E-Scores: Statistical, Decision & Predictive Methods

• E-scores are rigorously defined scoring functions based on e-values that quantify statistical evidence while controlling for Type I error even under optional stopping.
• They enable adaptive multiple hypothesis testing by transforming and aggregating p-values into e-values, ensuring effective false discovery control without strict dependency assumptions.
• Applications span conformal prediction, generative model evaluation, multi-criteria decision making, and educational forecasting, offering interpretable and prescriptive insights.

An E-score is a general term encompassing a suite of rigorous, theoretically motivated scoring functions arising in diverse settings, including statistical evidence assessment, uncertainty quantification, educational forecasting, multi-criteria decision making, and generative model evaluation. The defining property shared by contemporary E-score frameworks is a statistical or algorithmic construction grounded in e-values—quantities with controlled expectation under a null or reference model—or robust interval assignment based on formal outranking relations.

1. E-Values and Statistical E-Scores

E-values, central to modern E-score methodologies, are nonnegative random variables $E$ satisfying $\mathbb{E}_0[E] \le 1$ under a null hypothesis $H_0$ (Wang et al., 2020). Key examples include likelihood ratios, Bayes factors for simple nulls, and stopped betting martingales. The realized value $e$ can be directly interpreted as the multiplicative betting gain under $H_0$.

E-scores constructed from e-values exhibit several crucial properties:

• Evidence Scaling: Larger e-scores indicate stronger statistical evidence against the null.
• Robustness to Optional Stopping: Type I error is not inflated by data-driven or sequential stopping, in contrast to $p$-values.
• Multiplicativity and Adaptivity: Independent e-values multiply, supporting evidence accumulation across datasets or hypotheses.
• Dependence Robustness: Validity holds under arbitrary dependence as expectation control is the only required condition.

These properties provide the backbone for e-score–based approaches in multiple testing (e.g., in the e-Benjamini–Hochberg (e-BH) procedure) and uncertainty quantification in predictive modeling (Wang et al., 2020, Alami et al., 7 Dec 2025).

2. E-Scores in Multiple Hypotheses Testing

The e-BH procedure generalizes classical multiple testing frameworks by replacing $p$-values with e-scores (Wang et al., 2020). For $K$ hypotheses with e-values $e_1,\ldots,e_K$ and FDR target $\alpha$, hypotheses are sorted and rejected up to the largest $\mathbb{E}_0[E] \le 1$0 with $\mathbb{E}_0[E] \le 1$1. This thresholding is self-consistent and provides FDR control at level $\mathbb{E}_0[E] \le 1$2 under no assumptions on the joint dependency structure of the e-scores.

Compared to $\mathbb{E}_0[E] \le 1$3-value–based methods, e-score procedures:

• Eliminate the need for independence or positive dependence assumptions.
• Permit post-selection and structured inference, as arbitrary monotone rejection sets based on e-scores preserve error control.
• Enable sequential and adaptive testing, including applications to multi-armed bandits, where per-arm e-scores are combined at arbitrary stopping times without correction.

Furthermore, p-values can be calibrated into e-values via functions $\mathbb{E}_0[E] \le 1$4 with controlled expectation, unifying the $\mathbb{E}_0[E] \le 1$5- and $\mathbb{E}_0[E] \le 1$6-score frameworks.

3. E-Scores for Conformal Prediction and Uncertainty Aggregation

In conformal prediction and ensemble model aggregation, e-scores arise by transformation of nonconformity or anomaly scores into normalized e-values (Alami et al., 7 Dec 2025). The Symmetric Aggregated Conformal Prediction (SACP) framework demonstrates this approach for combining uncertainty across predictors:

• Each model's nonconformity scores are converted to e-values through a normalization exploiting exchangeability.
• Symmetric aggregation functions (arithmetic, geometric, harmonic mean, product, $\mathbb{E}_0[E] \le 1$7-sum) combine per-model e-values into an ensemble e-score for each candidate prediction.
• The resulting aggregated e-scores enable construction of prediction sets with guaranteed marginal coverage, achieved by empirical quantile thresholding.
• Experimental benchmarks show that symmetric e-score aggregation yields tighter and more efficient uncertainty sets compared to alternative conformal aggregation strategies, with maintained coverage rates across diverse datasets.

This approach leverages the closure properties of e-values and the symmetric exchangeability in model selection, extending to both regression and classification.

4. E-Scores for Output (In)Correctness in Generative Models

Recent advances address the evaluation of output correctness in generative models, especially LLMs, using e-scores computed from e-value constructions (Dhillon et al., 29 Oct 2025). The methodology targets major limitations of $\mathbb{E}_0[E] \le 1$8-score–based conformal set filtering, notably the lack of post-hoc validity when users adapt the significance threshold after inspecting the scores ("p-hacking").

