Causal Graph Learning in Observational Data

I. Introduction to Causal Graph Learning in Observational Data

Causal graph learning represents a pivotal subfield within machine learning, shifting the focus from mere predictive modeling to the identification of genuine cause-and-effect relationships between variables. This distinction is paramount for developing artificial intelligence systems that transcend simple prediction, enabling them to comprehend the underlying mechanisms of a system and make more informed, robust decisions.1 Unlike traditional statistical methods that often highlight correlations, causal learning aims to uncover the directional influences that drive observed phenomena.

Central to this endeavor are causal graph models, frequently depicted as Directed Acyclic Graphs (DAGs). These graphical representations utilize nodes to symbolize variables and directed edges to denote assumed causal influences. Such models are indispensable tools for explicitly mapping out hypothesized causal structures, which is crucial for predicting the outcomes of interventions and fostering a deeper understanding of complex systems.1 For instance, in health research, the primary objective often involves identifying and quantifying risk factors that exert a causal effect on health and social outcomes.2 This foundational understanding underscores that causal inference seeks to move beyond statistical associations to actionable insights, a prerequisite for effective intervention and scientific advancement.

 

The Fundamental Problem of Causal Inference

The core conceptual hurdle in causal inference is widely recognized as the “fundamental problem of causal inference”: the inherent impossibility of directly observing counterfactuals. For any given individual, only one of two potential outcomes can ever be observed—either what transpired under a specific exposure (e.g., a medical treatment) or what would have occurred under a different exposure (e.g., no treatment)—but never both simultaneously.3 This inherent limitation means that the individual causal effect, often denoted as

τi​, cannot be empirically measured directly.3

Consequently, causal inference in practice relies on estimating average causal effects across groups of individuals. This involves comparing the risk or outcome in an exposed group to that in an unexposed group.2 This task, however, remains distinct from comparing an individual to their own unobserved counterfactual state, necessitating meticulous methodological considerations. The pervasive nature of this impossibility transforms causal inference from a straightforward data-driven task into a rigorous exercise in assumption validation and sensitivity analysis. The entire framework of causal inference from observational data is constructed upon the imputation or approximation of these unobserved states, which inherently necessitates the introduction of strong, often untestable, assumptions.6 This emphasizes that causal inference is less about directly discovering an inherent truth from data and more about constructing a plausible causal narrative under specific, often stringent, assumptions, thereby shifting the focus from simple data analysis to rigorous assumption validation and sensitivity analysis.

Why Observational Data Poses Unique Challenges

Observational data, by its very nature, is collected without deliberate interventions or randomized assignments, rendering it intrinsically susceptible to various biases that can obscure or distort true causal relationships.2 These biases include confounding, selection bias, and measurement bias.2 Unlike randomized controlled trials (RCTs), which are considered the “gold standard” for causal inference due to their ability to balance confounders across groups through random assignment, observational studies inherently lack this balance.2 The absence of randomization implies that observed associations may stem from common causes (confounders) rather than direct causal links, making the inference of causation significantly more complex and requiring advanced techniques to mitigate bias.7

The impracticality, expense, or ethical constraints associated with conducting RCTs in many real-world scenarios often make observational data the only available option.2 This forces researchers to utilize observational data despite its inherent susceptibility to biases. This situation highlights an inherent, often unavoidable, trade-off between the “purity” of causal inference, which RCTs achieve through design-based bias minimization, and the “practicality” of data collection, where observational data is readily available but prone to biases. The decision to use observational data is thus frequently a pragmatic necessity rather than a choice of convenience. This necessity, in turn, drives the imperative for developing and applying sophisticated bias mitigation techniques and demanding careful assumption validation.2 The most effective methods in practice are often those that can robustly manage the specific biases present in the available observational data, even if it means accepting certain compromises on theoretical ideals.

The table below delineates the fundamental differences between correlation and causation, underscoring why causal graph learning is essential for moving beyond mere statistical associations to understanding underlying mechanisms and enabling effective interventions.

