E-value in Statistical Inference

• E-value is a nonnegative statistic defined so that its expected value under any null hypothesis is at most one, providing a unified framework for quantifying evidence.
• Constructed via likelihood ratios, Bayes factors, or p-value calibrators, E-values generalize classical evidence measures and support robust sequential and meta-analytic testing.
• E-values enable valid combination across studies and flexible error control in multiple testing, making them essential for sensitivity analysis and adaptive inference.

An E-value is a nonnegative statistic $E$ with the property that for every probability distribution $P$ in a null hypothesis set $\mathcal{P}$, the expectation satisfies $\mathbb{E}_P[E] \leq 1$. E-values are a central concept in modern statistical inference, providing a unified framework for evidence quantification, hypothesis testing, and robustness analysis, and admit both static (single-sample) and sequential (e-process) forms. E-values generalize classical likelihood ratios and Bayes factors for hypothesis evaluation, are optimally combinable across studies (even under dependence), and underpin error control under flexible stopping and loss structures.

1. Fundamental Definition and Mathematical Principles

E-values (or e-variables) are defined as follows: Given a null hypothesis $H_0$ (often a composite of probability distributions on a measurable space), a nonnegative random variable $E$ is an e-value for $H_0$ if for every $P \in H_0$,

$\mathbb{E}_P[E]\leq 1.$

This ensures that, under any null scenario, observing a large value $e \gg 1$ is rare and provides direct evidence against $P$0 (Ramdas et al., 2024, Chugg et al., 25 Mar 2026). Markov's inequality provides the key operational guarantee: $P$1 for any $P$2. The e-value thus quantifies the maximum expected multiplicative evidence against the null that a statistician or "bettor" can accrue—an interpretation formalized as the "betting interpretation."

E-values are generalized beyond simple hypotheses: for composite nulls, the requirement is that $P$3.

2. Operational Properties and Relationship to Other Evidence Measures

E-values combine three key features of major inferential quantities:

• Likelihood Ratios: For simple-vs-simple hypotheses, the likelihood ratio $P$4 is an e-value under $P$5 (Chugg et al., 25 Mar 2026).
• Bayes Factors: For simple nulls, the Bayes factor is an e-variable; for composites, Bayes factor e-values coincide with mixtures or "universal" e-values under certain priors (Vovk et al., 2019, Chugg et al., 25 Mar 2026).
• p-values: A p-value is probability-calibrated: $P$6. E-values are expectation-calibrated, but a simple mapping connects the two: $P$7 is a valid p-value if $P$8 is an e-value, and e-values can be constructed from p-values via integrable "calibrators" $P$9 with $\mathcal{P}$0 (Ramdas et al., 2024, Vovk et al., 2019).

E-values are post hoc valid: the rejection threshold can be chosen after seeing the data, supporting "roving $\mathcal{P}$1" testing robust to post-hoc threshold choices (Grünwald, 2022).

Table: Comparison of E-values, p-values, and Bayes Factors

Quantity	Calibration Criterion	Robust to Optional Stopping?	Multiplicative Combination Valid?
p-value	$\mathcal{P}$2	No	No
e-value	$\mathcal{P}$3	Yes	Yes (via product/averaging)
Bayes factor	$\mathcal{P}$4 (simple $\mathcal{P}$5)	Yes (simple only)	Yes (special cases)

3. Core Methodologies: Construction and Combination

Construction: E-values are constructed as likelihood ratios, Bayes factors, supermartingales (stopped or sequential), or transformations/calibrations of p-values (Vovk et al., 2019, Chugg et al., 25 Mar 2026). In maximum entropy models, growth-rate optimal (GRO) e-values take closed forms and can be constructed for microcanonical and canonical tests (Giuffrida et al., 1 Sep 2025). For robust evidence under composite nulls, universal inference and method-of-mixtures yield valid e-processes.

Combination: E-values can be multiplied across independent studies or sequential data batches, and combined by convex averaging or arithmetic mean under arbitrary dependency, maintaining validity for the global null (Vovk et al., 2019). This allows e-values to underpin sequential tests (e-processes) and meta-analytic frameworks, and to scale to large multiple-testing settings (Wang et al., 2020, Ignatiadis et al., 2022).

4. Advanced Applications: Sensitivity Analysis and Multiple Testing

Sensitivity to Unmeasured Confounding

The "E-value" in epidemiological sensitivity analysis is defined as the minimal association on the risk ratio scale ($\mathcal{P}$6) that an unmeasured confounder would need with both exposure and outcome to explain away the observed causal effect. The original E-value (VanderWeele & Ding) is

$\mathcal{P}$7

where $\mathcal{P}$8 is the lower 95% confidence limit (or $\mathcal{P}$9 if point estimate $\mathbb{E}_P[E] \leq 1$0) (McGowan et al., 2020). Extensions contextualize this with observed covariate E-values: $\mathbb{E}_P[E] \leq 1$1 where $\mathbb{E}_P[E] \leq 1$2 is the limiting bound ratio omitting a covariate. This duality allows comparison of hypothetical unmeasured confounding to real measured covariates, visualized via "observed bias plots" (McGowan et al., 2020).

