Causal Fidelity: Preserving Causal Evidence

• Causal fidelity is defined as the extent to which models, explanations, or synthetic datasets preserve true causal variables, intervention effects, or mechanism-level evidence rather than mere predictive correlations.
• Empirical studies reveal a predictive-causal gap where models with high predictive performance often exhibit low causal fidelity, highlighting challenges in reliably recovering system variables.
• Preserving causal fidelity is critical in domains such as time-series attribution, emotion reasoning, and recommendation, ensuring that interventions reflect accurate causal mechanisms.

Searching arXiv for papers on causal fidelity and closely related uses of the term. Causal fidelity denotes the extent to which a model, representation, explanation, simulator, or synthetic dataset preserves the causal variables, intervention effects, or mechanism-level evidence that are relevant to a target task, rather than merely matching predictive, associational, or perceptual regularities. Recent arXiv work uses the term in several technically distinct senses: as the fraction of encoder sensitivity allocated to system rather than environment degrees of freedom in predictive representation learning (Liu, 6 May 2026); as the preservation of causal segment effects in time-series attribution (Alcaraz et al., 2024); as counterfactual faithfulness to a supplied structural graph in action-unit-to-emotion reasoning (Huynh et al., 14 Jun 2026); as preservation of intervention estimands such as the average treatment effect in synthetic data and behavioral simulation (Xu, 26 Apr 2026, Li et al., 2 Apr 2026, Asiaee et al., 2 Mar 2026); as mechanism parity between real and synthetic imagery (Safaei et al., 18 Dec 2025); and as exact information conservation in counterfactual generation (Wu et al., 7 Nov 2025).

1. Definitions and formal scope

In predictive representation learning, causal fidelity is defined for an encoder $\phi:\mathbb{R}^{d_s+d_e}\to\mathbb{R}$, with $x_t=(s_t,e_t)$, by comparing its absolute partial sensitivities to system and environment coordinates: $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$ Here $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$ and $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$; $f_{\rm causal}=1$ corresponds to a purely system-only encoder, whereas $f_{\rm causal}=0$ corresponds to an encoder that depends purely on the environment (Liu, 6 May 2026).

In causal inference with synthetic data, causal fidelity is defined at the level of estimands. One formulation states that a synthetic distribution $P^\star$ has high causal fidelity if its plug-in ATE $\Psi(P^\star)$ is close to the true ATE $\Psi(P)$, with causal-fidelity error measured by

$x_t=(s_t,e_t)$0

A closely related behavioral-simulation formulation defines causal fidelity as the extent to which an estimated treatment effect $x_t=(s_t,e_t)$1 matches the true effect $x_t=(s_t,e_t)$2, quantified in practice by $x_t=(s_t,e_t)$3 (Xu, 26 Apr 2026, Li et al., 2 Apr 2026).

In explanation systems, the object of preservation is often a causal rationale rather than an estimand. In FACR, faithfulness is defined through counterfactual consistency between the rationale, the label, and a structural AU$x_t=(s_t,e_t)$4emotion graph $x_t=(s_t,e_t)$5, and it is trainable and measurable through intervention-based sensitivity and invariance metrics (Huynh et al., 14 Jun 2026). In CausalConceptTS, the framework does not introduce a separate named “fidelity score,” but it quantifies the divergence between causal and associational attributions through concept-wise attribution discrepancy, sign agreement, mean absolute discrepancy, and the fraction of sign-disagreements (Alcaraz et al., 2024).

In synthetic-versus-real perception studies, Decisive-Feature Fidelity (DFF) extends the fidelity spectrum to mechanism parity: the decisive features driving the system-under-test should agree across matched real and synthetic inputs. A synthetic image is DFF-fidelitous with respect to a real partner when

$x_t=(s_t,e_t)$6

and a pass-rate is computed across matched pairs (Safaei et al., 18 Dec 2025).

