Causal and Counterfactual Explanations

• Causal and counterfactual explanations are techniques that interpret AI model predictions by simulating feature interventions through structural causal models.
• They integrate methods such as neural network-based causal graphs, ASP frameworks, and latent-space counterfactual searches to generate actionable recourse.
• These approaches enhance transparency, fairness, and trust in AI while addressing challenges like computational efficiency and reliable causal discovery.

Causal and counterfactual explanations are formal approaches for interpreting predictions in artificial intelligence models, designed to provide insight into not only how inputs correlate with outputs, but also how deliberate changes to certain inputs would causally affect model predictions. This distinction has become central in modern explainable AI (XAI), as purely associational (observational) methods cannot reliably estimate the effect of feature interventions, particularly when input variables exhibit mutual dependencies or when recourse advice must map to feasible, actionable changes. Counterfactual explanations refer to statements about how the outcome of a model would differ if certain features or latent factors were changed, typically formulated within structural causal models or related frameworks that explicitly encode causal relationships.

1. Causal Models and Counterfactual Reasoning

A foundational premise is that explanations for model behavior should be grounded in a causal model of the data, not solely in observational associations. Structural Causal Models (SCMs) formalize this approach by positing that observed variables $\mathcal{V} = \{X_1, ..., X_n\}$ are generated through deterministic structural functions $f_i$ over their parents in a causal graph, together with independent exogenous (noise) variables $\mathcal{E} = \{E_1, ..., E_n\}$ (Parafita et al., 2019). This causal graph (a Directed Acyclic Graph, DAG) dictates the factorization of the data distribution:

$$P(X_1, \dots, X_n) = \prod_{i=1}^n P(X_i \mid Pa_i).$$

Counterfactual reasoning involves the application of interventions, typically represented as $do(X_i = x'_i)$, which either fix a variable’s value or replace its generating mechanism, thereby propagating changes through the causal system. Abduction-action-prediction is the three-step process for computing counterfactual outcomes: (1) abduce the latent “state of the world” (noise), (2) act by intervening on selected variables, (3) predict the new outcome by applying the SCM with these modifications (Crupi et al., 2021). Intervening only on variables receiving the “do” operation while re-sampling the noise for downstream variables (the “weak-stability” principle) ensures that changes remain consistent with the system’s causal semantics (Parafita et al., 2019).

2. Methodologies for Causal and Counterfactual Explanations

Technical approaches for providing causal and counterfactual explanations span various classes of AI models and data modalities:

• Distributional Causal Graphs (DCGs) encode causal relationships among latent factors, each implemented as a neural network estimating the conditional distribution parameters for each node (Parafita et al., 2019).
• Answer Set Programming (ASP) frameworks use logic programming to specify allowable interventions and minimality (i.e., the smallest set of feature changes necessary for outcome flips), and define responsibility scores for features (degree to which a feature value is involved in the explanation) (Bertossi, 2020).
• Prototype-based frameworks (ProCE) integrate SCMs with proto-typicality constraints and multi-objective optimization to generate counterfactuals that are valid for categorical and continuous variables and respect causal dependencies (Duong et al., 2021).
• Latent-space counterfactual search (e.g. CEILS) finds recourse in the space of exogenous variables or residuals, mapping interventions through the causal generative process to feature space for ensuring feasibility and respecting causal pathways (Crupi et al., 2021).
• Causal Generative Models such as VAEs or GANs paired with known causal attributes generate counterfactual images or samples that isolate the effect of specific interventions, enabling quantitative analysis of causal attributions at pixel or attribute-level granularity (Taylor-Melanson et al., 21 Jan 2024).
• ASP-based planning frameworks (CFGs, CoGS) model the counterfactual problem as traversing a state-space defined by both decision and causal rules, producing detailed “paths” of feature changes that preserve the entire system’s causal integrity (Dasgupta et al., 24 May 2024, Dasgupta, 13 Feb 2025).

A key distinction across methods is whether causal attribution occurs at the raw input level, in a latent representation, or via logic rules (explicit symbolic constraints). Integration of causal discovery—identifying the unknown causal graph from data via methods like PC, DirectLiNGAM, or NOTEARS—enables counterfactual explanations even when the true dependencies between features are not given (2402.02678). Embedding prior domain knowledge or semantic constraints in the programmatic specification of causal dependencies further ensures realism (Bertossi, 2020, Dasgupta et al., 24 May 2024).

3. Estimating and Interpreting Counterfactual Effects

Counterfactual explanations quantify the causal effect of interventions by comparing model predictions before and after the intervention. Key metrics and mathematical constructs include:

• Counterfactual probabilities: For variable $X$, $P(Y \mid do(X = x'))$ gives the post-intervention probability of outcome $Y$ (2402.02678).
• Counterfactual Explainability Measure: For black-box models, the variance-based metric

$$\xi_Y(\vee_{k\in S} W_k) = \frac{\text{Var}(Y(W) - Y(W'_S, W_{-S}))}{2 \cdot \text{Var}(Y(W))}$$

quantifies the contribution of changing a set of features $S$ to outcome variability, generalizing sensitivity analysis to the causal domain (Gao et al., 3 Nov 2024).