Feature	Correlation	Causation
Definition	Statistical association between variables where changes in one are related to changes in another.	A relationship where a change in one variable directly leads to a change in another, implying a cause-and-effect link.
Data Source	Can be observed in any type of data (observational, experimental).	Requires experimental/interventional data or observational data analyzed under strong, explicit assumptions.
Key Challenge	Spurious associations due to confounding or other biases.	The fundamental problem of causal inference (unobservable counterfactuals) and the presence of various biases.
Primary Goal	Prediction, description, pattern recognition.	Understanding underlying mechanisms, predicting outcomes of interventions, informing policy.
Example	Falling barometer and storm.9	Medicine and patient survival.10

 

II. Challenges and Biases in Causal Inference from Observational Data

 

Drawing valid causal inferences from observational data is fraught with challenges due to inherent biases. A comprehensive understanding of these biases and the stringent assumptions required for their mitigation is critical for robust causal analysis.

 

Confounding Bias

 

Confounding bias arises when an uncontrolled common cause, termed a “confounder,” influences both the exposure (treatment variable) and the outcome variable. This simultaneous influence creates a spurious, non-causal association between the exposure and outcome, leading to misleading conclusions.2 In the context of a Directed Acyclic Graph (DAG), confounding is visually represented by an “open back-door path” between the exposure (A) and the outcome (Y) that remains unblocked by conditioning on other variables.5

Confounders can manifest in various forms. “Observed confounders” are those for which measurements are available in the study data, allowing for statistical adjustment.2 However, a significant challenge arises from “unmeasured” or “unobserved confounders,” which lead to “residual confounding” even after adjusting for all known variables.2 This situation directly violates the crucial ignorability assumption, making unbiased causal estimation exceedingly difficult.3 Furthermore, confounders can be “time-varying,” changing over time for an individual, or “time-invariant,” remaining static (e.g., age).2 Handling time-varying confounders that are themselves affected by prior exposure often necessitates specialized methods, such as marginal structural models.2 While traditional statistical adjustments, including regression models and propensity score methods, aim to control for confounding, they frequently rely on the strong and often unrealistic assumption that all potential confounders are accurately measured and included.2 Fixed-effects regression models offer a partial solution by accounting for unobserved time-invariant confounders.2 The pervasive nature of confounding bias directly undermines the internal validity of causal claims, making it impossible to distinguish genuine cause-effect relationships from mere statistical associations.

 

Selection Bias and Collider Bias

 

Selection bias, within the framework of causal inference, is specifically understood as a type of “collider bias”.2 A “collider” is a variable that serves as a common effect of two or more other variables, graphically represented by two arrows “colliding” into it on a DAG.2 Critically, unlike confounders, which introduce bias if

not conditioned on, colliders can introduce bias if they are conditioned on.5 This conditioning, often occurring through the selection of a study sample based on the collider’s value, can inadvertently “open up” a spurious back-door path between its causes, thereby creating a non-causal association.

This bias can manifest in various ways, including differential loss to follow-up, non-response bias, or the inappropriate selection of participants.2 A classic example is “Berkson’s bias,” where restricting a sample to hospitalized patients can create spurious negative associations between otherwise unrelated risk factors.5 Collider bias often proves counter-intuitive for researchers accustomed to simply “controlling” for all available variables. This highlights that indiscriminately including covariates in a model can

introduce bias rather than remove it, potentially leading to what are termed “garbage-can regressions”.7 This underscores the non-mechanistic nature of causal inference and the imperative of utilizing causal graphs and domain knowledge to guide judicious covariate selection.3

 

Measurement Bias

 

