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arxiv: 2606.01960 · v1 · pith:O5XYLIH2new · submitted 2026-06-01 · 📊 stat.ME · math.ST· stat.TH

Return-to-Baseline Testing via Empirically Calibrated e-processes

Marta Regis , Paulo Serra This is my paper

Pith reviewed 2026-06-28 13:28 UTC · model grok-4.3

classification 📊 stat.ME math.STstat.TH
keywords return to baselinee-processuniversal inferencesequential testingdistribution-freeanytime-valid inferencesuper-martingalehigh-frequency monitoring
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The pith

A sequential distribution-free test detects when monitoring data returns to its pre-intervention baseline using an empirically calibrated e-process.

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

The authors develop a procedure to find the moment high-frequency data realigns with its earlier distribution after an intervention. They start with a discrepancy measure from universal inference, aggregate it into a non-negative super-martingale, and calibrate the result empirically using only the baseline observations to produce an e-process. This yields anytime-valid error control without requiring any parametric form for the data-generating process. Finite-sample bounds on the calibration error are derived under a flexible non-parametric condition. The method is illustrated on simulations and clinical data to show accurate detection of the return time.

Core claim

The central claim is that an e-process formed by empirically calibrating a super-martingale of a universal-inference discrepancy measure, using only the subject-specific baseline data, supplies a sequential, distribution-free test for return to baseline that controls error at every stopping time.

What carries the argument

Empirically calibrated e-process obtained by aggregating a universal-inference discrepancy measure into a non-negative super-martingale and tuning it with baseline observations.

If this is right

  • Detection of the return time is possible without ever specifying a null distribution.
  • Error control remains valid no matter when the test is stopped.
  • Calibration is performed once on the baseline data and is therefore tailored to each subject or series.
  • Finite-sample guarantees on calibration error hold without asymptotic approximations.

Where Pith is reading between the lines

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

  • The same calibration step could be applied to other sequential problems where only a reference sample is available.
  • The procedure might be combined with existing alert systems to flag return events in real time.
  • Extensions to multivariate or dependent observations would require checking whether the non-parametric bound still applies.

Load-bearing premise

The observations obey a flexible non-parametric condition that lets the calibration error be bounded in finite samples.

What would settle it

A sequence of observations in which the calibrated e-process crosses its threshold while the post-intervention distribution still differs from the baseline (or fails to cross when the distributions match), under the stated non-parametric condition.

Figures

Figures reproduced from arXiv: 2606.01960 by Marta Regis, Paulo Serra.

Figure 2
Figure 2. Figure 2: The line in Figure 2 corresponding [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: An example of data generated from the model. There are 200 observations, the [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A few conditional quantiles of the baseline and intervention. Under the [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: (log-)e-processes corresponding to sequentially inspecting sub-segments of [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scaling in n of the computational time. We plot on a log-log scale the multiplier for n and the multiplier for the corresponding execution time. So for instance the point at roughly (16, 100) means that running the 4The specific data distribution is not relevant when assessing the computation time but we took the same signal as in Section 3. 41 [PITH_FULL_IMAGE:figures/full_fig_p041_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scaling in p, s, df, and |T | (left to right and top to bottom) of the computational time. The scaling for p is as expected aligned with quadratic- or cubic (s + 1 = 3) growth. The scaling in s is by far the least favorable. Indeed, increasing s has a large impact on the number of sets S that have to be considered. If the computational cost becomes too 42 [PITH_FULL_IMAGE:figures/full_fig_p042_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effect of different scaling factors on the discrepancy measure. Rather than [PITH_FULL_IMAGE:figures/full_fig_p044_6.png] view at source ↗
read the original abstract

We consider the problem of detecting a Return to Baseline (RtB) in high-frequency monitoring data preceding and following an intervention, where the aim is to identify the time at which the data-generating distribution realigns with its pre-intervention distribution. We propose a sequential, distribution-free testing procedure that does not rely on specifying a null model and provides anytime-valid error control. The method relies on ideas from universal inference to define a discrepancy measure that is aggregated into a non-negative super-martingale, and is then empirically cal- ibrated to form an e-process. The calibration is performed using the baseline data, and is thus subject-specific. We establish finite-sample bounds for the calibration error (under a flexible non-parametric assumption), discuss the impact of tuning parameters and computational complexity, and illustrate through simulations and a clinical case study that the procedure accurately detects RtB from monitoring data.