In counterfactual generation, the term is pushed further toward exact structural recovery. “The Causal Round Trip” formalizes Structural Reconstruction Error,

$x_t=(s_t,e_t)$7

and defines Causal Information Conservation as the condition $x_t=(s_t,e_t)$8, equivalently $x_t=(s_t,e_t)$9, so that faithful abduction is lossless (Wu et al., 7 Nov 2025).

These formulations indicate that causal fidelity is not a single metric but a family of criteria tied to what is being preserved: variables, effects, rationales, decisive features, or structural information.

2. Predictive representation learning and the predictive-causal gap

The strongest impossibility result in the supplied literature appears in the study of predictive representation learning under linear-Gaussian dynamics. For a one-dimensional linear encoder $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$0, the population latent self-prediction risk is

$$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$1

with $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$2 solving $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$3. The “null-zero” encoder $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$4 selects the system coordinate. The main theorem establishes an explicit family of stable dynamics for which every minimizer $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$5 strictly favors the environment component, implying $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$6. The paper further states that enlarging the model class to arbitrarily large nonlinear encoders cannot recover $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$7: capacity only lowers the minimum risk, further entrenching the non-causal solution (Liu, 6 May 2026).

The empirical evidence is correspondingly large scale. Across 539 distinct dynamics and 5 random seeds per setting, totaling 2695 runs, the mean causal fidelity is 0.49, the median is 0.48, and only 2.5% of runs exceed 0.70 fidelity; 0.4% exceed 0.90. The same study reports that the neural network lowers the prediction risk by 99.3% over the optimal linear encoder, yet remains as non-causal, often more so. In a high-dimensional extension with environment dimension $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$8, causal fidelity collapses to $$f_{\rm causal}(\phi) = \frac{\lvert\partial\phi/\partial s\rvert} {\lvert\partial\phi/\partial s\rvert+\lvert\partial\phi/\partial e\rvert}\in[0,1].$$9 at $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$0, while the predictive-causal gap grows to 92% improvement in prediction error relative to the causal representation (Liu, 6 May 2026).

The nonlinear Duffing-GRU experiments generalize the point beyond the linear-Gaussian case. Under unconstrained prediction of full $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$1, 55% of tasks are environment-dominant; under operational grounding, 24% are environment-dominant, with Fisher $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$2. After shifting $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$3 and $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$4, the median OOD MSE inflation is 1.82$$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$5 for the unconstrained setting and $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$6 under grounding (Liu, 6 May 2026).

The broader significance is stated explicitly in that work. Minimizing prediction error on all observables without distinguishing system from environment drives representations toward the most predictable modes, often the environment. Operational grounding partially suppresses the gap, but causal fidelity is never recovered without an explicit system-environment boundary. A plausible implication is that predictive objectives alone cannot serve as reliable proxies for causal state recovery in self-supervised learning, world models, or scaling-based evaluation.

3. Explanations, attributions, and reasoning systems

In time-series classification, CausalConceptTS embeds causal fidelity in a Rubin-Pearl potential-outcomes framework with predefined segments $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$7. The Individual Treatment Effect for segment $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$8 is defined by a class-conditional do-intervention: $$\lvert\partial\phi/\partial s\rvert\equiv\|\partial\phi/\partial s\|_1$$9 The Average Treatment Effect averages this quantity over all test samples with true label $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$0. The framework contrasts these causal attributions with associational PredDiff attributions built from the observational conditional $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$1. It then quantifies divergence via

$$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$2

and the sign-agreement indicator $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$3, together with mean absolute discrepancy and fraction of sign-disagreements. Across datasets, 33% sign-disagreement is reported, including cases where associational attributions flip sign relative to the causal effect; the paper attributes this to Simpson’s paradox and correlated features, and provides no finite-sample bound beyond standard diffusion-approximation error and the usual ignorability assumptions (Alcaraz et al., 2024).

