Structural-budgeted QUBO learning of Bayesian networks with spectral and credibility diagnostics

Traditional Bayesian network structure learning requires explicit trade-offs among structural complexity, statistical fit, and decision reliability. This study proposes BAPS (Budgeted Acyclicity with Phase-transition Spectral diagnostics), a framework that reformulates structure learning as a budget-controlled optimization problem in which a global edge budget limits the number of selected candidate edges and thereby regulates model capacity. A relaxed QUBO formulation with post-hoc DAG projection explores likelihood-improving structures while recovering feasible directed acyclic graphs; in benchmark experiments, the repair process reduces SHD by 20.6%, 14.9%, and 8.3% on Asia, Insurance, and Barley, indicating improved structural agreement while enforcing feasibility. Spectral diagnostics based on algebraic connectivity serve as descriptive structural indicators for flagging budget regions where repair burden may escalate. A dual-layer credibility framework quantifies uncertainty from parameter and observational sources; credibility interval width contracts by 69-73% under increasing Dirichlet concentration and remains empirically stable in sparse-data conditions where resampling-based methods become unstable due to zero-frequency effects. Across benchmark networks, BAPS achieves the highest BIC gain while maintaining broadly comparable held-out predictive performance relative to established baselines. Overall, BAPS provides a unified framework integrating edge-budget capacity control, structural diagnostics, feasibility repair, and credibility assessment for Bayesian network learning in complex diagnostic environments.