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.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes a sequential, distribution-free procedure for detecting return-to-baseline (RtB) in high-frequency monitoring data. It defines a discrepancy measure via universal inference, aggregates it into a non-negative super-martingale, and empirically calibrates the result (using subject-specific baseline data) to obtain an e-process with claimed anytime-valid error control. Finite-sample bounds on calibration error are established under a flexible non-parametric assumption; the paper also discusses tuning parameters and computational complexity, and provides illustrations via simulations and a clinical case study.

Significance. If the supermartingale property is preserved after subject-specific calibration and the finite-sample bounds hold, the procedure would supply a practical, distribution-free tool for anytime-valid sequential testing without requiring a parametric null model. The use of external baseline data for calibration is a strength for individualized applications such as clinical monitoring. Reproducible simulation code and explicit finite-sample guarantees would further strengthen the contribution if supplied.

major comments (2)
  1. [§4] §4 (Finite-sample calibration bounds): the central validity claim rests on these bounds under the stated non-parametric assumption; the manuscript must explicitly verify that the assumption accommodates possible serial dependence between baseline and post-intervention periods, or provide a counter-example showing when the bound fails.
  2. [§3.2] §3.2 (Super-martingale construction after calibration): it is not immediately clear from the aggregation step whether the empirical calibration factor is a predictable process with respect to the filtration; if it is not, the supermartingale property (and thus anytime-valid control) may require an additional argument beyond the universal-inference discrepancy.
minor comments (3)
  1. The abstract states that tuning-parameter impact is discussed, yet the main text would benefit from a dedicated sensitivity table or figure showing how the detection time changes with the primary tuning constants.
  2. Notation for the calibrated e-process (e.g., the exact functional form of the calibration multiplier) should be introduced once and used consistently; currently it appears to vary between the method and simulation sections.
  3. Figure captions for the clinical case study should state the exact sample sizes and the value of the tuning parameter used, to allow direct replication.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and indicate the revisions made to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [§4] §4 (Finite-sample calibration bounds): the central validity claim rests on these bounds under the stated non-parametric assumption; the manuscript must explicitly verify that the assumption accommodates possible serial dependence between baseline and post-intervention periods, or provide a counter-example showing when the bound fails.

    Authors: The non-parametric assumption is that baseline observations are i.i.d. from the pre-intervention distribution and post-intervention observations are i.i.d. from their time-specific distributions, with the bound derived via concentration inequalities that depend only on the marginal empirical measures. This formulation imposes no restriction on dependence between the baseline and post-intervention periods. We have revised §4 to include an explicit verification that the finite-sample bound continues to hold under arbitrary serial dependence across the two periods, together with a brief discussion of the assumption's scope. revision: yes

  2. Referee: [§3.2] §3.2 (Super-martingale construction after calibration): it is not immediately clear from the aggregation step whether the empirical calibration factor is a predictable process with respect to the filtration; if it is not, the supermartingale property (and thus anytime-valid control) may require an additional argument beyond the universal-inference discrepancy.

    Authors: The empirical calibration factor is computed exclusively from the subject-specific baseline data, which is fully observed prior to the intervention and the commencement of sequential monitoring. It is therefore a fixed, F_0-measurable quantity and hence predictable with respect to the filtration generated by the post-intervention observations. The supermartingale property of the aggregated process follows directly from the corresponding property of the universal-inference discrepancy without further conditions. We have added a clarifying remark in §3.2 to make this predictability explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation relies on universal inference to construct a discrepancy measure aggregated into a supermartingale, followed by subject-specific empirical calibration using external baseline data and finite-sample bounds under a non-parametric assumption. No quoted step reduces a claimed result to a fitted parameter or self-citation by construction; the anytime-valid control follows directly from the supermartingale property plus the stated calibration bounds, which are independent of the target RtB detection outcome.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on universal inference for the discrepancy measure, aggregation into a super-martingale, and empirical calibration from baseline data under a non-parametric assumption. Tuning parameters are discussed but not enumerated as fitted values.

free parameters (1)
  • tuning parameters
    The abstract states that the impact of tuning parameters is discussed, implying they are chosen or calibrated as part of the procedure.
axioms (1)
  • domain assumption flexible non-parametric assumption
    Invoked to establish finite-sample bounds for the calibration error.

pith-pipeline@v0.9.1-grok · 5674 in / 1170 out tokens · 31838 ms · 2026-06-28T13:28:51.885609+00:00 · methodology

discussion (0)

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