Sequential E-Process Methods

• Sequential e-processes are nonnegative, adapted processes that enable anytime-valid inference by leveraging optional stopping and martingale properties.
• They are constructed using predictive multiplicative updates and applied in areas like multiple testing, changepoint detection, randomized trials, and agent monitoring.
• Recent advances address filtration dependencies, control of false discovery rates through adjusted running maxima, and asymptotic optimality under various testing regimes.

Sequential e-processes are nonnegative evidence processes designed for anytime-valid inference. In the standard formulation, a process $(E_t)_{t\ge 0}$ with $E_0=1$ is an e-process for a null $H_0$ if, for every stopping time $\tau$, $\mathbb E_P[E_\tau]\le 1$ for every $P\in H_0$. This yields the time-uniform guarantee

$$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$

so rejection can be based on threshold crossing rather than on a pre-specified analysis time. Recent work has expanded this object well beyond single-stream optional-stopping robustness, developing sequential e-process methodology for multiple testing, changepoint detection, randomized trials, agent monitoring, and asymptotic regimes in which exact e-values are unavailable (Tavyrikov et al., 31 Jan 2025, Massiani et al., 21 Apr 2026).

1. Definition, validity, and the role of filtration

An e-variable is a nonnegative random variable $E$ satisfying $\mathbb E[E]\le 1$ under the null. An e-process is its sequential analogue: a nonnegative adapted process such that every stopped value remains an e-variable. One equivalent formulation is that, for every null law $P$, the process is dominated by a nonnegative supermartingale with initial value $E_0=1$0. In this sense, e-processes formalize optional-stopping-robust evidence accumulation rather than fixed-sample evidence summaries (Wang et al., 12 Feb 2025).

The filtration is part of the definition. If $E_0=1$1 is an e-process on a filtration $E_0=1$2, then “$E_0=1$3 is an e-value” is guaranteed only for $E_0=1$4 that are stopping times with respect to $E_0=1$5. This filtration-relativity is routine in single-hypothesis sequential testing, but it becomes decisive in multi-stream settings, where local and global filtrations need not coincide (Wang et al., 12 Feb 2025).

In the finite-horizon setting, the same logic appears through test martingales. A nonnegative martingale with initial value $E_0=1$6 is an e-process by optional stopping, and an e-process is strictly stronger than a single terminal e-value because it remains valid under arbitrary bounded stopping times. This connection is central to the betting interpretation of sequential evidence: one monitors a capital process rather than a single terminal statistic (Vovk et al., 2020).

2. Canonical constructions and complete-class results

A basic construction starts from sequential e-values $E_0=1$7 satisfying

$E_0=1$8

Given a gambling system $E_0=1$9, the associated game martingale is

$H_0$0

Any such martingale merger is an se-merging function, and every se-merging function is dominated by a martingale merging function. More strongly, adapted, anytime-valid, precise constructions based on sequential e-values coincide with game martingales. In this finite-horizon sequential-merging setting, martingale betting is therefore not merely a convenient construction; it is the complete admissible class (Vovk et al., 2020).

A second complete-class result concerns conditional nonparametric hypotheses defined by finitely many conditional constraints. For

$H_0$1

every e-process is pointwise dominated by a predictable product of affine one-step e-variables. The one-step factors are

$H_0$2

and the dominating process has the form

$H_0$3

For this class of hypotheses, arbitrary e-processes can therefore be replaced without loss by test supermartingales, and sequential design reduces to choosing predictable coefficients $H_0$4 in a finite-dimensional action set (Clerico, 4 Jun 2026).

Taken together, these results identify a recurring structural theme: although e-processes are more general than supermartingales in full generality, large and practically important sequential-testing classes admit complete descriptions by predictable multiplicative updates.