FACR treats causal fidelity as counterfactual faithfulness to a supplied, polarity-aware AU$$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$4emotion graph $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$5. The model learns a disentangled AU latent $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$6, making $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$7 a defined intervention. Training combines classification, AU grounding, a sensitivity hinge loss for $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$8, and an invariance loss for $$\lvert\partial\phi/\partial e\rvert\equiv\|\partial\phi/\partial e\|_1$$9. Evaluation mirrors these terms through zero-one metrics: counterfactual sensitivity is the fraction of relevant AU-class pairs for which $f_{\rm causal}=1$0, and counterfactual invariance is the fraction of irrelevant pairs for which $f_{\rm causal}=1$1. On UNBC-PAIN, PSPI-agreement rises from $f_{\rm causal}=1$2 for the no-objective baseline to $f_{\rm causal}=1$3 for FACR, while invariance rises from $f_{\rm causal}=1$4 to $f_{\rm causal}=1$5. On cross-dataset seven-class emotion transfer, graph-causal agreement rises from $f_{\rm causal}=1$6 to $f_{\rm causal}=1$7. The paper is explicit that the metric tests fidelity to the supplied structure rather than its rediscovery; with a noisy learned $f_{\rm causal}=1$8, active-AU agreement remains low, whereas replacing the graph with verified EMFACS edges raises active-AU agreement from 0.24 to 0.59 (Huynh et al., 14 Jun 2026).

A related post-hoc notion appears in recommendation. There, Fidelity@$f_{\rm causal}=1$9 is the fraction of recommendations for which the causal-rule miner can produce at least one valid top-$f_{\rm causal}=0$0 rule whose antecedent lies in the user’s true history: $f_{\rm causal}=0$1 The reported causal Fidelity@1 values are 96.50%, 98.51%, and 97.03% on MovieLens for FPMC, GRU4Rec, and Caser, with corresponding Amazon values 95.11%, 95.94%, and 95.99%; association-rule fidelity remains in the single- to teens-percent range (Xu et al., 2020).

Reasoning systems in retrieval-augmented generation use yet another variant. Causal-Counterfactual RAG assesses “reasoning fidelity” with the Causal Chain Integrity Score,

$f_{\rm causal}=0$2

and the Counterfactual Robustness Score, defined identically over counterfactual explanations. On the reported benchmark, Regular RAG obtains Precision $f_{\rm causal}=0$3, Recall $f_{\rm causal}=0$4, CCIS $f_{\rm causal}=0$5, and CRS $f_{\rm causal}=0$6, whereas Causal-Counterfactual RAG obtains Precision $f_{\rm causal}=0$7, Recall $f_{\rm causal}=0$8, CCIS $f_{\rm causal}=0$9, and CRS $P^\star$0 (Khadilkar et al., 17 Sep 2025).

Across these systems, the common pattern is that explanation quality is evaluated not only by plausibility or coverage but by intervention-sensitive agreement with the mechanism the model is supposed to use.

4. Counterfactual generation, information conservation, and semantic closure

“The Causal Round Trip” recasts causal fidelity as lossless abduction and exact counterfactual transport. A diffusion-based SCM is written as an encoder $P^\star$1 and decoder $P^\star$2, with the ideal round trip $P^\star$3. The paper argues that standard samplers such as DDIM incur a nonzero one-step reconstruction error of order $P^\star$4, and introduces a BELM-based framework with an analytically invertible update rule so that

$P^\star$5

Under the usual identifiability assumptions, if $P^\star$6 and $P^\star$7, the recovered code is isomorphic to the true exogenous noise, and exact counterfactuals follow. Empirically, BELM-MDCM is reported to achieve the highest CIC-Score, approximately 0.37 versus near zero for DDIM variants, and a 44% lower PEHE than DDIM in the non-invertible stress test (Wu et al., 7 Nov 2025).

The same work couples zero-SRE design with Targeted Modeling and a Hybrid Training Objective

$P^\star$8

arguing that this hybrid objective is equivalent to a weighted score-matching objective that forces more accurate scores in causally salient regions, while the auxiliary task loss encourages a division of labor in the latent code (Wu et al., 7 Nov 2025). This formulation treats causal fidelity as a property of the entire abduction-action-prediction cycle, not just of the final sample quality.

