Attribution and Variables

Updated 6 July 2026

Attribution and Variables is a class of formal problems that map changes in outcomes or predictions to explanatory quantities using defined allocation rules.
It spans applications from climate detection and causal distribution shifts to deep neural network explanations and multi-touch conversion analysis.
Methodologies employ techniques such as Shapley values, gradient-based explanations, and robust regression to ensure completeness, causal interpretability, and predictive faithfulness.

Searching arXiv for the cited papers to ground the article. Searching for climate detection-and-attribution, attribution theory, and feature attribution papers. “Attribution and variables” denotes a family of formal problems in which an observed outcome, distributional shift, prediction, or event is assigned to one or more explanatory quantities. Across the literatures represented here, the attributed object may be a detected climate signal, a counterfactual change in a distribution, a model prediction, a conversion event, an anomalous interval, or a LLM response; the candidate variables may be external forcings, graph nodes, input features, event tokens, channels, or causal mechanisms (Székely et al., 2019, Budhathoki et al., 2021, Deng et al., 11 Aug 2025). The central technical question is how to map a change in an outcome to variables while preserving some combination of completeness, robustness, causal interpretability, or predictive faithfulness.

1. Conceptual scope

In climate detection and attribution, detection asks whether observed change exceeds internal variability alone, whereas attribution asks whether the detected change can be assigned to a particular external forcing or combination of forcings such as anthropogenic greenhouse gases, solar variability, or volcanic aerosols (Székely et al., 2019). In causal distribution-shift analysis, the object of attribution is not a single prediction but the change from one joint distribution to another, decomposed across causal mechanisms of the form $P(X_j \mid Pa_j)$ (Budhathoki et al., 2021). In deep learning, attribution explanation assigns a score $a_i$ to each input variable $x_i$ for a specified model output, and the dominant concern is whether those scores faithfully reflect the actual contribution of input variables to the decision-making process (Deng et al., 11 Aug 2025). In online multi-touch attribution, the attributed object is conversion credit along a customer journey, with touch events as the explanatory units (Yang et al., 2020). In social-psychological auditing of LLMs, attribution is operationalized as a forced choice among effort, ability, task difficulty, and luck, and the explanatory variables are grouped into internal and external causes (Raj et al., 28 May 2025).

This variety does not eliminate a shared structure. Each formulation introduces: a target quantity to be explained; a representation of candidate variables; and an allocation rule. A plausible implication is that “attribution” is less a single method than a class of allocation operators defined over different statistical objects.

2. Variable representations and attribution formalisms

Different traditions define “variables” at different levels of granularity. Some use raw inputs such as pixels, tokens, or climate fields; others use events, relations, coalitions, or mechanisms. The attributed quantity changes accordingly.

Setting	Variables	Attributed object
Climate D&A	Spatial climate field $X$ , target forcing $Y$ , anchor forcing $A$	Fingerprint $\beta$ and test statistic
Causal change attribution	Nodes $X_j$ and mechanisms $P(X_j\mid Pa_j)$	Change in joint or marginal distribution
DNN / LLM explanation	Input features or tokens $x_i$	Local attribution score $a_i$ 0 or $a_i$ 1
Multi-touch attribution	Journey events and channels	Event- or channel-level conversion credit

Several formalisms recur. Cooperative-game attribution uses the Shapley value

$a_i$ 2

which appears in classical feature attribution, multi-touch attribution, and coalition analysis (Sun et al., 2011, Yang et al., 2020, Zheng et al., 2023). For differentiable models, the Aumann–Shapley integral assigns change along a path from baseline to input,

$a_i$ 3

and is uniquely characterized on the class of multilinear-plus-additive characteristic functions by Dummy, Additivity, Conditional Nonnegativity, Affine Scale Invariance, and Anonymity (Sun et al., 2011).

Causal-mechanism formulations replace variable-level credit with mechanism-level credit. With a DAG over $a_i$ 4,

$a_i$ 5

and hybrid distributions are constructed by replacing only selected mechanisms:

$a_i$ 6

This supports either a KL decomposition of the full joint shift or a Shapley allocation of a marginal or functional change (Budhathoki et al., 2021, Quintas-Martinez et al., 2024).

Probabilistic token attribution in LLMs uses a different object: the conditional response probability. If $a_i$ 7 is a prompt and $a_i$ 8 a response, the attribution of prompt token $a_i$ 9 is

$x_i$ 0

where $x_i$ 1 denotes the prompt with token $x_i$ 2 marginalized away via Bayes inversion over the model’s own next-token probabilities (Shilpika et al., 20 May 2026).

3. Climate attribution: forcings, fingerprints, and counterfactual worlds

Traditional optimal fingerprinting projects high-dimensional observations and model responses onto empirical orthogonal functions derived from control runs, then estimates forcing-specific scaling factors. The direct critique is that EOFs are unsupervised and their truncation is somewhat arbitrary, because they maximize variance rather than relevance to a target forcing (Székely et al., 2019). A supervised alternative learns a mapping from a climate field $x_i$ 3 directly to the amplitude of an external forcing $x_i$ 4. In the linear case,

$x_i$ 5

and the learned coefficient map $x_i$ 6 functions as a fingerprint map (Székely et al., 2019).