Measurement bias, also known as measurement error, occurs when the observed values of a variable systematically deviate from its true, unobserved value.2 This can stem from imprecise data collection methods or misreporting.2 Both differential (where error is related to another variable) and non-differential measurement errors can lead to biased causal estimates.2 For instance, measurement error in a mediating variable can result in an underestimation of the indirect effect and an overestimation of the direct effect.2 Furthermore, differential measurement error can inadvertently “open” backdoor pathways between exposure and outcome, effectively inducing confounding.2 The potential outcomes framework conceptualizes measurement error as a form of missing data, where the true values remain unobserved.6 One approach to mitigate measurement bias involves employing “latent variables,” which are statistical constructs estimated from the covariation among strongly related observed variables.2 If these observed variables are assessed using multiple methods with different sources of bias, the latent variable approach can help in removing variability attributable to shared biases.2 Measurement bias is a universal challenge in empirical research; even with a perfectly understood causal structure and identified confounders, inaccurate measurement can still lead to flawed causal conclusions, emphasizing the need for robust data collection and methods that account for measurement uncertainty.

The interconnectedness of these biases is a critical aspect of causal inference. For instance, selection bias is fundamentally a type of collider bias 5, and measurement error can, under certain conditions (e.g., differential error), open up backdoor pathways, thereby inducing confounding.2 This implies that a single flawed methodological decision, such as conditioning on a collider, can cascade into multiple forms of bias, creating a complex web of distortions. Directed Acyclic Graphs (DAGs) become indispensable not just for identifying individual biases but for diagnosing the interplay of biases and understanding how a chosen adjustment strategy might inadvertently introduce new ones. They serve as a “gold standard tool” for assessing bias likelihood and guiding appropriate adjustment sets.13 This highlights that causal inference is not a mechanical process of simply “throwing in control variables”.7 Instead, it demands a deep, graph-theoretic understanding of how variables relate to avoid “garbage-can regressions” 7 and ensure the validity of causal conclusions. Researchers must be rigorously trained in graphical causal models to effectively navigate the complexities of observational data and make defensible causal claims.

 

Key Identification Assumptions

 

For valid causal inference from observational data, it is imperative to translate real-world causal questions into statistical parameters that can be causally interpreted. This translation hinges on satisfying a set of strong identification assumptions.4 While these assumptions are generally guaranteed by design in a well-executed randomized controlled trial, they must be meticulously considered and, whenever possible, validated in observational studies.

• Exchangeability: This assumption posits that, conditional on observed covariates, the treatment groups are comparable and well-balanced with respect to the distribution of both measured and unmeasured confounders.4 In essence, it implies that the exposed and unexposed groups are “exchangeable” in terms of their potential outcomes, as if treatment had been randomly assigned.
• Consistency: This assumption requires that the observed treatment levels in the collected data correspond to well-defined versions of the treatment.4 It ensures that the observed outcome for an individual who received a specific treatment is indeed the outcome that would have been observed if that individual had received that specific treatment.
• Positivity: This assumption dictates that there must be a non-zero positive probability of receiving every level of treatment for all individuals in the study population.4 This ensures that all subgroups of the population have a chance to be exposed or unexposed, allowing for meaningful comparisons. Violation can occur if data for a subpopulation is entirely absent.12
• Causal Sufficiency (Markovianity): This is a strong assumption, particularly for constraint-based algorithms like PC. It states that the set of observed variables includes all common causes for any pair of variables, effectively implying the absence of unmeasured confounders.14 This assumption is frequently violated in real-world observational settings.8
• Causal Markov Condition: This condition assumes that each variable in the causal graph is conditionally independent of its non-descendants, given its direct causes (parents).14 It links the causal structure of the graph to the observed conditional independencies in the data.
• Faithfulness: This assumption is the converse of the Causal Markov Condition. It implies that every conditional independence observed in the data is exactly represented by the causal graph.14 This means the causal graph captures all conditional independencies without including “extra” ones. This assumption is a “critical bottleneck” for neural causal discovery methods, as it is frequently violated across reasonable dataset sizes, undermining their performance.19
• Acyclicity: This assumption requires that the causal graph is a Directed Acyclic Graph (DAG), meaning there are no directed cycles or feedback loops.12
• Large Sample Size: For statistical tests used in causal discovery algorithms (e.g., conditional independence tests), a sufficiently large sample size is required to accurately detect relationships.14
• Appropriate Statistical Test: The choice of the statistical test for independence must be suitable for the nature of the data (e.g., continuous, categorical, mixed) to ensure reliable detection of conditional independence relationships.14

