Learning Bayesian and Markov Networks with an Unreliable Oracle

Learning Bayesian network (BN) structure from sparse discrete data is hard: when each instance records only a few variables, most variable pairs lack the joint observations needed for reliable scoring, and data-only methods recover little structure. Imperfect domain knowledge, expressible as a weighted directed knowledge graph (KG), is often available. KG-SoftMAP encodes such a KG as a finite-strength, confidence-weighted edge prior and maximizes a MAP objective that adds this logit-form prior to the BDeu score. With an informative but imperfect KG, KG-SoftMAP recovers partial directed structure even at observation rate rho=0.05, with directed F1 (DF1) of 0.19-0.32 across benchmarks. At higher observation rates within this sparse grid, DF1 reaches 0.44-0.66 at rho=0.20 and 0.46-0.64 at rho=0.40. Across the same three rates, KG-SoftMAP without the KG prior averages DF1 0.00, 0.19, and 0.21. Stress tests that corrupt, remove, or blur the KG signal, together with checks on LLM-extracted graphs beyond canonical benchmarks, show that recovery rises and falls with KG quality. On three real sparse educational datasets without ground-truth DAGs, we evaluate prediction, calibration, and KG-consistency. On Short Answer Feedback (SAF), KG-SoftMAP+VE reaches Fail-class F1 0.75 versus 0.78 for logistic regression while also providing an inspectable concept graph, calibrated Fail probabilities, and posterior queries from partially observed concept evidence. The remaining datasets sharpen the operating picture: weak heuristic KG signal leaves prediction unchanged, while an independent expert ontology moves the learned graph toward expert relatedness.

The problem of learning the graphical Markov blanket (MB) of a variable from data has applications in many areas such as structure learning for Bayesian networks and Markov random fields, causal discovery, and feature selection. However, a common assumption most methods make is that the conditional independencies in the distribution imply the same separation in the graphical structure -- also known as the faithfulness assumption. Unfortunately, this assumption can be violated by higher-order dependencies such as XOR and parity-type relations, and -- on finite samples -- by empirical violations that, in extreme cases, even induce spurious dependencies absent from the true distribution. Therefore, in this paper we propose a "k-order" relaxation of the faithfulness assumption that captures parity type relationships between k+2 variables. We then propose a proof of concept algorithm called k-order Markov blanket (kOMB) that uses this relaxation for MB discovery. Finally, we empirically show how kOMB can recover the MB of a variable under both true and empirical violations of faithfulness. Code available at: https://github.com/lklee9/k-order-Markov-blanket

Large language models are increasingly used as natural-language interfaces to structured data, yet they remain unreliable when answers require consistent evidence conditioning, dependency-aware reasoning, and uncertainty estimation. Bayesian networks provide an explicit probabilistic reasoning layer, but learning useful structures from data remains costly and fragile at scale. We introduce ABSOL, a hybrid LLM-guided Bayesian network structure-learning framework that uses LLMs as bounded semantic guides. Across five discrete BN benchmarks spanning 27 to 1041 nodes, ABSOL is the only evaluated method to produce a viable graph on every benchmark, and achieves the highest Edge F_1 on every benchmark larger than 27 nodes with GPT-5.4. The four LLM augmentations, which contribute complementary semantic evidence to the statistical backbone, improve Edge F_1 over the non-LLM aggregation backbone by +0.23 on average. Complementary post-hoc refinement experiments suggest that these gains depend in part on limiting the LLM's authority over the final structure. Together, these results show that language-derived semantic knowledge can substantially improve scalable probabilistic structure learning when used as bounded guidance within a statistically grounded reasoning pipeline. The code for ABSOL is available at github.com/megagonlabs/absol-bn.