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We also provide an algorithm for sampling CPDAGs efficiently from their posterior given the learned tree. That is, not from the full posterior, but from a reduced space of CPDAGs encoded in the learned tree.

Learning causal structure from observational data is especially challenging when latent variables or selection bias are present. The Fast Causal Inference (FCI) algorithm addresses this setting but performs exhaustive conditional independence tests across many subsets, often leading to spurious independences, missing or extra edges, and unreliable orientations. We present a family of score-guided mixed-strategy causal search algorithms that extend this framework. First, we introduce BOSS-FCI and GRaSP-FCI, variants of GFCI (Greedy Fast Causal Inference) that substitute BOSS (Best Order Score Search) or GRaSP (Greedy Relaxations of Sparsest Permutation) for FGES (Fast Greedy Equivalence Search), preserving correctness while trading off scalability and conservativeness. Second, we develop FCI Targeted-Testing (FCIT), a novel hybrid method that replaces exhaustive testing with targeted, score-informed tests guided by BOSS. FCIT guarantees well-formed PAGs and achieves higher precision and efficiency across sample sizes. Finally, we propose a lightweight heuristic, LV-Dumb (Latent Variable "Dumb"), which returns the PAG of the BOSS DAG (Directed Acyclic Graph). Though not strictly sound for latent confounding, LV-Dumb often matches FCIT's accuracy while running substantially faster. Simulations and real-data analyses show that BOSS-FCI and GRaSP-FCI provide robust baselines, FCIT yields the best balance of precision and reliability, and LV-Dumb offers a fast, near-equivalent alternative. Together, these methods demonstrate that targeted and score-guided strategies can dramatically improve the efficiency and correctness of latent-variable causal discovery.

Causal discovery methods can identify valid adjustment sets for causal effect estimation for a pair of target variables, even when the underlying causal graph is unknown. Global causal discovery methods focus on learning the whole causal graph and therefore enable the recovery of optimal adjustment sets, i.e., sets with the lowest asymptotic variance, but they quickly become computationally prohibitive as the number of variables grows. Local causal discovery methods offer a more scalable alternative by focusing on the local neighborhood of the target variables, but are restricted to statistically suboptimal adjustment sets. In this work, we propose Local Optimal Adjustments Discovery (LOAD), a sound and complete causal discovery approach that combines the computational efficiency of local methods with the statistical optimality of global methods. First, LOAD identifies the causal relation between the targets and tests if the causal effect is identifiable by using only local information. If it is identifiable, it then finds the optimal adjustment set by leveraging local causal discovery to infer the mediators and their parents. Otherwise, it returns the locally valid parent adjustment sets based on the learned local structure. In our experiments on synthetic and realistic data LOAD outperforms global methods in scalability, while providing more accurate effect estimation than local methods.

To exploit the PDAGs for achieving interventional fairness, previous methods have been built on variable selection or causal effect identification, but limited to reduced prediction accuracy or strong assumptions. In this paper, we propose a general min-max optimization framework that can achieve interventional fairness with promising prediction accuracy and can be extended to maximally oriented PDAGs (MPDAGs) with added background knowledge.

Learning about the directed acyclic graph (DAG) underlying a system's data-generating process from observational data under causal sufficiency is a fundamental causal discovery task (Pearl, 2009). Score-based algorithms address this task by assigning penalized likelihood scores to each DAG and seeking graphs whose scores are optimal. Identifiability theory asks when such score-optimal graphs identify the target graph (or its equivalence class) in the infinite-sample limit, with various results under different assumptions and scores (Chickering, 2002; Nandy et al., 2018). Exact algorithms, that are guaranteed to find a score-optimal graph, have exponential run-time and are feasible up to roughly 30 variables (Koivisto & Sood, 2004; Silander & Myllym aki, 2006). For larger graphs, local search must be employed, which evaluates neighbouring graphs to find graphs with better scores; canonical moves for this hill climbing are single edge insertions, deletions, or reversals (Heckerman et al., 1995). In the sample limit, greedy discrete search with a neighbourhood notion that respects score equivalence provably finds a graph with optimal score (Chickering, 2002). In finite samples, scores are inexact and local search may get stuck in local optima or, as we demonstrate, even find graphs with better scores than the true graph. Finite-sample performance is a practical challenge, despite the mature identifiability theory and asymptotic guarantees. Continuous optimization methods have emerged as a popular alternative.

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We introduce CIfly, a framework for efficient algorithmic primitives in graphical causal inference that isolates reachability as a reusable core operation. It builds on the insight that many causal reasoning tasks can be reduced to reachability in purpose-built state-space graphs that can be constructed on the fly during traversal. We formalize a rule table schema for specifying such algorithms and prove they run in linear time. We establish CIfly as a more efficient alternative to the common primitives moralization and latent projection, which we show are computationally equivalent to Boolean matrix multiplication. Our open-source Rust implementation parses rule table text files and runs the specified CIfly algorithms providing high-performance execution accessible from Python and R. We demonstrate CIfly's utility by re-implementing a range of established causal inference tasks within the framework and by developing new algorithms for instrumental variables.

Recent advances have established the identifiability of a directed acyclic graph (DAG) under additive noise models (ANMs), spurring the development of various causal discovery methods. However, most existing methods make restrictive model assumptions, rely heavily on general independence tests, or require substantial computational time. To address these limitations, we propose a sequential procedure to orient undirected edges in a completed partial DAG (CPDAG), representing an equivalence class of DAGs, by leveraging the pairwise additive noise model (PANM) to identify their causal directions. We prove that this procedure can recover the true causal DAG assuming a restricted ANM. Building on this result, we develop a novel constraint-based algorithm for learning causal DAGs under nonlinear ANMs. Given an estimated CPDAG, we develop a ranking procedure that sorts undirected edges by their adherence to the PANM, which defines an evaluation order of the edges. To determine the edge direction, we devise a statistical test that compares the log-likelihood values, evaluated with respect to the competing directions, of a sub-graph comprising just the candidate nodes and their identified parents in the partial DAG. We further establish the structural learning consistency of our algorithm in the large-sample limit. Extensive experiments on synthetic and real-world datasets demonstrate that our method is computationally efficient, robust to model misspecification, and consistently outperforms many existing nonlinear DAG learning methods.