These assumptions form the theoretical bedrock of causal inference. Their validity is paramount, as violations can lead to incorrect or misleading causal conclusions, irrespective of the sophistication of the statistical methods employed. The challenge lies in their untestability from data alone, often requiring substantial domain knowledge and careful reasoning. The pervasive nature of assumption violations, particularly concerning unmeasured confounding and faithfulness, reveals a fundamental “fragility” in drawing definitive causal conclusions from single observational studies. This inherent fragility necessitates the adoption of “triangulation of evidence”.2 Triangulation, defined as integrating results from several different approaches—each with different and largely unrelated sources of potential bias—provides a “stronger basis” for causal inference.2 It is not about finding a single perfect method, but about using multiple, diverse methods to cross-validate findings, acknowledging that each method has inherent weaknesses. This proactive approach, which can involve pre-registering triangulation strategies, enhances the robustness of findings. This suggests a paradigm shift from seeking a single “true” causal graph to understanding the robustness and sensitivity of causal claims across different modeling assumptions and methods. A comprehensive causal analysis should therefore include rigorous sensitivity analyses to assumption violations and, ideally, integrate findings from diverse methodological paradigms to build a more compelling body of evidence.

The following table provides a concise overview of the biases and identification assumptions discussed, highlighting their mechanisms, impacts, and associated challenges.

Bias/Assumption	Definition	DAG Representation/Mechanism	Impact on Inference	Key Challenge/Limitation	Relevant Sources
Confounding Bias	An uncontrolled common cause influencing both exposure and outcome, creating a spurious association.	Open back-door path.	Leads to spurious associations and biased estimates.	Unmeasured confounders (residual confounding); often unrealistic assumption that all are measured.	2
Selection Bias (incl. Collider Bias)	Occurs when conditioning on a common effect (collider) of two variables, creating a spurious association between them.	Conditioning on a common effect (collider) opens a path.	Can introduce bias if conditioned on; counter-intuitive.	Often introduced by improper control; difficult to identify without domain knowledge.	2
Measurement Bias	Observed values deviate systematically from true values due to imprecise methods or misreporting.	True (unobserved) and measured values as distinct variables; differential error can open back-door paths.	Distorts true relationships, can induce confounding.	Difficult to quantify and correct for; requires robust data collection.	2
Exchangeability	Treatment groups are comparable regarding measured and unmeasured confounders.	Assumed comparability of groups.	Enables causal interpretation of statistical parameters.	Often unrealistic in observational data; difficult to verify.	4
Consistency	Observed treatment levels correspond to well-defined versions of treatment.	Well-defined treatment assignments.	Ensures observed outcome is the true potential outcome under treatment.	Requires precise definition of interventions.	4
Positivity	Non-zero probability of receiving every level of treatment for all individuals.	All subgroups have a chance to be exposed/unexposed.	Ensures meaningful comparisons across groups.	Violation if data for a subpopulation is absent.	4
Causal Sufficiency	All common causes for any pair of variables are observed.	No unmeasured confounders.	Guarantees identifiability of causal structure.	Often unrealistic in real-world data; limits applicability of some algorithms.	14
Causal Markov Condition	Each variable is independent of its non-descendants, given its direct causes (parents).	Implied by d-separation patterns in the graph.	Links causal structure to observed conditional independencies.	Assumed property of the data-generating process.	14
Faithfulness	All observed conditional independencies are exactly represented by the causal graph.	Graph captures all conditional independencies without “extra” ones.	Essential for accurate structure recovery.	Can be violated in practice, especially for neural methods; critical bottleneck.	14
Acyclicity	The causal graph contains no directed cycles or feedback loops.	Directed Acyclic Graph (DAG).	Ensures a well-defined causal ordering.	May not hold in systems with feedback loops (requires specialized methods).	12

