_{kg}$ follows a Bernoulli distribution with success probability $p_{i,11}$ if $Z^{(i)}_{kg}=1$ and $p_{i,10}$ if $Z^{(i)}_{kg}=0$. The constraint $p_{i,11} > p_{i,10}$ ensures the signal state is distinguishable from noise.
• Prior Specifications:
  • Shared Pattern Matrix ($\mathbf{H}$): A hierarchical mixture prior is used to induce sparsity. Each entry $H_{rg}$ is governed by a Beta hyper-prior, with a binary indicator $\psi_g$ determining whether feature $g$ is broadly relevant (slab) or shrunk towards exclusion (spike).
  • Loading Matrices ($\mathbf{W}^{(1)}, \mathbf{W}^{(2)}$): To capture dependence between paired datasets, a conditional prior $p(\mathbf{W}^{(2)} | \mathbf{W}^{(1)})$ is introduced. This models the transition of factor activity from condition 1 to condition 2 using parameters $\gamma_{11}$ (persistence of active factors) and $\gamma_{00}$ (probability of remaining inactive).
  • Conjugacy: All priors are conjugate (Beta for probabilities, Bernoulli for binary entries), yielding closed-form full conditional distributions.
• Inference: Posterior inference is performed via an efficient Gibbs sampler. The algorithm iteratively samples all unknown quantities (loadings, patterns, noise parameters, and hyperparameters) from their full conditional distributions. The method includes a label-switching correction using the Hungarian algorithm and Jaccard similarity to align sampled factors with a reference.
• Model Selection: The rank $R$ is selected by running multiple Gibbs chains and choosing the value that maximizes the log-posterior, treating the selected rank as a working choice rather than a definitive inference.

3. Key Contributions

• Joint Probabilistic Framework: JBBMF is the first method to combine joint Boolean matrix factorization with a fully Bayesian treatment, allowing for the simultaneous estimation of shared patterns and dataset-specific usages.
• Uncertainty Quantification: Unlike deterministic approaches, JBBMF provides posterior distributions for latent factors, reconstructed matrices, and noise parameters, enabling analysts to distinguish stable findings from noise-driven artifacts.
• Conditional Dependence Modeling: The introduction of a conditional prior between loading matrices allows the model to explicitly capture how latent factors persist or change across conditions (e.g., disease stages), rather than treating datasets as independent.
• Computational Tractability: The use of conjugate priors ensures that the Gibbs sampler scales to moderately large binary datasets, with closed-form updates for all parameters.

4. Results

The paper evaluates JBBMF through three simulation experiments and an application to real multiple myeloma CNA data.

• Simulation Studies:
  • Experiment I (Progression): Simulates a scenario where shared factors exist alongside condition-specific factors. JBBMF recovered shared latent factors more accurately than applying the Asso algorithm (a state-of-the-art deterministic BooMF method) independently to each matrix. While Asso performed slightly better on the simple diagnosis matrix, JBBMF significantly outperformed Asso on the complex relapse matrix.
  • Experiment II (Conditional Dependence): Designed to encode conditional dependence between matrices. JBBMF substantially outperformed Asso on both matrices across all metrics (F1, MCC, reconstruction error), demonstrating the value of exploiting shared structure.
  • Experiment III (Synthetic CNA): Generated data from posterior samples to mimic realistic CNA structures. JBBMF achieved significantly lower reconstruction error rates (approx. 1/5th of Asso) and provided accurate posterior inclusion probabilities with low uncertainty for most features.
• Real Data Application (Multiple Myeloma):
  • Applied to paired CNA profiles ($K=62$ patients, $G=44$ chromosomal arms) at diagnosis and relapse.
  • The model identified four shared latent signatures (recurring sets of co-altered chromosomal arms) and quantified how their usage changed between stages.
  • Biological Insights: The analysis revealed stable genomic configurations common to both stages and stage-specific changes in activation. It identified a set of chromosomal arms consistently altered across the cohort and highlighted coherent patient subgroups.
  • Uncertainty: Posterior inclusion probabilities showed high confidence (near 0 or 1) for most arm-factor assignments, validating the stability of the inferred signatures.

5. Significance and Claims

The paper claims that JBBMF offers a flexible and interpretable Bayesian model for the joint analysis of related binary datasets. Its primary significance lies in:

1. Improved Recovery: Jointly modeling related datasets substantially improves the recovery of shared latent factors compared to independent analysis, particularly when datasets exhibit conditional dependence or when patterns are weak but consistent.
2. Robustness: The explicit noise model and Bayesian framework make the method robust to measurement error and biological variability, providing a principled basis for model comparison and uncertainty quantification.
3. Interpretability: By maintaining binary constraints throughout the factorization, the model yields factors that are directly interpretable as combinatorial patterns of features, avoiding the ambiguity of continuous loadings found in PCA or NMF.

The authors acknowledge limitations, noting that the rank $R$ is fixed and selected via a heuristic rather than inferred nonparametrically, and the current implementation focuses on two matrices (though the framework extends to $M$ matrices). They suggest future work could involve nonparametric extensions (e.g., Indian Buffet Process) and broader implementation for $M > 2$ matrices.