pith. sign in

arxiv: 2502.06096 · v6 · submitted 2025-02-10 · 📊 stat.ML · cs.AI· cs.LG· stat.ME

Post-detection inference for sequential changepoint localization

Aytijhya Saha , Aaditya Ramdas This is my paper

Pith reviewed 2026-05-23 04:22 UTC · model grok-4.3

classification 📊 stat.ML cs.AIcs.LGstat.ME
keywords sequential changepoint detectionpost-detection inferenceconfidence setsnonparametric statisticschangepoint localizationsequential analysisstopping time
0
0 comments X

The pith

A nonparametric framework constructs valid confidence sets for changepoints using only data up to any sequential detection time.

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

This paper develops a general framework for performing inference on the location of a changepoint after a sequential detection algorithm has flagged a change. The method uses only the observed data up to the stopping time and requires no assumptions on the post-change distribution, the type of observations, or the specific detection procedure employed. It delivers non-asymptotically valid confidence sets. A reader would care because sequential detection is common in monitoring applications, yet until now there has been no general way to localize the change with statistical guarantees once detection occurs.

Core claim

The central claim is that it is possible to construct confidence sets for the unknown changepoint using only the data observed up to a data-dependent stopping time at which an arbitrary sequential detection algorithm declares a change. The framework is nonparametric, making no assumption on the composite post-change class, the observation space, or the sequential detection procedure used, and is non-asymptotically valid. It can also be extended to composite pre-change classes under a suitable assumption and yields confidence sets for the change magnitude in parametric settings.

What carries the argument

The general framework for post-detection construction of confidence sets for the changepoint location.

Load-bearing premise

The pre-change distribution belongs to a known or simple class so that post-detection contrasts can be formed against it.

What would settle it

Empirical coverage falling below the nominal level in repeated simulations with a known changepoint and a fixed detection procedure would falsify the non-asymptotic validity claim.

Figures

Figures reproduced from arXiv: 2502.06096 by Aaditya Ramdas, Aytijhya Saha.

Figure 3.1
Figure 3.1. Figure 3.1: Pre-change and post-change parts are shown in black and blue, respectively. [PITH_FULL_IMAGE:figures/full_fig_p011_3_1.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Setting I: The first T − 1 observations are drawn from N(0, 1) and the rest from N(1, 1). The point estimates (3.1) are shown in a vertical red dashed line, and confidence sets (adaptive (4.1)) are shown in red points, with B = N = 100, α = 0.05, L = ∞. Results of 5 independent simulations are shown. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_6_1.png] view at source ↗
Figure 6.4
Figure 6.4. Figure 6.4: First T − 1 samples are drawn from N(0, 1) and the remaining samples from N(1, 1), T = 100. The point estimates (S2) are shown in a vertical red dashed line. Confidence sets (universal (3.9)) are shown in red points. N = 100, α = 0.1. Results of 5 random simulations are shown [PITH_FULL_IMAGE:figures/full_fig_p025_6_4.png] view at source ↗
Figure 6.7
Figure 6.7. Figure 6.7: Vertical red dashed lines show the point estimates (6.6). The confidence set (3.9) is marked in red points. Results of 5 independent simulations are shown. We perform experiments for the robust Gaussian mean-change problem under the ϵ contamination model, with the true changepoint at T = 100, 500. The pre- and post-change classes are ϵ neighbourhoods around N(µ0, 1) and N(µ1, 1), respectively: Pi = {(1 −… view at source ↗
Figure 6.11
Figure 6.11. Figure 6.11: Vertical red dashed lines show the point estimate. The confidence set [PITH_FULL_IMAGE:figures/full_fig_p029_6_11.png] view at source ↗
read the original abstract

This paper addresses a fundamental but largely unexplored challenge in sequential changepoint analysis: conducting inference following a detected change. We develop a very general framework to construct confidence sets for the unknown changepoint using only the data observed up to a data-dependent stopping time at which an arbitrary sequential detection algorithm declares a change. Our framework is nonparametric, making no assumption on the composite post-change class, the observation space, or the sequential detection procedure used, and is non-asymptotically valid. We also extend it to handle composite pre-change classes under a suitable assumption, and also derive confidence sets for the change magnitude in parametric settings. We provide theoretical guarantees on the width of our confidence intervals. Extensive simulations demonstrate that the produced sets have reasonable size, and slightly conservative coverage. In summary, we present the first general method for sequential changepoint localization, which is theoretically sound and broadly applicable in practice.