 

III. Causal Discovery Algorithms for Observational Data

 

The complexities and biases inherent in observational data necessitate sophisticated algorithmic approaches for causal graph learning. These methods broadly fall into constraint-based, score-based, and hybrid categories, each with distinct principles, assumptions, and limitations.

 

Constraint-Based Methods

 

Constraint-based algorithms infer causal structure by identifying conditional independence (CI) relationships within the observed data. They typically commence with a fully connected undirected graph and iteratively remove edges that are found to be conditionally independent given a subset of other variables. Following this “skeleton discovery” phase, a set of orientation rules is applied to determine the direction of causal links, such as identifying v-structures (where two variables cause a third, but are not directly connected themselves) and avoiding cycles.15 The ultimate objective is to recover the Markov equivalence class of the true causal graph, which represents a set of graphs that imply the same conditional independencies.15

 

PC Algorithm

 

The PC (Peter-Clark) algorithm stands as a foundational constraint-based method. It initiates with a complete undirected graph and systematically performs conditional independence tests. If two variables (X, Y) are found to be conditionally independent given a subset of other variables (S), the edge between X and Y is removed. This process is iterative, progressively increasing the size of the conditioning set (n) in each step.14 After identifying the “skeleton” (the undirected graph representing adjacencies), the algorithm applies a series of orientation rules. These rules include identifying “v-structures” (e.g., X → Z ← Y where X and Y are not adjacent) and applying Meek’s rules to orient remaining edges and prevent the formation of cycles.14

The PC algorithm relies on several strong assumptions for its correctness. These include Causal Sufficiency (the absence of latent confounders, meaning all common causes are observed), the Causal Markov Condition, Faithfulness (all observed conditional independencies are implied by the graph), Causal Ordering (the existence of a causal order among variables), the availability of a Large Sample Size for reliable CI tests, and the use of an Appropriate Statistical Test for independence.14 To address some limitations of the original PC algorithm, several variants have been developed. These include

PC-Stable, designed to mitigate order dependence in test results; Conservative-PC (CPC) and PC-Max, which employ more cautious strategies for orienting colliders; and Copula-PC, specifically engineered to handle mixed continuous and ordinal datasets by inferring rank correlation.23 The PC algorithm is a cornerstone of causal discovery, demonstrating the power of conditional independence tests to infer causal structure. Its explicit reliance on strong assumptions, such as causal sufficiency, highlights the ideal, often unattainable, conditions under which a definitive causal graph can be learned, thus setting a benchmark for more robust algorithms.

 

FCI Algorithm (Fast Causal Inference)

 

The Fast Causal Inference (FCI) algorithm is a significant extension of the PC algorithm, specifically designed to infer causal relationships from observational data even in the presence of hidden variables (latent confounders) and selection bias.16 Unlike PC, which outputs a Completed Partially Directed Acyclic Graph (CPDAG) representing a Markov equivalence class under causal sufficiency, FCI generates a

Partial Ancestral Graph (PAG). A PAG represents the common features of all DAGs that are observationally equivalent given the observed variables, even if unmeasured confounders or selection bias are present.26