A separate mathematical line appears in reversible causal nets. There, rollback-relevant meaning is represented by a monotone semantic closure $P^\star$9, and closure-preserving fidelity is defined by a Jaccard-type similarity

$\Psi(P^\star)$0

A deletion scan decomposes the log into an irredundant core $\Psi(P^\star)$1 and redundant remainder $\Psi(P^\star)$2, with $\Psi(P^\star)$3; under admissibility, every redundant fact is information-theoretically invisible. The resulting rate-distortion function factors through the core, and at the perfect-fidelity endpoint $\Psi(P^\star)$4, the minimum rate is governed by the confusability hypergraph via hypergraph entropy. In the medium instance with 4 branches of depth 3, $\Psi(P^\star)$5, $\Psi(P^\star)$6, $\Psi(P^\star)$7, $\Psi(P^\star)$8 bits/symbol, and $\Psi(P^\star)$9 bits/symbol (Xu, 15 Jun 2026).

These two strands define fidelity at a stricter level than effect agreement alone. In one case, the target is faithful recovery of exogenous noise and exact counterfactuals; in the other, it is preservation of the semantic closure relevant for rollback.

5. Synthetic data, behavioral simulation, and mechanism parity

For synthetic data in causal inference, the central claim is that predictive fidelity is insufficient. One formulation states that ATE preservation requires control of both the generated covariate law and the treatment-effect contrast in the outcome regression: $\Psi(P)$0 Motivated by this bound, the hybrid framework generates covariates from $\Psi(P)$1, samples treatment from $\Psi(P)$2, and sets outcome by $\Psi(P)$3, while monitoring covariate synthesis with Distance-to-Closest-Record. Across experiments, fully generative GAN- and LLM-based models can achieve strong TSTR AUC and low mean DCR yet substantially distort ATE; hybrid models keep TSTR AUC and mean DCR essentially unchanged while reducing ATE MSE by an order of magnitude, for example from IPW MSE $\Psi(P)$4 to $\Psi(P)$5 for LLM full versus LLM hybrid, and from TMLE MSE to $\Psi(P)$6 (Xu, 26 Apr 2026).

CausalWrap approaches the same problem by imposing Partial Causal Knowledge $\Psi(P)$7 on samples from a frozen base generator. Structural fidelity is encoded by a violation functional

$\Psi(P)$8

and downstream causal fidelity is measured by ATE error or, when ground truth is unavailable, by an ATE-agreement score computed from an estimator ensemble. The wrapper learns a differentiable correction map $\Psi(P)$9 under an augmented-Lagrangian schedule. Reported gains include up to 63% reduction in ATE error on the ACIC-style suite and an increase in ATE agreement from 0.00 to approximately 0.38 on the ICU cohort for TabDDPM; the paper also reports that gains are nonmonotonic in the amount of partial knowledge, and that over-constraining can hurt (Asiaee et al., 2 Mar 2026).

Behavioral simulation with LLMs sharpens the distinction between descriptive fit and causal fidelity. In that setting, the true ATE is $x_t=(s_t,e_t)$00, the simulated ATE is $x_t=(s_t,e_t)$01, and causal fidelity is quantified by $x_t=(s_t,e_t)$02. Across three LLMs on 11 climate-psychology interventions, baseline absolute ATE error is 9.5 percentage points for GPT-4o-mini, 8.8 for Gemini 2.5 Flash Lite, and 9.2 for Claude 3 Haiku, whereas supervised OLS and LASSO baselines are approximately 4.3 and 4.2. VBN-CoT reduces mean ATE error by 34–41% for belief and policy but provides no systematic benefit for action. The descriptive-causal divergence is quantified directly: country-level correlations between MAE and ATE error are weak, with $x_t=(s_t,e_t)$03 for belief, $x_t=(s_t,e_t)$04 for policy, and $x_t=(s_t,e_t)$05 for action, and the top and bottom 10 countries overlap by at most 3 out of 10 (Li et al., 2 Apr 2026).

