These results are then generalized to general models and likelihoods, where the same claims hold.

Understanding the structural relationships among different variables provides critical insights in manyreal-worldapplications, suchasmedicine,economics andeducation [42,62]. Thus,learning graphs from observed data, known as structure learning, has recently made remarkable progress [10,61,63,64]. Formanyapplications, variables inthedata can begathered into semantically meaningful groups, where useful insights are at group level. For example, in finance, one may be interested in how a financial situation influences different industries (i.e.

This paper re-examines a continuous optimization framework dubbed NOTEARS for learning Bayesian networks. We first generalize existing algebraic characterizations of acyclicity to a class of matrix polynomials. Next, focusing on a one-parameter-per-edge setting, it is shown that the Karush-Kuhn-Tucker (KKT) optimality conditions for the NOTEARS formulation cannot be satisfied except in a trivial case, which explains a behavior of the associated algorithm. We then derive the KKT conditions for an equivalent reformulation, show that they are indeed necessary, and relate them to explicit constraints that certain edges be absent from the graph. If the score function is convex, these KKT conditions are also sufficient for local minimality despite the non-convexity of the constraint. Informed by the KKT conditions, a local search post-processing algorithm is proposed and shown to substantially and universally improve the structural Hamming distance of all tested algorithms, typically by a factor of 2 or more. Some combinations with local search are both more accurate and more efficient than the original NOTEARS.

Causal discovery from observational data is a fundamental task in artificial intelligence, with far-reaching implications for decision-making, predictions, and interventions. Despite significant advances, existing methods can be broadly categorized as constraint-based or score-based approaches. Constraint-based methods offer rigorous causal discovery but are often hindered by small sample sizes, while score-based methods provide flexible optimization but typically forgo explicit conditional independence testing. This work explores a third avenue: developing differentiable $d$-separation scores, obtained through a percolation theory using soft logic. This enables the implementation of a new type of causal discovery method: gradient-based optimization of conditional independence constraints. Empirical evaluations demonstrate the robust performance of our approach in low-sample regimes, surpassing traditional constraint-based and score-based baselines on a real-world dataset. Code and data of the proposed method are publicly available at https://github$.$com/PurdueMINDS/DAGPA.

Portfolio managers rely on correlation-based analysis and heuristic methods that fail to capture true causal relationships driving performance. We present a hybrid framework that integrates statistical causal discovery algorithms with domain knowledge from two complementary sources: a financial knowledge graph extracted from SEC 10-K filings and large language model reasoning. Our approach systematically enhances three representative causal discovery paradigms, constraint-based (PC), score-based (GES), and continuous optimization (NOTEARS), by encoding knowledge graph constraints algorithmically and leveraging LLM conceptual reasoning for hypothesis generation. Evaluated on a synthetic financial dataset of 500 firms across 18 variables, our KG+LLM-enhanced methods demonstrate consistent improvements across all three algorithms: PC (F1: 0.622 vs. 0.459 baseline, +36%), GES (F1: 0.735 vs. 0.367, +100%), and NOTEARS (F1: 0.759 vs. 0.163, +366%). The framework enables reliable scenario analysis with mean absolute error of 0.003610 for counterfactual predictions and perfect directional accuracy for intervention effects. It also addresses critical limitations of existing methods by grounding statistical discoveries in financial domain expertise while maintaining empirical validation, providing portfolio managers with the causal foundation necessary for proactive risk management and strategic decision-making in dynamic market environments.