FCI also begins with an undirected graph and removes edges based on conditional independence tests. However, it employs more complex rules for identifying conditioning sets, such as using “possibly d-separating” sets, and for orienting edges, which explicitly account for the potential influence of unobserved variables.16 It can identify “maximal ancestral graphs” (MAGs), which represent independence relations in the presence of latent and selection variables.27 A notable feature of FCI is its “anytime” property: the algorithm can be interrupted at any stage, and the output will still be correct in the large sample limit, though potentially less informative.26 Despite its robustness to hidden variables, FCI can be computationally very expensive. The number of conditional independence tests performed grows exponentially with the number of variables in the worst case, impacting both speed and accuracy, especially with smaller sample sizes, as tests conditional on many variables have low statistical power.26 FCI is crucial for real-world applications where the strong causal sufficiency assumption of the PC algorithm is almost always violated. It provides a more realistic and robust framework for causal discovery by acknowledging the inherent limitations of observational data and providing a less definitive but more trustworthy output (PAG) that accounts for unobserved factors.

 

Score-Based Methods

 

Score-based algorithms conceptualize causal discovery as an optimization problem. They define a “score function” that quantifies how well a given causal graph fits the observed data, typically incorporating a penalty for model complexity to prevent overfitting.17 The algorithm then systematically searches the vast space of possible graphs to identify the one that maximizes this score. A key advantage of this approach is that it inherently avoids the multiple testing problem that is often encountered in constraint-based methods.29

 

Greedy Equivalence Search (GES)

 

Greedy Equivalence Search (GES) is a widely used heuristic score-based algorithm that navigates the space of causal Bayesian networks to find the model with the highest Bayesian score. It operates in two principal phases:

• Forward Search: Beginning with an empty graph, GES iteratively and greedily adds edges between nodes. Each potential addition is evaluated based on whether it increases the Bayesian score of the model. This process continues until no single edge addition can further improve the Bayesian score.17
• Backward Search: Following the forward phase, GES transitions to a backward search, during which edges are removed from the graph. Similar to the forward phase, an edge is removed only if its removal leads to an increase in the Bayesian score. This backward process persists until no single edge removal can further enhance the score.17

GES operates under several critical assumptions: i.i.d. (independent and identically distributed) observational samples, linear relationships between variables with Gaussian noise terms, adherence to the Causal Markov condition, Faithfulness, and, importantly, the assumption of no hidden confounders.17 A notable limitation is that standard implementations of GES do not support multi-processing.17 Furthermore, edge orientation in tabular data is generally less reliable compared to time series data, where the inherent temporal order provides valuable additional causal information that can aid in directing edges.17 GES offers a compelling alternative to constraint-based methods, particularly when the data characteristics align with its underlying assumptions (e.g., linearity, absence of hidden confounders). Its two-phase search strategy efficiently explores the graph space to find a locally optimal structure, showcasing the utility of a score-based optimization approach.

 

NOTEARS (Non-linear Optimization for Causal Discovery)

 

NOTEARS (Non-linear Optimization for Causal Discovery) represents a significant innovation by transforming the traditionally discrete and combinatorial problem of structural learning into a continuous, differentiable numerical optimization problem.22 This transformation enables the use of powerful gradient-based optimization techniques to identify the globally optimal causal graph (DAG).22 A key innovation lies in its use of a smooth, differentiable acyclicity constraint function, specifically

h(W) = tr(exp(W ∘ W)) – d = 0, which mathematically ensures that the learned graph is a Directed Acyclic Graph (DAG) by penalizing cycles.22 The optimization problem is typically solved using augmented Lagrangian methods, converting a constrained problem into an unconstrained one solvable by numerical methods.

While the core NOTEARS framework is flexible, its basic applications often implicitly or explicitly assume Acyclicity (which it enforces), smoothness/differentiability of the loss function and constraint for gradient-based optimization, and sometimes linearity and Gaussian noise for simplicity. Causal sufficiency is also typically assumed in its basic application.14 A critical limitation of NOTEARS is its

lack of scale-invariance. Research has demonstrated that a simple rescaling of input variables can significantly alter the derived DAG, suggesting that its results may not reflect true causal relationships but rather depend on arbitrary data scaling.31 This raises serious concerns about its robustness and generalizability in practical applications. Despite this, NOTEARS has found application in

Feature Selection (FSNT), where it identifies direct causal relationships between features and a target variable and then ranks features by “causal strength” to select an optimal subset.22 NOTEARS pioneered a new class of causal discovery algorithms by framing the problem as a continuous optimization task. However, its scale-invariance issue underscores the importance of robust algorithmic properties that extend beyond mere mathematical tractability for reliable causal discovery.

