arxiv: 2605.12720 · v1 · submitted 2026-05-12 · 🧮 math.ST · math.PR· stat.ML· stat.TH

Recognition: no theorem link

Optimal sequential tests yield log-optimal e-processes

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Pith reviewed 2026-05-14 19:46 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.MLstat.TH

keywords sequential testinge-processesWAIT e-processesasymptotic optimalitylog-optimalityhypothesis testingstopping timesmartingales

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The pith

Asymptotically optimal sequential tests can be aggregated into asymptotically log-optimal e-processes using WAIT e-processes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that asymptotically optimal sequential tests can be converted into asymptotically log-optimal e-processes. This conversion uses a new construction called WAIT e-processes, formed as weighted aggregates of indicators of stopping times. These e-processes start at zero, remain nondecreasing, and grow to infinity at the optimal rate under the alternative hypothesis. A sympathetic reader would care because the result completes a two-way equivalence: optimal tests produce optimal e-processes and vice versa, unifying two perspectives on sequential hypothesis testing while highlighting nuances in how asymptotic optimality is defined for each.

Core claim

It is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. The result proves the converse to the known implication from log-optimal e-processes to optimal tests, completing the equivalence while discussing several nuances in the varied definitions of asymptotic optimality.

What carries the argument

WAIT e-processes, which are weighted aggregates of indicators of stopping times that start at zero, stay nondecreasing, and grow at the optimal rate under the alternative.

If this is right

Thresholding any e-process at 1/alpha produces a level-alpha sequential test.
Log-optimal e-processes produce asymptotically optimal sequential tests via appropriate thresholding.
The WAIT construction preserves the optimal growth rate when aggregating a family of optimal tests.
Compatibility of the two notions of asymptotic optimality is necessary for the equivalence to hold.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Practitioners could design optimal tests first and then automatically obtain log-optimal e-processes for anytime-valid inference.
The approach may extend to composite alternatives or nonparametric settings where stopping-time optimality is easier to characterize than direct e-process growth.
Similar aggregation ideas could apply to other sequential procedures such as confidence sequences or change-point detection.

Load-bearing premise

The asymptotic optimality rates defined for stopping times in sequential tests translate directly into the asymptotic log-growth rates required for e-processes under the alternative.

What would settle it

An explicit counterexample sequence of tests that are asymptotically optimal in stopping time yet whose WAIT aggregation fails to achieve the log-optimal growth rate for the resulting e-process under any fixed alternative.

Figures

Figures reproduced from arXiv: 2605.12720 by Aaditya Ramdas, Ashwin Ram.

**Figure 2.** Figure 2: Finite-horizon size check for the Gaussian one-sided SPRT. The diagonal is the nominal [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** Figure 3: First-passage ratio τb/b under Q. The target is 1/I = 2. Experiment 4: The e-power under the alternative The main experiment here estimates the e-power as 1 t EQ [log{η + (1 − η)Mt}] 25 [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗

**Figure 4.** Figure 4: Analysis of growth metrics under Q: e-power curves (a) and mean log-growth (b). Experiment 5: Finite-time speed among the full-rate schemes In this experiment, we go a bit further by noting that the finite-time approach of the schemes to I may vary significantly. This is because the schemes only share the asymptotic rate I. We compare the gap I − PbM(t), the first grid times at which the full-rate scheme r… view at source ↗

**Figure 5.** Figure 5: Performance analysis of full-rate schedules: e-power gaps (a), reaching times (b), and final [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗

**Figure 6.** Figure 6: Validation of null properties: fixed-time expectations (a) and stopped expectations (b). [PITH_FULL_IMAGE:figures/full_fig_p027_6.png] view at source ↗

**Figure 7.** Figure 7: KL scaling for the weighted dyadic full-rate aggregate. The vertical axis is e-power divided [PITH_FULL_IMAGE:figures/full_fig_p028_7.png] view at source ↗

**Figure 8.** Figure 8: Delayed tests transfer slower base-test speed into slower aggregate [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗

**Figure 9.** Figure 9: Analysis of the weak-optimality counterexample: deterministic branches (a) and final-time [PITH_FULL_IMAGE:figures/full_fig_p029_9.png] view at source ↗

**Figure 10.** Figure 10: Partial validity budgets P k≤K wkαk. All curves stay at or below 1. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_10.png] view at source ↗

read the original abstract

It has been recently shown that e-processes are sufficient for sequential testing in the following sense: every level-$\alpha$ sequential test can be obtained by thresholding an e-process at $1/\alpha$. However, in the above result, neither does the test have to be asymptotically optimal (in terms of stopping times) nor does the e-process have to be asymptotically log-optimal. It has separately been shown that asymptotically log-optimal e-processes yield asymptotically optimal sequential tests. In this paper, we prove the converse, arguably completing the story: it is possible to aggregate asymptotically optimal sequential tests into asymptotically log-optimal e-processes. This is accomplished by using a new class of WAIT e-processes: those that are Weighted Aggregates of Indicators of stopping Times that begin at zero, are nondecreasing and increase to infinity under the alternative at the optimal rate. Importantly, the paper discusses several nuances in the varied definitions of asymptotic (log-)optimality.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper proves the missing converse by turning asymptotically optimal stopping times into log-optimal e-processes via a new WAIT construction.

read the letter

The key point here is that the authors close the circle: if you start with asymptotically optimal sequential tests, you can aggregate them into e-processes that achieve the optimal growth rate under the alternative. They do this with a new class they call WAIT e-processes—weighted aggregates of indicators of those stopping times, starting at zero and nondecreasing to infinity at the right rate. The proof uses standard martingale properties and is a direct construction rather than an existence argument. That unifies the two directions that were previously only one way each. They also spend time sorting out the different flavors of asymptotic optimality that earlier papers left a bit loose, which is useful. The math looks clean on the surface—no obvious circularity or free parameters—and the citation pattern ties back to the relevant e-process and sequential testing literature without padding. For composite alternatives the weighting step could in principle dilute the rate if the individual tests are not uniformly optimal, but the paper claims to handle the nuances in the definitions so that the envelope rate is recovered; the abstract sketch supports that the construction works as stated. This is the kind of result that makes it easier to build anytime-valid procedures without starting from scratch each time. It is aimed at people working on sequential inference, e-values, and anytime-valid methods in stats and ML. A reader who already knows the e-process literature will get the most out of it, but the clarifications on optimality make it accessible enough for a broader audience. The work is grounded enough to deserve a serious referee; I would send it out rather than desk-reject.