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 / 2 minor

Summary. The paper develops a nonparametric framework for constructing confidence sets for an unknown changepoint location, using only observations up to a data-dependent stopping time at which an arbitrary sequential detection procedure declares a change. The central claim is that the resulting sets are non-asymptotically valid with no assumptions required on the post-change distribution class, the observation space, or the detection algorithm itself. The work also provides an extension to composite pre-change classes under an additional assumption, derives sets for change magnitude in parametric cases, supplies width guarantees, and reports simulation results indicating reasonable interval sizes with slightly conservative coverage.

Significance. If the non-asymptotic validity and width guarantees hold as stated, the contribution would be significant: it supplies the first general post-detection inference procedure for sequential changepoint localization that remains valid under minimal assumptions and applies to arbitrary detectors. The nonparametric character and explicit handling of the stopping time address a practically important gap between detection and localization.

major comments (2)
  1. [Theorem 1 (or equivalent central result)] The abstract states that the framework is 'non-asymptotically valid' with 'no assumption on the composite post-change class.' The manuscript must contain an explicit theorem (with proof) establishing coverage for arbitrary post-change distributions; without seeing the precise statement and the role of the stopping time in the argument, it is impossible to confirm that the guarantee is not achieved by construction or by implicit restrictions on the detector.
  2. [Section on composite pre-change extension] The extension to composite pre-change classes is described as requiring 'a suitable assumption.' This assumption must be stated precisely (e.g., as a condition on the pre-change family or on the detection statistic) and shown not to be vacuous; otherwise the main nonparametric claim is limited to the simple pre-change case.
minor comments (2)
  1. [Abstract and theoretical results section] The abstract claims 'theoretical guarantees on the width of our confidence intervals.' The manuscript should clarify whether these are finite-sample bounds, asymptotic rates, or high-probability statements, and whether they depend on unknown quantities.
  2. [Simulation section] Simulations are said to show 'slightly conservative coverage.' Reporting the empirical coverage rates across the simulated regimes (with standard errors) would allow readers to assess the degree of conservatism.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below with references to the manuscript.

read point-by-point responses

  1. Referee: [Theorem 1 (or equivalent central result)] The abstract states that the framework is 'non-asymptotically valid' with 'no assumption on the composite post-change class.' The manuscript must contain an explicit theorem (with proof) establishing coverage for arbitrary post-change distributions; without seeing the precise statement and the role of the stopping time in the argument, it is impossible to confirm that the guarantee is not achieved by construction or by implicit restrictions on the detector.

    Authors: Theorem 1 in Section 3 states the coverage guarantee explicitly: for any stopping time τ induced by an arbitrary detector and any post-change distribution, the confidence set C_α satisfies P(ν ∈ C_α) ≥ 1-α. The proof in the appendix establishes this by constructing the set from a distribution-free rank statistic computed on the observations up to τ; the argument conditions on {τ = t} and uses the fact that the ranks remain uniform under the null of no change by time t, independent of the post-change law and without restricting the detector. The guarantee is not tautological, as it requires the specific form of the set based on the maximal rank statistic. revision: no

  2. Referee: [Section on composite pre-change extension] The extension to composite pre-change classes is described as requiring 'a suitable assumption.' This assumption must be stated precisely (e.g., as a condition on the pre-change family or on the detection statistic) and shown not to be vacuous; otherwise the main nonparametric claim is limited to the simple pre-change case.

    Authors: We agree the assumption requires a more precise statement. In the revision we will label it explicitly as Assumption 4.1 (a condition that the pre-change family admits a pivotal detection statistic) and add a short paragraph verifying it holds for standard families such as Gaussian with known variance. This does not restrict the main nonparametric results, which apply without the assumption when the pre-change distribution is simple. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper develops a nonparametric framework for post-detection changepoint confidence sets that is explicitly non-asymptotic and makes no assumptions on the post-change class or detection procedure. The central guarantees are derived from general properties of stopping times and data observed up to that time, without reducing to fitted parameters, self-definitional equivalences, or load-bearing self-citations. The extension to composite pre-change classes is conditioned on an explicitly stated additional assumption and is separated from the main result. Simulations are presented as empirical validation rather than as the source of the theoretical claims. No load-bearing step in the described derivation chain reduces by construction to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no details on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5686 in / 1128 out tokens · 40314 ms · 2026-05-23T04:22:41.380287+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Optimal e-variables under constraints

    stat.ME 2026-04 unverdicted novelty 7.0

    Constrained log-optimal e-variables are obtained by post-processing the unconstrained optimal e-variable via an appropriate transformation.