3. Multiple testing, running suprema, and stopped e-BH

In multiple testing with parallel e-processes $H_0$5, pointwise application of e-BH to the current vector $H_0$6 yields fixed-time FDR control, but it is not “carefree” in a genuinely sequential sense. A hypothesis rejected at time $H_0$7 may no longer be rejected at time $H_0$8 after collecting more data for one or more unrelated streams. The proposed sequential target is therefore FDR based on running suprema,

$H_0$9

This is the paper’s FDR-sup criterion, and it yields an accept-to-reject monotonicity property: the rejection set is nondecreasing in time. However, raw running maxima are generally not e-variables, and e-BH applied directly to $\tau$0 does not in general control FDR-sup under arbitrary dependence. The correction is to pass each running maximum through an admissible adjuster $\tau$1 satisfying

$\tau$2

so that $\tau$3 becomes an e-process. The main theorem then states that e-BH applied to adjusted running maxima controls FDR-sup at level $\tau$4. Concrete admissible adjusters include

$\tau$5

(Tavyrikov et al., 31 Jan 2025).

A distinct subtlety arises when e-BH is stopped at a global stopping time. In a multi-stream problem, each stream often has its own local filtration $\tau$6, while the analyst stops using the global filtration

$\tau$7

An adaptively stopped local e-process is an e-value only for local stopping times; this does not automatically imply validity at global stopping times. The resulting “stopped e-BH procedure” can therefore fail under arbitrary dependence. A sufficient condition for globalization is the causal condition

$\tau$8

under which standard products of one-step local e-values become nonnegative supermartingales on the global filtration, and stopped e-BH regains finite-sample, nonasymptotic FDR control for any global stopping time. When that condition is doubtful, the paper also gives a filtration-agnostic fallback via adjusters applied to running maxima (Wang et al., 12 Feb 2025).

These developments show that “sequential multiple testing with e-processes” is not merely fixed-time e-BH rerun at successive looks. The correct object is evidence accumulated over time and across streams in a way that respects monotonicity, filtration, and dependence.

4. Major instantiations

In randomized clinical trials, the randomization e-process (e-RT) constructs a patient-by-patient wealth process

$\tau$9

with updates

$\mathbb E_P[E_\tau]\le 1$0

where $\mathbb E_P[E_\tau]\le 1$1 is the known randomization probability and $\mathbb E_P[E_\tau]\le 1$2 is chosen after observing the current outcome but before revealing treatment assignment. Under the sharp null of no treatment effect, $\mathbb E_P[E_\tau]\le 1$3, the conditional expected multiplier is $\mathbb E_P[E_\tau]\le 1$4, so $\mathbb E_P[E_\tau]\le 1$5 is a test martingale and rejection at $\mathbb E_P[E_\tau]\le 1$6 is anytime-valid. In simulations with $\mathbb E_P[E_\tau]\le 1$7, type I error was between $\mathbb E_P[E_\tau]\le 1$8 and $\mathbb E_P[E_\tau]\le 1$9, e-RT power was about $P\in H_0$0 for designs built for $P\in H_0$1 power and about $P\in H_0$2 for designs built for $P\in H_0$3 power, and the median crossing occurred between $P\in H_0$4 and $P\in H_0$5 of full enrollment (Zampieri, 4 Dec 2025).

In agent verification, verifier-score prefixes $P\in H_0$6 are turned into sequential evidence through the density-ratio process

$P\in H_0$7

where $P\in H_0$8 is the score-sequence law on successful trajectories and $P\in H_0$9 is the corresponding law on unsuccessful trajectories. Under the null $$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$0, $$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$1 is a test martingale with

$$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$2

The practical method estimates the ratio with a classifier,

$$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$3

and calibrates either the theory-motivated threshold $$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$4 or a PAC threshold based on $$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$5 over held-out successful trajectories (Sadhuka et al., 2 Dec 2025).

In sequential change detection, the primary object is an e-detector rather than a single e-process. If $$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$6 is an $$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$7-process started at candidate changepoint time $$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$8, then the Shiryaev–Roberts and CUSUM-style e-detectors are

$$P\!\left(\sup_{t\in\mathbb N} E_t \ge \frac1\alpha\right)\le \alpha,$$9

An e-detector satisfies

$E$0

under the no-change model, so thresholding at $E$1 yields average run length at least $E$2. This framework recovers classical likelihood-based CUSUM and Shiryaev–Roberts procedures in parametric settings and extends them to nonparametric composite pre-change classes (Shin et al., 2022).

In stratified count data, per-block conditional e-variables $E$3 are multiplied into cumulative e-processes

$E$4

yielding sequential tests of the global null

$E$5

The same machinery is inverted into anytime-valid confidence sequences. The framework allows adaptive block sizes, arbitrary stratum arrival patterns, convex mixtures across strata, switching, and “cross-talk” in which alternative-side estimators for one stratum borrow information from others while preserving e-validity (Turner et al., 2023).