The robust variant is anchor regression. With anchor variables $x_i$ 7 representing non-target forcings, the estimator solves

$x_i$ 8

or, with ridge regularization,

$x_i$ 9

This penalizes residual alignment with anchor forcings and targets robustness to shift interventions on exogenous variables (Székely et al., 2019, Székely et al., 2022). In CMIP5 proof-of-concept experiments using 82 simulations from 21 models and annual $X$ 0 near-surface temperature maps, increasing $X$ 1 reduced residual–anchor correlation from about $X$ 2 at $X$ 3 to about $X$ 4 at $X$ 5, at the cost of degraded RMSE and $X$ 6 (Székely et al., 2019). In the CMIP6 formulation, attribution to greenhouse gases while anchoring against aerosols achieved mean type I error $X$ 7 and mean power $X$ 8, while attribution to aerosols while anchoring against $X$ 9 achieved average $Y$ 0 and $Y$ 1 with linear anchors (Székely et al., 2022).

A second development replaces single-step forcing-to-impact attribution with conditional multi-step attribution. For the 1991 Mt. Pinatubo eruption, the framework models pathways through FLNT, T050, FSDS, and TREFHT using peak-impact features and a discrete forcing posterior over eight $Y$ 2 levels. Under well specified priors, the posterior for the true 10 Tg forcing rises from about $Y$ 3 in the single-step stratospheric analysis to about $Y$ 4 in the multi-step stratospheric pathway, to about $Y$ 5 in the surface pathway, and to about $Y$ 6 when both pathways are combined (Wentland et al., 2024). This shows that conditioning on intermediary variables with higher signal-to-noise can improve forcing attribution where single-step temperature-only approaches fail.

Extreme-event attribution introduces counterfactual probabilities of necessary, sufficient, and necessary-and-sufficient causation. Under identification assumptions, with $Y$ 7 and $Y$ 8,

$Y$ 9

In the multivariate peaks-over-threshold setting, events are defined on a high-dimensional climate vector and modeled with a multivariate generalized Pareto distribution, while a linear projection $A$ 0 is optimized to maximize causal evidence (Kiriliouk et al., 2019).

4. Feature, event, and token attribution in machine learning

A major theoretical synthesis reformulates fourteen attribution methods as weighted allocations of independent effects $A$ 1 and interaction effects $A$ 2 in a Taylor-interaction system. The essential difference among methods lies in how they allocate those effects, and three principles are proposed for faithful allocation: low approximation error, no-unrelated-allocation, and complete allocation (Deng et al., 2023). A later theoretical review generalizes this unification and argues that attribution methods can be compared through theoretical unification, theoretical rationale, and theoretical evaluation; it also summarizes that only a subset of methods, including IG, Expected Gradients, DeepLIFT-Res/Rev, and Shapley/DeepSHAP, satisfy the full Taylor-interaction allocation criteria, whereas DTD, LRP- $A$ 3, Occ-1/PDiff, Grad $A$ 4Input, and Grad-CAM violate at least one allocation principle (Deng et al., 11 Aug 2025).

Within gradient-based explanation, Disentangled Masked Backpropagation learns per-layer masks $A$ 5 to separate positive, negative, and nuisance components of a ReLU network’s piecewise linear mapping. The optimization target is

$A$ 6

On PASCAL VOC and ImageNet with ResNet50 and VGG16, DMBP achieved the best overall average insertion metric, $A$ 7, and near-zero sanity-check rank correlations under layer reinitialization, such as $A$ 8 on VOC/VGG16 and $A$ 9 on VOC/ResNet50 (Ruiz et al., 2021).

Relational Local Explanations depart from independent per-variable scoring by representing homogeneous inputs as graphs. The local explanation is a symmetric matrix $\beta$ 0 over pairwise relations, and individual attribution is recovered by row-wise means,

$\beta$ 1

On ImageNet with ResNet-50, RLE achieved an IROF score of $\beta$ 2, compared with $\beta$ 3 for LIME, $\beta$ 4 for SHAP, and $\beta$ 5 for IG (Borisov et al., 2022).

Sequence attribution exhibits domain-specific variable structure. DeepMTA models customer journeys with event-level inputs $\beta$ 6, uses Phased-LSTM to capture event order, frequency, and time-decay, and then allocates conversion credit with an additive explanation model $\beta$ 7. Its conversion predictor was reported as achieving $\beta$ 8 accuracy in the abstract, while the detailed evaluation reports AUC $\beta$ 9 on eBay data (Yang et al., 2020). Probabilistic attribution for LLMs instead computes exact prompt-token influence through the response likelihood ratio and couples it with conditional entropies. Across eight models and seven prompts, near-zero attribution often co-occurred with low contextual entropy, whereas large positive or negative attribution tended to coincide with higher entropy and larger KL divergences (Shilpika et al., 20 May 2026).