Mechanism parity in vision is operationalized by Decisive-Feature Fidelity. DFF compares counterfactual explanation heatmaps $x_t=(s_t,e_t)$06 and $x_t=(s_t,e_t)$07 for matched synthetic-real pairs, with a pass-rate over the evaluation set. On 2126 KITTI-VirtualKITTI2 pairs, the reported Spearman correlations between output-value fidelity and DFF are low, $x_t=(s_t,e_t)$08, indicating that DFF detects discrepancies missed by output consistency. DFF-guided calibration improves decisive-feature and input-level fidelity without sacrificing output-value fidelity, with held-out $x_t=(s_t,e_t)$09DFF ranging from $x_t=(s_t,e_t)$10 to $x_t=(s_t,e_t)$11 across systems under test (Safaei et al., 18 Dec 2025).

A common conclusion across these studies is that causal fidelity is an intervention-sensitive property: it depends on preserving effect contrasts or decisive evidence, not simply on reproducing marginals, predictions, or appearance.

6. Recurring limitations, misconceptions, and methodological tensions

A recurring misconception in the cited literature is that low predictive error, strong descriptive fit, or high perceptual realism imply causal fidelity. The predictive-causal-gap results reject this directly: lower latent prediction loss can coincide with weaker alignment to the system variables of interest, and the gap intensifies with dimension (Liu, 6 May 2026). The behavioral-simulation results reject the same inference at the level of interventions: prompting refinements can improve descriptive fit while leaving causal errors structurally distinct (Li et al., 2 Apr 2026). The synthetic-data studies similarly show that TSTR AUC, mean DCR, or joint reconstruction quality do not guarantee ATE preservation (Xu, 26 Apr 2026, Asiaee et al., 2 Mar 2026).

A second tension concerns associational versus causal explanations. CausalConceptTS shows that unconditional imputation can produce sign flips or magnitude distortions, and reports 33% sign-disagreement across datasets (Alcaraz et al., 2024). FACR makes the distinction sharper by separating plausible AU naming from faithful AU$x_t=(s_t,e_t)$12emotion reasoning under $x_t=(s_t,e_t)$13 interventions, and shows that simply naming action units accurately is not the same as reasoning through them (Huynh et al., 14 Jun 2026). In recommendation, causal-rule coverage substantially exceeds association-rule coverage, again separating causal explanatory reach from correlational co-occurrence (Xu et al., 2020).

A third tension is dependence on supplied structure. FACR states that faithfulness to $x_t=(s_t,e_t)$14 only reflects reality if $x_t=(s_t,e_t)$15 is correct, and replacing noisy learned edges with verified EMFACS edges materially changes active-AU agreement (Huynh et al., 14 Jun 2026). CausalWrap likewise assumes trusted edges, forbidden edges, and monotonicities, and reports that performance is nonmonotonic in the knowledge fraction (Asiaee et al., 2 Mar 2026). In closure-preserving rate-distortion, admissibility is defined relative to a chosen proof system and closure operator (Xu, 15 Jun 2026). This suggests that causal fidelity is often relative to an interface, graph, estimand, or semantic closure specified in advance.

Finally, multiple papers identify open technical gaps. CausalConceptTS provides no finite-sample bound beyond diffusion-approximation error and the usual ignorability assumptions (Alcaraz et al., 2024). The predictive-causal-gap work states that operational grounding partially suppresses environment dominance but does not recover causal fidelity without an explicit system-environment boundary (Liu, 6 May 2026). FACR identifies structure discovery as an open challenge, and notes that no existing affective-reasoning benchmark allows direct evaluation against known causal structure outside the supplied compositions (Huynh et al., 14 Jun 2026). “The Causal Round Trip” argues that standard diffusion designs optimized for perceptual generation rather than logical inference introduce a structural reconstruction barrier that must be removed by construction (Wu et al., 7 Nov 2025).

Taken together, these results establish causal fidelity as a stricter criterion than accuracy, realism, or plausibility. Its operational content depends on the domain, but the shared requirement is consistent: the object under study must preserve the causal variables, effects, or mechanisms that remain valid under intervention.