For a set of model outputs, an oracle estimator $\mathbb{E}_0[E] \le 1$9 predicts correctness. Scores are computed as:

$H_0$0

where $H_0$1 is a monotonic transform of $H_0$2, and $H_0$3 aggregates the worst-case "incorrect" score on calibration samples. The e-score for $H_0$4 is $H_0$5. The key theoretical guarantee is that for any (even post-hoc chosen) threshold $H_0$6,

$H_0$7

a post-hoc size-distortion control not shared by p-scores. Applications to mathematical factuality and property-constrained LLM outputs demonstrate that e-scores preserve selectivity while closing the "p-hacking" loophole. The use of e-scores thus ensures the statistical validity of filtering procedures under adaptively chosen tolerance levels.

5. E-Scores in Multi-Criteria Decision Analysis

The ELECTRE-Score methodology (also referenced as "E-score") assigns interval-valued scores to alternatives based on formal outranking relations among reference sets, departing from compensatory additive models such as Multi-Attribute Value Theory (MAVT) (Figueira et al., 2019). The method proceeds as follows:

• For alternatives $H_0$8 and criteria $H_0$9 (with possible imprecision), define pairwise outranking $e$0 using per-criterion thresholds for indifference, strict preference, and veto.
• Reference sets of limiting profiles are constructed and ordered, each accorded a score via a deck-of-cards protocol reflecting the decision-maker's perceived gaps between reference bands.
• For each alternative, lower and upper bounds of its E-score interval are determined as the maximal reference score it outranks and the minimal one that outranks it, respectively.
• E-scores are thus robust intervals $e$1 rather than precise cardinal values, mitigating overcompensation and enhancing robustness to parameter or data perturbations.

This non-compensatory, outranking-based E-score paradigm supports decision contexts with qualitative or uncertain data and provides theoretical guarantees including monotonicity, stability, and conformity.

6. E-Scores in Educational Prediction: The Embibe Score Quotient

In the Embibe Score Quotient (ESQ) framework, E-scores refer to the predicted next-test score of a learner, computed from a high-dimensional summary of academic, behavioral, effort, and test-taking features (Donda et al., 2020):

• Feature engineering encompasses Bayesian Knowledge Tracing outputs, DeepFM concept mastery embeddings, and fine-grained behavioral signals.
• Sequential RNNs or bucketed random forests generate test score predictions, with the RNN achieving median absolute error of 4.57% and Pearson correlation of $e$2.
• Shapley values calculated per prediction enable actionable, individualized feedback by attributing score contributions to academic, effort, behavioral, and test-taking features.
• Quantile regression provides calibrated prediction intervals, functioning as confidence gauges for decision support.
• The framework supports counterfactual what-if analyses, enabling prescriptive nudges by inversely optimizing features for a target score increment.

The Embibe Score Quotient operationalizes E-scores in a production educational platform, unifying prediction, interpretation, uncertainty quantification, and prescriptive analytics.

7. Interpretations, Strengths, and Theoretical Guarantees

E-scores, as instantiated in the above settings, are distinguished by the following features:

• Expectation Control: E-scores are designed to control first moments or produce interval bands, with rigorous guarantees against error inflation, even under complex dependence (e.g., e-BH FDR control (Wang et al., 2020), e-score post-hoc error control (Dhillon et al., 29 Oct 2025), ELECTRE interval stability (Figueira et al., 2019)).
• Non-Compensatory and Robust: Outranking-based E-scores do not allow poor performance on one criterion to be offset by high performance elsewhere (Figueira et al., 2019).
• Adaptivity: E-score methodologies retain their guarantees under adaptive, post-hoc thresholding, unlike conventional $e$3-value or value-function approaches (Dhillon et al., 29 Oct 2025).
• Interpretability and Prescriptiveness: Modern applications, as in ESQ, leverage feature attribution for actionable guidance (Donda et al., 2020).
• Unified Framework: Both $e$4-based and e-based multiple testing procedures are subsumed via calibration, illustrating the generality of the e-score principle (Wang et al., 2020).

A plausible implication is that e-value–driven E-score methodologies are likely to see further adoption across fields requiring robust, adaptive, and interpretable decision-support or uncertainty quantification.

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References

• (Wang et al., 2020) "False discovery rate control with e-values"
• (Alami et al., 7 Dec 2025) "Symmetric Aggregation of Conformity Scores for Efficient Uncertainty Sets"
• (Dhillon et al., 29 Oct 2025) "E-Scores for (In)Correctness Assessment of Generative Model Outputs"
• (Figueira et al., 2019) "Electre-Score: A first outranking based method for scoring actions"
• (Donda et al., 2020) "A framework for predicting, interpreting, and improving Learning Outcomes"