Can an e-value exceed one?

Yes; large values are the evidence-bearing end of its scale.

Is twenty always a universal scientific threshold?

No; it corresponds to alpha = 0.05 for the basic Markov-calibrated rule, while scientific decisions need context.

Can valid e-values be averaged under dependence?

Fixed-weight averaging for the same null preserves the expectation bound; other combinations need separate justification.

AI Glossary / Statistics and probability / Advanced

E-values

Definition

E-values are nonnegative statistical evidence values obtained from random variables whose expected value is at most one under every distribution allowed by the null hypothesis. Large values provide evidence against that null under the specified construction. They offer an alternative framework to p-values and are useful in some sequential and multiple-testing methods. They are not probabilities that a hypothesis is true.

What it means

A p-value is calibrated through probabilities of small values under a null, whereas an e-value is calibrated through an expectation bound. Its scale can extend above one without an upper limit. A value of twenty is not a 20% probability or a claim that an effect is twenty times larger.
The name overlaps with other concepts. In epidemiology, an E-value can refer to sensitivity to unmeasured confounding, and sequence-search software uses expectation values for matches. This entry follows the supplied term's statistical-testing meaning, not those separate quantities.

How it works

Specify the null hypothesis and construct a nonnegative statistic E satisfying E_P(E) <= 1 for every null distribution P. Under a simple null, a likelihood ratio comparing a properly specified alternative distribution with the null provides a basic construction. Composite nulls require validity across the whole null family, not just a convenient fitted member.
Markov's inequality implies P(E >= 1/alpha) <= alpha under the null. Thus a prespecified level-0.05 test can reject when E reaches twenty. For a fixed valid e-value, min(1, 1/E) provides a valid p-value, with the zero case interpreted as one. This calibration is generally conservative and does not make the two concepts identical.
Averaging valid e-values for the same null with fixed nonnegative weights summing to one preserves validity, even under dependence. Multiplication needs additional conditions, such as independence or appropriate sequential conditional validity. Selecting the largest of several values is not generally valid without a suitable procedure.
For continuous monitoring, use a valid e-process or another construction justified at stopping times. A sequence of statistics that is valid separately at each fixed time is not automatically safe to stop when one becomes large. Proper likelihood-ratio martingales illustrate how sequential validity can be obtained.
Evaluate power and practical assumptions. The chosen alternative or betting strategy affects which departures are detected efficiently. E-values do not correct poor measurements, invalid null assumptions, or selective reporting outside the procedure.

Three practical examples

An experimenter monitors a stream using an e-process designed to permit a valid data-dependent stopping decision.
A research team combines separately obtained evidence for the same null using a justified averaging rule.
An AI evaluation studies many hypotheses with an e-value-based false-discovery procedure appropriate to the analysis.

Think of it this way

Think of an e-value as the capital of a carefully constrained statistical betting strategy. If the null is correct, the strategy cannot increase its expected capital beyond the allowed starting amount. Large capital is therefore evidence against the null, but the metaphor only works when the strategy follows the required rules.

Worked example

Consider independent coin tosses. The null specifies head probability 0.5, and a fixed alternative specifies 0.75. A head multiplies the likelihood ratio by 0.75/0.5 = 1.5; a tail multiplies it by 0.25/0.5 = 0.5.
Under the null, the expected multiplier is 0.5 x 1.5 + 0.5 x 0.5 = 1. With independent tosses and the fixed alternative, the product is a nonnegative likelihood-ratio martingale starting at one.
After ten tosses with eight heads and two tails, the e-value is 1.5 to the eighth power times 0.5 squared, approximately 6.40723. This is below the level-0.05 rejection threshold of twenty. After ten heads and no tails, it is 1.5 to the tenth power, approximately 57.66504, above that threshold.
For the ten-head outcome, the reciprocal calibration gives approximately 0.01734. This is a valid calibrated p-value under the stated model, not necessarily the usual exact binomial p-value. The e-process can be monitored with a valid threshold-crossing rule because its sequential construction supports that use; arbitrary repeated testing would need its own justification.
The example targets an excess of heads through the chosen alternative. Choosing the alternative after seeing all results would not preserve the same argument without an additional valid construction.

Illustrative case study

Fictional case: Rowan Experiments wants to inspect an AI feature trial frequently. Analysts reject the idea that any evidence number can be checked repeatedly without consequences. They specify a sequential procedure whose e-process remains valid under the null and define the decision threshold in advance.
The team documents assumptions about the data stream, response delays, and treatment assignment. It also reports the estimated effect and uncertainty rather than using the evidence threshold as a measure of practical benefit. Monitoring becomes more flexible within the defined procedure, while the responsibilities for study design and interpretation remain.

Common misconceptions and mistakes

An e-value is not an effect size, posterior odds, or a probability of the null. Not every Bayes factor for a composite null is an e-value for every member of that null. Multiplying dependent e-values or selecting a favorable one can break validity. A small e-value does not automatically prove the null. Optional-stopping validity requires the appropriate process-level construction, not merely the e-value label.

Business use

E-values can support evidence accumulation and multiple-testing workflows where their calibration and combination properties are useful. Choose them because the method fits the design, not because they guarantee more discoveries. Specify the null, construction, monitoring policy, and interpretation in language the decision maker understands. Pair evidence with effect magnitude, costs, and uncertainty about operational impact.

Check your understanding

What is the null expected multiplier in the coin example?

Answer: One.

Does the eight-head result cross twenty?

Answer: No.

Why can the illustrated product be monitored sequentially?

Answer: Its conditional expected multiplier under the null is one, giving the required martingale structure.

Related terms

P-value
False discovery rate
E-process
likelihood ratio
test martingale
optional stopping
Markov's inequality
multiple testing

Learn next

Derive the likelihood-ratio process for a simple discrete model and check its null expectation. Then compare fixed-time e-values with stopping-time-valid e-processes and study valid combination rules before applying them to several hypotheses.

Stopping time

Sources

Vladimir Vovk and Ruodu Wang, E-values: Calibration, combination, and applications: https://arxiv.org/abs/1912.06116 Ruodu Wang and Aaditya Ramdas, False Discovery Rate Control with E-values, original research: https://academic.oup.com/jrsssb/article/84/3/822/7056146

Previous: E-optimal design Next: Eager learning