5. Asymptotic, approximate, and optimal e-processes

Exact finite-sample e-processes are not always available. Asymptotic e-processes address this by introducing a doubly indexed process $E$6, where $E$7 is an approximation index and $E$8 is monitoring time. For a horizon sequence $E$9, a uniformly strongly $\mathbb E[E]\le 1$0-asymptotic e-process satisfies

$\mathbb E[E]\le 1$1

This yields the asymptotic Ville inequality

$\mathbb E[E]\le 1$2

For cumulative-product constructions with approximation error $\mathbb E[E]\le 1$3, one may take any horizon $\mathbb E[E]\le 1$4 such that

$\mathbb E[E]\le 1$5

The framework formalizes sequential validity when e-values are available only approximately, for example because of nuisance estimation or model misspecification (Massiani et al., 21 Apr 2026).

A converse optimality theory starts from sequential tests rather than from e-processes. Given valid level-indexed stopping times $\mathbb E[E]\le 1$6 with $\mathbb E[E]\le 1$7, the paper defines a WAIT e-process,

$\mathbb E[E]\le 1$8

with budget $\mathbb E[E]\le 1$9 and profile

$P$0

If

$P$1

and

$P$2

then

$P$3

Thus asymptotically optimal sequential tests can be aggregated into asymptotically log-optimal e-processes; full log-optimality corresponds to $P$4 (Ram et al., 12 May 2026).

A complementary result studies wealth processes of the form

$P$5

under composite alternatives. If the process satisfies a deterministic sublinear portfolio regret bound

$P$6

then for every alternative $P$7,

$P$8

The associated rejection time

$P$9

satisfies the matching asymptotic bound

$E_0=1$00

This links betting regret, Kelly-style log growth, and first-order optimal sequential testing (Waudby-Smith et al., 3 Apr 2025).

6. Tradeoffs, misconceptions, and open directions

Several recurring misconceptions are explicitly ruled out by the recent literature. Pointwise-in-time FDR control is not the same as order-invariant sequential multiple testing, and local optional-stopping validity is not the same as global optional-stopping validity. In multi-stream settings, “an adaptively stopped e-process is an e-value” can fail once stopping depends on a larger filtration. Likewise, in agent monitoring, marginal calibration of a verifier score does not by itself control false alarm rate, and still less the probability of ever crossing a threshold over time (Tavyrikov et al., 31 Jan 2025, Sadhuka et al., 2 Dec 2025).

Conservativeness is a second recurring theme. Adjusting running maxima restores FDR-sup control but costs power, sometimes appreciably. In clinical trials, e-RT is a conservative, assumption-free complement to model-based sequential analyses and is usually less powerful than model-based methods when parametric assumptions are credible. In approximate settings, asymptotic e-processes are only valid up to a horizon $E_0=1$01 tied to approximation quality, rather than on an unrestricted infinite horizon at fixed $E_0=1$02 (Tavyrikov et al., 31 Jan 2025, Zampieri, 4 Dec 2025, Massiani et al., 21 Apr 2026).

Current scope limitations are also explicit. The randomization e-process has been developed for binary outcomes; extension to continuous endpoints is suggested but not yet developed, and application to time-to-event outcomes is unclear and under development. The conditional complete-class theorem for finitely many conditional constraints does not cover hypotheses like conditional sub-Gaussianity that naturally involve infinitely many inequalities. Estimated density-ratio procedures for agent monitoring are not exact e-processes, and long trajectories can stress estimation quality (Zampieri, 4 Dec 2025, Clerico, 4 Jun 2026, Sadhuka et al., 2 Dec 2025).

These limitations do not weaken the central methodological conclusion. Sequential e-processes provide a unified language for evidence accumulation under optional stopping, but valid deployment requires careful attention to filtration, dependence, lookback calibration, and approximation error. The recent theory shows both where naive transplants from fixed-time inference fail and how principled constructions—game martingales, affine one-step factors, adjusted running maxima, e-detectors, asymptotic horizons, and regret-optimal wealth processes—repair those failures within a coherent sequential framework.