Constraint-Based Causal Discovery

Variables can predict an outcome jointly while each is individually independent of it, causing structure learners to discard informative links. We investigate how to operationalise weaker faithfulness through two approaches: selective protection in an adjacency learner and explicit retention of alternative directed graph explanations. Matched experiments show that changes at both an early candidate-removal stage and later pruning can be necessary. In noisy XOR with a descendant (N = 10,000, 20 matched samples), combining reactivation and protection raises complete parent-pair recovery from 0/20 for either change alone to 20/20, while adding false edges. A confirmation study shows that the family estimator can preserve explanations lost by orienting the protected learner's selected skeleton: in one weaker signal XOR condition (N = 10,000) it retains the entire reference family in 48/50 samples, compared with 0/50 for the skeleton adapter. These gains are conditional. Protection can add only error, weaker effects produce false exclusions, and an ordinary faithful-chain comparison favours the PC learner. Complete nonempty family outputs decline at six/eight variables, and all 400 twelve/twenty-variable searches remain unfinished at the reporting cutoff. Neural controls show engineered retention, while the family estimator returns no accepted graph within its application budget. This operationalisation identifies two deletion stages that must be coordinated in the motivating example and shows when direct family search preserves explanations lost by selecting one skeleton. Finite evidence and computation remain substantial barriers to reliable general inference.

Conditional Independence (CI) tests are the statistical engine of constraint-based causal discovery: in algorithms such as PC (Peter-Clark) and FCI (Fast Causal Inference), skeleton pruning and key orientations follow directly from CI decisions. This survey reviews CI testing with emphasis on assumptions, robustness, and scalability in high-dimensional and mixed-type settings common in biomedical domains. The survey organizes widely used CI methods into six families: partial-correlation, contingency-table, regression, nearest-neighbor, kernel, and machine-learning-based. Special emphasis is provided on the robustness layers that address the limitations of these families. For each family, the survey examines when CI decisions reflect the data-generating distribution and when they fail. By this, we link test-level properties, including power decay with conditioning set size and asymmetric type I/II error consequences, to graph-level errors in skeleton recovery and v-structure orientation. The survey also compares adoption across major R and Python libraries and summarizes open challenges, including mixed-type CI testing without discretization, small-sample error control, and strategies for improving scalability of CI-testing.

We propose CEDAR (Causal Edge Discovery for Autoregressive Processes), a constraint-based method for lagged causal edge discovery in sparse autoregressive time series. CEDAR screens candidate cross-variable lags using AR(1)-residualized, U-centered distance correlation, then applies two targeted conditional-independence tests per significant cross-variable lag candidate and accepts at most one lag per ordered pair. A stable MCI pruning step removes indirect edges, and optional deterministic C-nodes adjust for specified trend-like nonstationarity. In sparse regimes where few lags survive screening, CEDAR requires $O(d^2)$ CI tests after screening while retaining edge-level interpretability. CEDAR is most effective when data are scarce and variables exhibit lag-1 self-dynamics; methods with richer conditioning sets become preferable as $T$ grows or when higher-order autoregressive or simultaneous multi-lag effects are common.

Causal discovery methods have shown strong performance in temporal systems, but they typically rely on regular and discrete lag structures, limiting their applicability to regularly sampled data. However, many real-world tasks require dealing with irregularly sampled streams of events, such as sensor streams, healthcare data, and financial transactions. In this work, we propose an extension of PCMCI+, a state-of-the-art method for causal discovery on regular multivariate time series, to allow for handling irregular time series. Instead of modelling causal relations through fixed-lag dependencies, our method aggregates causal influence over predefined temporal windows. We evaluate our method on synthetic irregular event streams with known causal structures under different signal-to-noise ratios, showing that it consistently recovers the underlying causal graph and substantially outperforms the standard PCMCI+ on irregularly sampled data.

Biological systems exhibit a hierarchical structure, characterised by directed flow from upstream regulators to downstream effects. Although this ordering provides a natural scaffold for causal inference, most causal discovery and GRN methods either ignore the tiered organisation or condition on all upstream variables, which becomes infeasible for high-dimensional omics data. We present ASCEND (Ancestral Scalable Causal discovEry via iNherited Descent), a constraint-based framework that leverages known two-tiered structure to enable genome-scale causal discovery. ASCEND introduces a divide-and-conquer strategy that maintains dynamically updated ancestral conditioning sets for each downstream variable, dramatically reducing the number of conditional independence tests required, and achieves polynomial-time complexity where traditional approaches face exponential blow-up. Through extensive simulations and real biological data, we demonstrate that ASCEND accurately recovers ancestral relationships, scales properly and much faster, and outperforms existing gene regulatory network inference methods in both causal precision and computational efficiency. The algorithm's ability to resolve directionality makes it particularly suited for integrating multi-omic data where upstream regulators (e.g., SNPs, methylation sites) and downstream responses (e.g., gene expression) are measured jointly.