• Necessity, Sufficiency, and Nesuf scores: Quantify the probability that an intervention on a feature is required or sufficient for outcome changes (derived from counterfactual probabilities) (2402.02678).
• Interaction Effects: The explanation algebra supports allocation of importance to both main effects and feature interactions using inclusion–exclusion style decompositions (Gao et al., 3 Nov 2024).

These formalizations support users in identifying both which individual feature changes could be made for recourse, and how combinations of features interact causally to support or block desired outcomes.

4. Practical Implementations and Limitations

Causal and counterfactual explanations are implemented in practice using a mix of optimization and logic programming:

• Optimization in feature or latent space: Traditional approaches optimize a cost function under a label constraint, but may violate causal constraints. Recent approaches optimize in latent space under the SCM mapping to ensure changes are realizable and respectful of the feature dependencies (Crupi et al., 2021, Fatemi et al., 5 May 2025).
• Logic programming-based planning: ASP-based solvers search for minimal intervention paths under causal constraints encoded as program rules, delivering both interventions and justifications as explainable “proof trees” (Dasgupta et al., 24 May 2024, Dasgupta, 13 Feb 2025).
• Gradient-free algorithms: For mixed data types, model-agnostic genetic algorithms (such as NSGA-II) identify Pareto-optimal sets of counterfactuals balancing validity, proximity, prototypicality, and causal compliance. Autoencoders and class prototypes are used to ensure constructed counterfactuals are regionally plausible (Duong et al., 2021).
• Limitations of generative models: Image generators used for creating visual counterfactuals (e.g. Fader Networks, AttGAN) may fail to produce plausible outputs for low-density or counter-intuitive factor configurations, limiting trustworthiness of the explanations when intervening far from the training distribution (Parafita et al., 2019).

These methods provide rigorous mechanisms for generating explanations amenable to user recourse or scientific insight, but often depend on assumptions about the causal model’s accuracy, robustness to sample size, and the representational power of generative or logic-based components.

5. Role in Trust, Fairness, and User Understanding

Causal and counterfactual explanations play a nuanced role in model transparency and perceived trust:

• Transparency and Accountability: By clarifying the pathways through which feature changes propagate to outcomes, these explanations support regulatory requirements and individual recourse in applications such as finance, healthcare, and hiring (Crupi et al., 2021, Dasgupta et al., 24 May 2024).
• Bias Mitigation: Frameworks such as CausaLM leverage counterfactual representation learning to produce models whose decisions are invariant to unwanted biasing concepts (e.g., gender or race), supporting fairness through explicit causal “purging” of confounders in representations (Feder et al., 2020).
• User Studies: Objective improvements in predictive accuracy are only modest when using counterfactual as opposed to causal explanations, but user-reported satisfaction and trust are higher for counterfactuals (Warren et al., 2022). Laypeople may conflate counterfactual explanations with causation, potentially misunderstanding the causal impact of features; explicit disclaimers are effective in correcting these misconceptions (Tesic et al., 2022).
• Human-centered design: Generating explanations with LLMs from counterfactual sets improves end-user comprehension, provided the underlying counterfactuals are themselves causally valid and actionable (Fredes et al., 27 Aug 2024).

These aspects underline the importance of aligning technical rigor in causal reasoning with the cognitive and ethical dimensions of explanation delivery.

6. Contemporary Challenges and Future Directions

Despite significant advances, several open issues remain:

• Causal Discovery Reliability: Robust causal structure estimation, especially under finite sample and measurement error, remains a challenge, as causal discovery algorithms are sensitive to the presence of unmeasured confounders and limited by Markov equivalence classes (2402.02678, Smith, 2023).
• Computational Efficiency: Approaches relying on combinatorial search over intervention sets are often computationally expensive. Recent frameworks (e.g., BRACE) employ novel optimizations in latent space to reduce computation while preserving causal fidelity (Fatemi et al., 5 May 2025).
• Generalizability: Ensuring that generated counterfactuals are “reachable” in real-world domains and reflective of true causal constraints—especially in high-dimensional or structurally complex systems—remains an area for further methodological advances (Crupi et al., 2021, Dasgupta, 13 Feb 2025).
• Evaluation: Metrics for validity, proximity, sparsity, interpretability, and causal constraint satisfaction are all used to benchmark and compare methods, but no universal standard yet exists. Closed-loop, human-in-the-loop, and recourse-focused assessments are emerging as best practices (Fredes et al., 27 Aug 2024).

Promising directions include developing more expressive causal models (hyper-relational knowledge graphs (Jaimini et al., 2022)), adaptive balancing of objectives in visual counterfactuals (Qiao et al., 14 Jul 2025), and expanding recourse explanations for multi-class and temporally evolving systems.

7. Integration with Broader XAI and AI Practice

Causal and counterfactual explanations now form a core pillar of state-of-the-art XAI, unifying inferential transparency, actionable recourse, and domain-grounded auditability. Their formalization enables not only attribution of responsibility (through minimal adequate counterfactual sets), but also supports principled approaches to fairness and reliability by tying explanations to processes reflecting how real-world interventions would play out within the modeled system. The mathematical and algorithmic diversity of contemporary methods attests to the field’s richness, with workflows grounded variously in logic, optimization, generative modeling, and causal inference, each offering unique strengths and limitations shaped by the modeling context. Ultimately, causal and counterfactual explanations represent a transition from static, descriptive model summaries to dynamic, intervention-aware narratives that inform both technical and societal interactions with AI.