Referee Report

1 major / 2 minor

Summary. The paper proves the converse to a prior result: asymptotically optimal sequential tests (in terms of stopping times) can be aggregated into asymptotically log-optimal e-processes. This is accomplished via a new class of WAIT e-processes—weighted aggregates of indicators of stopping times that begin at zero, are nondecreasing, and increase to infinity under the alternative at the optimal rate. The manuscript also discusses nuances in the definitions of asymptotic optimality for tests and log-optimality for e-processes.

Significance. If the central construction holds, the result completes the bidirectional link between asymptotically optimal sequential tests and log-optimal e-processes, strengthening the theoretical foundation for using e-processes in sequential hypothesis testing. The explicit construction via WAIT e-processes and the careful treatment of optimality definitions are strengths; the paper provides a direct mathematical argument rather than relying on parameter fitting or self-referential definitions.

major comments (1)

[Construction of WAIT e-processes and composite-alternative case] The skeptic concern on composite alternatives is load-bearing: for composite hypotheses, pointwise optimal stopping times may have different rates, and any fixed weighting in the WAIT aggregation risks producing a growth rate strictly below the envelope of individual rates. The proof sketch does not exhibit an explicit weighting scheme that recovers the pointwise optimum uniformly (see the discussion of optimality definitions and the construction of WAIT e-processes).

minor comments (2)

[Abstract and Introduction] The abstract states that the paper 'discusses several nuances' in optimality definitions; enumerating the key distinctions (e.g., pointwise vs. uniform optimality) already in the introduction would improve readability.
[Definition of WAIT e-processes] Notation for the weighting scheme in the WAIT definition could be clarified with an explicit formula or example for a simple composite case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful and constructive report. The central concern regarding the composite-alternative case is well-taken, and we address it directly below. We will revise the manuscript to make the weighting construction fully explicit.

read point-by-point responses

Referee: [Construction of WAIT e-processes and composite-alternative case] The skeptic concern on composite alternatives is load-bearing: for composite hypotheses, pointwise optimal stopping times may have different rates, and any fixed weighting in the WAIT aggregation risks producing a growth rate strictly below the envelope of individual rates. The proof sketch does not exhibit an explicit weighting scheme that recovers the pointwise optimum uniformly (see the discussion of optimality definitions and the construction of WAIT e-processes).

Authors: We agree that an explicit weighting rule is needed for the composite case. In the manuscript the WAIT e-process is defined by taking a countable dense subset of the composite parameter space, assigning positive weights that sum to one (e.g., via a fixed but arbitrary summable sequence such as w_i = 6/π² i^{-2}), and forming the weighted sum of the indicator processes of the corresponding pointwise-optimal stopping times. Because each individual process grows at its own optimal rate and the weights are strictly positive, the aggregate grows at a rate that is asymptotically at least the infimum of the pointwise rates (up to an arbitrarily small additive constant that can be absorbed into the definition of asymptotic log-optimality used in the paper). The proof in Section 3 proceeds by lower-bounding the log of the WAIT process by a convex combination of the individual log-growth rates and then taking the limit inferior. We concede, however, that the current write-up leaves the concrete choice of weights and the passage to the envelope implicit. In the revision we will add an explicit construction (including the choice of weights and a short verification that the resulting growth rate equals the envelope) together with a worked example for a simple composite alternative (testing a normal mean belonging to a closed interval). revision: yes

Circularity Check

0 steps flagged

Direct construction from optimal tests to log-optimal e-processes with no reduction to inputs

full rationale

The paper defines WAIT e-processes explicitly as weighted aggregates of indicators of stopping times that begin at zero and increase to infinity at the optimal rate under the alternative. It then proves that aggregating asymptotically optimal sequential tests via this construction yields e-processes whose growth rate matches the information-theoretic optimum. This is a forward construction from the test stopping times to the e-process, not a fit or redefinition. Definitions of asymptotic optimality for tests and log-optimality for e-processes are stated independently and shown to be compatible by explicit rate translation, without self-referential equations or load-bearing self-citations that would force the result. No step reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proof rests on standard properties of e-processes as nonnegative supermartingales under the null and the existence of asymptotically optimal stopping times under the alternative. No new free parameters or invented physical entities are introduced.

axioms (2)

standard math e-processes are nonnegative supermartingales under the null hypothesis
Invoked when defining validity of the e-process obtained by aggregation.
domain assumption asymptotically optimal stopping times exist for the sequential tests under consideration
Required for the WAIT construction to achieve the log-optimal growth rate.

invented entities (1)

WAIT e-process no independent evidence
purpose: Weighted aggregate of indicators of stopping times that is nondecreasing and increases to infinity at the optimal rate under the alternative
New construction introduced to convert optimal tests into a log-optimal e-process; no independent empirical evidence is provided beyond the mathematical definition.

pith-pipeline@v0.9.0 · 5463 in / 1354 out tokens · 68063 ms · 2026-05-14T19:46:39.543688+00:00 · methodology

discussion (0)

Reference graph

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