A Bayesian Boolean Matrix Factorization with Application to Copy Number Analysis in Cancer

Deep learning models for multi-modal breast cancer diagnosis achieve high predictive accuracy but remain clinically unacceptable without actionable, counterfactual explanations. Attribution-based methods (LIME, SHAP) are categorically inapplicable to this purpose, as they generate no alternative instances and thus cannot be evaluated on counterfactual quality metrics. This investigation provides empirical evidence that FCA-Guided Counterfactual (FCA-CF) framework that uses a Formal Concept Analysis (FCA) concept lattice as a hard structural constraint on counterfactual search, operating over a multi-modal TCGA-BRCA dataset. We benchmark against four genuine counterfactual methods: Wachter-style CF, DiCE, FACE, and NICE, evaluated on 60 benign-predicted TCGA-BRCA instances. The FCA-CF framework achieves Validity = 1.0000 (100% of counterfactuals successfully flip the prediction), Sparsity = 2.37 features changed (best among all valid methods), and Proximity = 0.900 (normalised L2-based, matching NICE as joint best). The classifier achieves Accuracy = 0.980, F1 = 0.976, ROC-AUC = 0.9947. Ablation analysis confirms that the FCA lattice constraint is the primary sparsity driver (removing it increases sparsity by +40%, p < 0.001, Cohen's d = 0.78), while Phase C greedy refinement accounts for the largest individual contribution (+113% sparsity increase when disabled, p < 0.001, d = 5.01). FCA-guided counterfactual generation achieves a clinically important Pareto-dominant outcome; it is simultaneously the sparsest and among the most proximate of all valid methods, with perfect validity. The emergent sparsity property arising from lattice topology rather than numerical penalty terms constitutes a structurally novel contribution to the counterfactual explanation literature.

Learning causal graphs from interventional data is a challenging problem with broad applications. In molecular biology, for example, a central goal is to uncover gene regulatory networks from large-scale perturbation data. An ideal algorithm for this task should scale to thousands of nodes, incorporate interventions even when their targets are unknown, quantify uncertainty, and provide identifiability guarantees. However, existing approaches---e.g. approaches using score-based optimization or approximate Bayesian inference---often fail to meet all of these criteria. To address these limitations, we develop Amortized Bayesian Causal Discovery of Extended Factor Graphs (ABCDEFG). Our method guarantees exact acyclicity, scales to graphs with thousands of nodes, and naturally handles interventions even when their targets are unknown. Additionally, ABCDEFG estimates a posterior distribution whose maximum a posteriori estimate provably identifies the true causal graph up to an equivalence class. On simulated datasets, ABCDEFG achieves state-of-the-art accuracy, producing a well-calibrated posterior distribution while outperforming previous score-based and approximate Bayesian methods. Applied to large-scale single-cell perturbation data, ABCDEFG identifies both established and novel gene targets of growth factors.

Pathology foundation models are reported to encode molecular programmes in tissue morphology, but the evidence is usually a cohort-wide ranked gene list rather than a prediction for a held-out patient. We rebuild such an analysis with the patient as the unit of evidence and ask which pipeline component carries signal. Across 11 frozen backbones, four pre-specified gene programmes and 285 TCGA-BRCA patients with paired slides and RNA-seq (44 cells; GroupKFold by patient, all preprocessing fitted inside the fold), ridge regression on mean-pooled embeddings predicts held-out programme scores at Spearman rho = 0.25-0.56, UNI2 strongest on all four (immune 0.556). A matched permutation null gives raw p ~ 1e-4 at 10,000 permutations for every cell; Holm-adjusted p = 0.0044. The signal is real but not uniformly morphological. Against competing models on the same patients and folds, embeddings beat tissue composition for ER/luminal, proliferation and immune (+0.280, +0.284, +0.479; p <= 0.003) but not basal, where compartment fractions alone reach 0.469 against the embedding's 0.493 (p = 0.77). Fifty-four interpretable cell-count features come within 0.043-0.085 on every programme. The geometric machinery contributes nothing measurable, and we identify why: the geodesic graph selects neighbours by Euclidean nearest-neighbour search and only reweights edges already chosen, so the topology is Euclidean by construction (Riemannian minus Euclidean = +0.0010, 95% CI [-0.0007, +0.0029]). Applied consistently the geometry is worse (-0.0117). Ridge regression beats the graph-and-metric decoder by +0.097 (CI [+0.069, +0.127]). The driver-count metric common in this literature is near-uninformative here: 91.8% of random six-gene panels recover >=5/6 drivers.