 

IV. Conclusion

 

Causal graph learning from observational data is a field of immense importance, driven by the need to move beyond mere correlations to understand the fundamental cause-and-effect relationships that govern complex systems. This understanding is critical for informed decision-making, effective interventions, and scientific discovery across diverse domains, from healthcare and economics to social sciences.1

The pursuit of causal knowledge from observational data is inherently challenging due to the “fundamental problem of causal inference”—the impossibility of observing counterfactuals directly.3 This necessitates reliance on strong, often untestable, identification assumptions such as exchangeability, consistency, positivity, causal sufficiency, the causal Markov condition, faithfulness, and acyclicity.4 The pervasive nature of biases—including confounding, selection bias (often a form of collider bias), and measurement bias—further complicates this endeavor.2 These biases are not isolated phenomena but often intricately interconnected, where a single flawed methodological choice can cascade into multiple distortions. Directed Acyclic Graphs (DAGs) are indispensable tools for diagnosing this interplay of biases and guiding appropriate covariate selection, moving beyond simplistic “control” practices to ensure the validity of causal conclusions.3

The inherent “fragility” of relying on strong assumptions, which are frequently violated in real-world observational settings, underscores the limitations of drawing definitive causal conclusions from single studies. This fragility necessitates the adoption of “triangulation of evidence,” integrating results from multiple, diverse approaches, each with different and largely unrelated sources of potential bias, to build a stronger and more robust body of causal evidence.2 This approach shifts the focus from seeking a single “true” causal graph to understanding the robustness and sensitivity of causal claims across various modeling assumptions and methods.

Algorithmic advancements in causal discovery offer diverse strategies to tackle these challenges. Constraint-based methods like the PC algorithm leverage conditional independence tests to infer graph structures, while the FCI algorithm extends this to handle hidden variables and selection bias, providing a more realistic framework for complex real-world data.14 Score-based methods, such as GES, frame causal discovery as an optimization problem, searching for graphs that best fit the data according to a defined score.17 Innovations like NOTEARS have transformed structural learning into a continuous optimization problem, though its lack of scale-invariance highlights the importance of robust algorithmic properties beyond mere mathematical tractability.22

Despite significant progress, particularly with the integration of machine learning and the emerging role of Large Language Models (LLMs) in direct causal extraction from text, integrating domain knowledge, and refining causal structures 38, several challenges persist. The difficulty in evaluating the quality of discovered causal structures in real-world datasets due to the absence of known ground truth remains a primary limitation, often necessitating reliance on synthetic data for evaluation.19 Furthermore, neural causal discovery methods still face accuracy issues in reliably distinguishing between existing and non-existing causal relationships in finite sample regimes, with the faithfulness property identified as a critical bottleneck.19

Future directions in causal inference research are poised to address these limitations. Key areas include developing methods for high-dimensional data, particularly in confounding and mediation, where complex variable selection and interaction effects pose significant hurdles.37 There is a growing emphasis on precision medicine, causal machine learning (aiming to predict outcomes under interventions), enriching randomized experiments with real-world data, and addressing algorithmic fairness and social responsibility through a causal lens.37 Distributed learning, interference and spillover effects, and transportability of causal effects across populations are also active areas of research.37 The integration of explainability techniques into causal discovery, as seen in methods like ReX, represents a promising new direction for enhancing interpretability and bridging the gap between predictive modeling and causal inference.44 Ultimately, advancing causal graph learning in observational data will require continued innovation in algorithmic design, rigorous validation against realistic benchmarks, and a deep understanding of the underlying assumptions and their implications.