Recent work extends E-values to time-varying confounder settings, defining the per-occasion E-value as

$\mathbb{E}_P[E] \leq 1$3

for $\mathbb{E}_P[E] \leq 1$4 time points, revealing that distributing confounding over time increases study vulnerability compared to the single-timepoint E-value (Sium, 27 Feb 2026).

Multiple Testing

E-values support advanced multiple testing procedures under arbitrary dependence:

• The e-BH procedure extends Benjamini–Hochberg for False Discovery Rate (FDR) control, directly applying the rejection rule to e-values, with no dependence correction needed (Wang et al., 2020, Ramdas et al., 2024).
• e-value closed testing provides strong familywise error rate (FWER) control, including weighted and graphical Bonferroni strategies, leveraging e-value averaging and dynamic programming for efficiency (Hartog et al., 15 Jan 2025).
• Discovery e-matrix and e-GWGS approaches provide simultaneous confidence bounds for the number of true discoveries in any subset, using arithmetic averaging and partial-sum algorithms (Vovk et al., 2019).

No normalization of e-value weights is required for FDR control if e-values are independent of the associated p-values, enabling substantial power gains in high-throughput testing (Ignatiadis et al., 2022).

5. Sequential Inference, Anytime-Valid Testing, and Conformal Prediction

E-values admit powerful sequential analogues ("e-processes"). For a sequence $\mathbb{E}_P[E] \leq 1$5 adapted to increasing data granularity, if $\mathbb{E}_P[E] \leq 1$6 has the supermartingale property under all $\mathbb{E}_P[E] \leq 1$7, validity is assured at arbitrary stopping times (Ville's inequality), supporting anytime-valid inference: $\mathbb{E}_P[E] \leq 1$8 This robustness is critical in adaptive clinical trials (Sokolova et al., 6 Feb 2026), classifier two-sample tests (E-C2ST) (Pandeva et al., 2022), and conformal prediction, where batch-anytime e-prediction, data-dependent coverage, and martingale-based conformal sets enable new guarantees that p-value methods cannot provide (Gauthier et al., 17 Mar 2025).

6. Bayesian E-values and the FBST

The Full Bayesian Significance Test (FBST) defines an e-value as the posterior probability of a "tangential set" of parameter values less surprising than the best-fitting on the hypothesis, using a reference density $\mathbb{E}_P[E] \leq 1$9: $H_0$0 where $H_0$1, and $H_0$2 (Stern et al., 2020, Stern et al., 2022). The FBST maintains the likelihood principle, finite-sample exactness, invariance under reparametrization, and is logically compositional—facilitating coherent inference for sharp nulls, unions, intersections, and epistemic logic of evidence.

7. Limitations and Open Challenges

E-values may entail conservativeness (the bound $H_0$3 in some models), and explicit log-optimal e-values can be intractable for complex or high-dimensional parametric nulls (Ramdas et al., 2024). Theoretical and computational research continues into

• generic, high-power e-value construction in irregular/nonparametric settings (Grünwald, 2022);
• further integration with Bayesian/posterior frameworks;
• empirical comparison under adaptive trial, meta-analysis, and data-driven weighting regimes.

• Contextualizing E-values for Interpretable Sensitivity to Unmeasured Confounding Analyses (McGowan et al., 2020)
• Testing maximum entropy models with e-values (Giuffrida et al., 1 Sep 2025)
• E-Valuating Classifier Two-Sample Tests (Pandeva et al., 2022)
• E-Values Expand the Scope of Conformal Prediction (Gauthier et al., 17 Mar 2025)
• The e-value: A Fully Bayesian Significance Measure for Precise Statistical Hypotheses and its Research Program (Stern et al., 2020)
• Confidence and discoveries with e-values (Vovk et al., 2019)
• Family-wise Error Rate Control with E-values (Hartog et al., 15 Jan 2025)
• Bayesian Sensitivity Analysis for Missing Data Using the E-value (Xue et al., 2021)
• E-values: Calibration, combination, and applications (Vovk et al., 2019)
• E-values as statistical evidence: A comparison to Bayes factors, likelihoods, and p-values (Chugg et al., 25 Mar 2026)
• False discovery rate control with e-values (Wang et al., 2020)
• Valid sequential inference on probability forecast performance (Henzi et al., 2021)
• Hypothesis testing with e-values (Ramdas et al., 2024)
• Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha (Grünwald, 2022)
• Quantifying Robustness to Unmeasured Confounding in Time-Varying Treatment Confounder Settings: An Extension of E-value Approach (Sium, 27 Feb 2026)
• E-values as unnormalized weights in multiple testing (Ignatiadis et al., 2022)
• E-values for Adaptive Clinical Trials: Anytime-Valid Monitoring in Practice (Sokolova et al., 6 Feb 2026)
• The e-value and the Full Bayesian Significance Test: Logical Properties and Philosophical Consequences (Stern et al., 2022)