A distinct attribution mode analyzes latent atmospheric foundation models. In AuroraSmallPretrained, layer-wise relevance propagation with $X_j$ 0 showed relevance concentrated on surface wind, 2 m temperature, and mean sea-level pressure during the Great Storm of 1987, with vertically coherent relevance peaking near the upper troposphere around $X_j$ 1 hPa; masking the top $X_j$ 2 most relevant pixels degraded MSL-pressure forecasts $X_j$ 3 more than random masking (Kasteleyn et al., 24 Jun 2026).

5. Causal mechanism attribution and coalition structure

Causal change attribution replaces feature scores with mechanism substitutions. For two environments with shared DAG but different mechanisms, the joint KL divergence decomposes as

$X_j$ 4

so contributions are non-negative and sum to total joint change (Budhathoki et al., 2021). When the target is not the full joint but a marginal or functional, hybrid distributions $X_j$ 5 are combined with Shapley averaging over mechanism replacements. A multiply-robust extension estimates these counterfactual functionals by combining nested regressions and density-ratio reweighting, with consistency and asymptotic normality under partial nuisance misspecification; the method is implemented as part of DoWhy (Quintas-Martinez et al., 2024).

Coalition attribution addresses a separate inconsistency: the attribution of a coalition $X_j$ 6 need not equal the sum of attributions of variables in $X_j$ 7. Using Harsanyi AND and OR interactions, the coalition value is defined as

$X_j$ 8

The conflict arises exactly from interactions that overlap the coalition only partially, and vanishes when such partial-overlap interactions are absent (Zheng et al., 2023). This mechanism-level explanation parallels the earlier axiomatic result that Aumann–Shapley and Shapley–Shubik coincide if and only if the characteristic function is the sum of a multilinear and an additive function (Sun et al., 2011).

This suggests a sharp distinction between additive attribution and interaction-aware attribution. Whenever the underlying system contains interaction terms, allocation depends not only on variable relevance but also on the rule used to split joint effects.

6. Robustness, evaluation, and recurring controversies

A recurrent controversy is faithfulness. The 2025 theoretical review argues that faithfulness is difficult to evaluate because attribution methods are heterogeneous, many lack solid theoretical underpinning, and empirical benchmarks lack accepted ground truth (Deng et al., 11 Aug 2025). The same review summarizes that nonnegative backpropagation families such as LRP- $X_j$ 9, DTD, Deconv, GBP, and RectGrad can violate output and parameter sensitivity, with explanations drifting toward near rank-1 directions that are largely independent of output class and later-layer parameters (Deng et al., 11 Aug 2025). By contrast, RLE identifies a different failure mode: independent masking often produces out-of-distribution samples, so it perturbs only global arrangement while preserving local content (Borisov et al., 2022).

Robustness is equally central in climate applications. Anchor regression only guarantees robustness to the class of shift interventions aligned with chosen anchors; true independence of residuals from all non-target forcings is not claimed, and observational deployment is explicitly left for future work in the CMIP5 proof-of-concept (Székely et al., 2019). The CMIP6 extension distinguishes linear-anchor decorrelation from nonlinear-anchor mean independence, but still assumes stable mechanisms and source-node anchors (Székely et al., 2022).

Data-driven attribution can also be undermined by confounding and structural change. In the lightning-strike study, two problems are emphasized: unknown confounding variables influencing strike frequency, and the possibility that climate change may lead to qualitatively different climate patterns with a different strike–temperature relationship (Webster, 2015). The study therefore standardizes within month and distinguishes an expected-attribution scheme from an event-proportional scheme whose expected attributed claims exactly match the expected increase in claims, a property the paper states is not shared by the epidemiological fraction (Webster, 2015).

Fairness-oriented attribution introduces another limitation: absence of gold truth. The LLM bias framework measures attribution through four Weiner causes, defines internal and external probabilities as $P(X_j\mid Pa_j)$ 0 and $P(X_j\mid Pa_j)$ 1, and studies $P(X_j\mid Pa_j)$ 2 and $P(X_j\mid Pa_j)$ 3 across approximately 140k prompts in ten societal domains (Raj et al., 28 May 2025). Yet the paper explicitly notes that attributions lack objective correctness, that the four-cause menu narrows attribution bandwidth, and that identity proxies via names may carry tokenization or frequency confounds (Raj et al., 28 May 2025).

Across these literatures, the common technical lesson is that attribution quality depends on how variables are defined, which counterfactual or marginalization operator is used, how interactions are handled, and whether the allocation rule is robust under distribution shift. A plausible implication is that the most defensible attribution schemes are those that make their variable semantics, intervention class, and completeness criterion explicit rather than treating attribution as a purely visual or heuristic post-processing step.