From climate teleconnections to gene regulation, modern time-series datasets encompass tens or hundreds of interacting variables, making causal discovery increasingly challenging. Constraint-based methods offer statistical rigor but their nonlinear CI tests are infeasible at scale, while score-based alternatives avoid CI testing but require arbitrary thresholds to binarize continuous edge scores. We propose GRACE ($\textbf{G}$ated $\textbf{R}$efinement for $\textbf{A}$ccurate $\textbf{C}$ausal $\textbf{E}$dge discovery), which refines constraint-based discovery using Hard Concrete gates with $L_0$ regularization: each candidate edge has an independent gate whose values concentrate near 0 or 1, yielding a clean bimodal separation that makes the binary decision robust, unlike the narrow, overlapping score distributions produced by $L_1$ and attention-based methods. A fast linear CI skeleton provides high-recall candidates; a single gated model then prunes false positives by learning which edges genuinely improve prediction, with automatic regularization adapted to problem dimensions and skeleton density. Systematic experiments on synthetic benchmarks, spanning diverse graph topologies (scale-free, Erdős-R'enyi, small-world) and dimensionalities up to $d=100$, show that GRACE substantially improves F1 over its base CI method while maintaining high precision, and outperforms attention-based and score-based alternatives. GRACE matches or exceeds expensive nonlinear CI tests at a fraction of the cost ($75\times$ faster). On a real-world river flow dataset, where rainfall confounders, variable propagation lags, and distributional shifts violate standard assumptions, a temporal bootstrap variant of GRACE recovers 9 of 11 causal edges along the Elbe River with only 1 false positive ($F_1 = 0.86$, AUROC${} = 0.99$), reducing the skeleton's 106 false positives by 99%.

Constraint-based causal discovery relies on repeated conditional independence tests, but fast nonparametric tests often sacrifice calibration, especially when variables depend on the conditioning set through nonlinear relationships. We introduce BLITZ (Broad-to-Local Independence Testing via residualiZation), a nonparametric conditional independence test designed to run well under a second while maintaining the accuracy needed for the thousands of queries performed by constraint-based causal discovery algorithms. BLITZ first removes broad smooth dependence on the conditioning set using low-order polynomial regression, then applies a small nonlinear feature map and residualizes those features with shallow tree regressions. The resulting statistic tests residual cross-covariance, with a moment-matched chi-square approximation to the null distribution. We show theoretically that the two-stage design reduces the effective complexity faced by the tree residualizers, allowing shallow trees to control residual conditional-mean bias while avoiding excessive overfitting. In simulations, BLITZ provides better null calibration than fast kernel, random-feature, and regression-based competitors while remaining among the fastest methods tested. In causal discovery experiments on synthetic graphs and flow-cytometry data, BLITZ yields more reliable endpoint orientations among retained adjacencies and competitive structural recovery. These results suggest that broad-to-local residualization is a practical route to calibrated, scalable nonparametric conditional independence testing for causal discovery.

Constraint-based causal discovery is widely used for learning causal structures, but heavy reliance on conditional independence (CI) testing makes it computationally expensive in high-dimensional settings. To mitigate this limitation, many divide-and-conquer frameworks have been proposed, but most assume causal sufficiency, i.e., no latent variables. In this paper, we show that divide-and-conquer strategies can be theoretically generalized beyond causal sufficiency to settings with latent variables. Specifically, we propose a recursive decomposition framework, termed DiCoLa, that enables divide-and-conquer causal discovery in the presence of latent variables. It recursively decomposes the global learning task into smaller subproblems and integrates their solutions through a principled reconstruction step to recover the global structure. We theoretically establish the soundness and completeness of the proposed framework. Extensive experiments on synthetic data demonstrate that our approach significantly improves computational efficiency across a range of causal discovery algorithms, while experiments on a real-world dataset further illustrate its practical effectiveness.

Constraint-based causal discovery is brittle in finite-sample regimes because erroneous conditional-independence (CI) decisions can cascade into substantial structural errors. We propose Quantitative Argumentation for Causal Discovery (QACD), a semantics-driven framework that represents CI outcomes as graded, defeasible arguments rather than irreversible constraints. QACD maps statistical test outcomes to argument strengths and aggregates conflicting evidence through connectivity-mediated witness propagation, producing a fixed-point acceptability labeling over candidate adjacencies. Experiments on standard benchmark Bayesian networks suggest that QACD improves structural coherence and interventional reliability in several noisy or inconsistent CI regimes, while remaining competitive with classical constraint-based, hybrid, and prior argumentation-based baselines.

Learning causal relations from observational data is a fundamental problem with wide-ranging applications across many fields. Constraint-based methods infer the underlying causal structure by performing conditional independence tests. However, existing algorithms such as the prominent PC algorithm need to perform a large number of independence tests, which in the worst case is exponential in the maximum degree of the causal graph. Despite extensive research, it remains unclear if there exist algorithms with better complexity without additional assumptions. Here, we establish an algorithm that achieves a better complexity of $p^{\mathcal{O}(s)}$ tests, where $p$ is the number of nodes in the graph and $s$ denotes the maximum undirected clique size of the underlying essential graph. Complementing this result, we prove that any constraint-based algorithm must perform at least $2^{Ω(s)}$ conditional independence tests, establishing that our proposed algorithm achieves exponent-optimality up to a logarithmic factor in terms of the number of conditional independence tests needed. Finally, we validate our theoretical findings through simulations, on semi-synthetic gene-expression data, and real-world data, demonstrating the efficiency of our algorithm compared to existing methods in terms of number of conditional independence tests needed.

Causal discovery seeks to uncover causal relations from data, typically represented as causal graphs, and is essential for predicting the effects of interventions. While expert knowledge is required to construct principled causal graphs, many statistical methods have been proposed to leverage observational data with varying formal guarantees. Causal Assumption-based Argumentation (ABA) is a framework that uses symbolic reasoning to ensure correspondence between input constraints and output graphs, while offering a principled way to combine data and expertise. We explore the use of large language models (LLMs) as imperfect experts for Causal ABA, eliciting semantic structural priors from variable names and descriptions and integrating them with conditional-independence evidence. Experiments on standard benchmarks and semantically grounded synthetic graphs demonstrate state-of-the-art performance, and we additionally introduce an evaluation protocol to mitigate memorisation bias when assessing LLMs for causal discovery.