---
title: E-value in Statistical Inference
url: https://www.emergentmind.com/topics/e-value
type: topic
---

# E-value in Statistical 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 $H_0$ [2410.23614, 2603.24421]. Markov's inequality provides the key operational guarantee:
\[
\mathbb{P}_P(E \geq 1/\alpha) \leq \alpha
\]
for any $\alpha\in(0,1)$. 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 $\sup_{P\in H_0} \mathbb{E}_P[E] \leq 1$.

## 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 $q(x)/p(x)$ is an e-value under $H_0$ [2603.24421].
- **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 [1912.06116, 2603.24421].
- **p-values**: A p-value is probability-calibrated: $P(P \leq \alpha) \leq \alpha$. E-values are expectation-calibrated, but a simple mapping connects the two: $p = \min(1, 1/E)$ is a valid p-value if $E$ is an e-value, and e-values can be constructed from p-values via integrable "calibrators" $f$ with $\int_0^1 f(p) dp \leq 1$ [2410.23614, 1912.06116].

E-values are **post hoc valid**: the rejection threshold can be chosen after seeing the data, supporting "roving $\alpha$" testing robust to post-hoc threshold choices [2205.00901].

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

| Quantity      | Calibration Criterion                | Robust to Optional Stopping? | Multiplicative Combination Valid?   |
|---------------|-------------------------------------|:----------------------------:|:-----------------------------------:|
| p-value       | $\sup_{P \in H_0} \mathbb{P}(P \leq \alpha) \leq \alpha$ | No                         | No                                 |
| e-value       | $\sup_{P \in H_0} \mathbb{E}_P[E] \leq 1$          | Yes                        | Yes (via product/averaging)        |
| Bayes factor  | $\mathbb{E}_{H_0}[BF] = 1$ (simple $H_0$)         | 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 [1912.06116, 2603.24421]. In maximum entropy models, growth-rate optimal (GRO) e-values take closed forms and can be constructed for microcanonical and canonical tests [2509.01064]. 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 [1912.06116]. This allows e-values to underpin sequential tests (e-processes) and meta-analytic frameworks, and to scale to large multiple-testing settings [2009.02824, 2204.12447].

## 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 ($RR$) 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
\[
\text{E-value} = LB_{\rm obs} + \sqrt{LB_{\rm obs}\,(LB_{\rm obs}-1)}
\]
where $LB_{\rm obs}$ is the lower 95% confidence limit (or $1/UCL_{\rm obs}$ if point estimate $<1$) [2011.07030]. Extensions contextualize this with observed covariate E-values:
\[
\text{E-value}_{OC}(X_j) = R + \sqrt{R\,(R-1)}
\]
where $R = \max(LB_{\rm obs} / LB_{\rm adj}(j), LB_{\rm adj}(j) / LB_{\rm obs})$ 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" [2011.07030].

Recent work extends E-values to time-varying confounder settings, defining the per-occasion E-value as
\[
\text{E-value}_{\text{equal}} = (RR_{\text{obs}})^{1/T} + \sqrt{(RR_{\text{obs}})^{1/T}\,\left[(RR_{\text{obs}})^{1/T} - 1\right]}
\]
for $T$ time points, revealing that distributing confounding over time increases study vulnerability compared to the single-timepoint E-value [2602.24261].

### 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 [2009.02824, 2410.23614].
- **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 [2501.09015].
- **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 [1912.13292].

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 [2204.12447].

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

E-values admit powerful sequential analogues ("e-processes"). For a sequence $(E_t)$ adapted to increasing data granularity, if $(E_t)$ has the supermartingale property under all $P\in H_0$, validity is assured at arbitrary stopping times (Ville's inequality), supporting *anytime-valid* inference:
\[
\mathbb{P}_{H_0}\left( \sup_{t\geq 1} E_t \geq 1/\alpha \right) \leq \alpha
\]
This robustness is critical in adaptive clinical trials [2602.06379], classifier two-sample tests (E-C2ST) [2210.13027], 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 [2503.13050].

## 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 $r(\theta)$:
\[
e(H) = W(s^*) = \int_{T(s^*)} p_n(\theta) d\theta
\]
where $s^* = \sup_{\theta \in H} p_n(\theta)/r(\theta)$, and $T(s^*) = \{\theta: p_n(\theta)/r(\theta) \leq s^*\}$ [2001.10577, 2205.08010]. 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 $\mathbb{P}(E\geq 1/\alpha)\ll \alpha$ in some models), and explicit log-optimal e-values can be intractable for complex or high-dimensional parametric nulls [2410.23614]. Theoretical and computational research continues into 
- generic, high-power e-value construction in irregular/nonparametric settings [2205.00901];
- further integration with Bayesian/posterior frameworks;
- empirical comparison under adaptive trial, meta-analysis, and data-driven weighting regimes.

## References

- Contextualizing E-values for Interpretable Sensitivity to Unmeasured Confounding Analyses [2011.07030]
- Testing maximum entropy models with e-values [2509.01064]
- E-Valuating Classifier Two-Sample Tests [2210.13027]
- E-Values Expand the Scope of Conformal Prediction [2503.13050]
- The e-value: A Fully Bayesian Significance Measure for Precise Statistical Hypotheses and its Research Program [2001.10577]
- Confidence and discoveries with e-values [1912.13292]
- Family-wise Error Rate Control with E-values [2501.09015]
- Bayesian Sensitivity Analysis for Missing Data Using the E-value [2108.13286]
- E-values: Calibration, combination, and applications [1912.06116]
- E-values as statistical evidence: A comparison to Bayes factors, likelihoods, and p-values [2603.24421]
- False discovery rate control with e-values [2009.02824]
- Valid sequential inference on probability forecast performance [2103.08402]
- Hypothesis testing with e-values [2410.23614]
- Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha [2205.00901]
- Quantifying Robustness to Unmeasured Confounding in Time-Varying Treatment Confounder Settings: An Extension of E-value Approach [2602.24261]
- E-values as unnormalized weights in multiple testing [2204.12447]
- E-values for Adaptive Clinical Trials: Anytime-Valid Monitoring in Practice [2602.06379]
- The e-value and the Full Bayesian Significance Test: Logical Properties and Philosophical Consequences [2205.08010]

Source: https://www.emergentmind.com/topics/e-value