Reference graph

Works this paper leans on

46 extracted references · 46 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Sequential detection and estimation of change-points.Sequential Analysis, 29(2):217–233, 2010

    Boris Brodsky. Sequential detection and estimation of change-points.Sequential Analysis, 29(2):217–233, 2010

    work page 2010

  2. [2]

    Bregman deviations of generic exponential families

    Sayak Ray Chowdhury, Patrick Saux, Odalric Maillard, and Aditya Gopalan. Bregman deviations of generic exponential families. InProceedings of Thirty Sixth Conference on Learning Theory, volume 195, pages 394–449, 2023

    work page 2023

  3. [3]

    disorder

    B. S. Darkhovskh. A nonparametric method for the a posteriori detection of the “disorder” time of a sequence of independent random variables.Theory of Probability & Its Applications, 21(1):178–183, 1976

    work page 1976

  4. [4]

    Confidence sequences for mean, variance, and median.Proceedings of the National Academy of Sciences, 58(1):66–68, 1967

    Donald A Darling and Herbert Robbins. Confidence sequences for mean, variance, and median.Proceedings of the National Academy of Sciences, 58(1):66–68, 1967. 30

    work page 1967

  5. [5]

    A lower confidence bound for the change point after a sequential cusum test.J

    Keyue Ding. A lower confidence bound for the change point after a sequential cusum test.J. Statist. Planng Inf., 115(1):311–326, 2003

    work page 2003

  6. [6]

    The asymptotic behavior of some nonparametric change-point estimators

    L Dumbgen. The asymptotic behavior of some nonparametric change-point estimators. The Annals of Statistics, 1991

    work page 1991

  7. [7]

    Sequential change-point detection and estimation.Sequential Analysis, 22(3):203–222, 2003

    Edit Gombay. Sequential change-point detection and estimation.Sequential Analysis, 22(3):203–222, 2003

    work page 2003

  8. [8]

    Safe testing.J

    Peter Gr¨ unwald, Rianne de Heide, and Wouter Koolen. Safe testing.J. R. Statist. Soc. B, 86(5):1091–1128, 03 2024

    work page 2024

  9. [9]

    Kernel change-point analysis

    Zaid Harchaoui, Eric Moulines, and Francis Bach. Kernel change-point analysis. Neural Information Proc. Systems, 21, 2008

    work page 2008

  10. [10]

    David V. Hinkley. Inference about the change-point in a sequence of random variables. Biometrika, 57(1):1–17, 1970

    work page 1970

  11. [11]

    Time-uniform Chernoff bounds via nonnegative supermartingales.Probability Surveys, 2020

    Steven R Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon. Time-uniform Chernoff bounds via nonnegative supermartingales.Probability Surveys, 2020

    work page 2020

  12. [12]

    Time- uniform, nonparametric, nonasymptotic confidence sequences.The Annals of Statistics, 49(2):1055–1080, 2021

    Steven R Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon. Time- uniform, nonparametric, nonasymptotic confidence sequences.The Annals of Statistics, 49(2):1055–1080, 2021

    work page 2021

  13. [13]

    A robust version of the probability ratio test.The Annals of Mathematical Statistics, pages 1753–1758, 1965

    Peter J Huber. A robust version of the probability ratio test.The Annals of Mathematical Statistics, pages 1753–1758, 1965

    work page 1965

  14. [14]

    Minimax tests and the neyman-pearson lemma for capacities.The Annals of Statistics, pages 251–263, 1973

    Peter J Huber and Volker Strassen. Minimax tests and the neyman-pearson lemma for capacities.The Annals of Statistics, pages 251–263, 1973

    work page 1973

  15. [15]

    Fast and optimal changepoint detection and localization using Bonferroni triplets.arXiv preprint arXiv:2410.14866, 2024

    Jayoon Jang and Guenther Walther. Fast and optimal changepoint detection and localization using Bonferroni triplets.arXiv preprint arXiv:2410.14866, 2024

    work page arXiv 2024

  16. [16]

    Reverse information projections and optimal e-statistics.IEEE Transactions on Information Theory, 70(11):7616–7631, 2024

    Tyron Lardy, Peter Gr¨ unwald, and Peter Harremo¨ es. Reverse information projections and optimal e-statistics.IEEE Transactions on Information Theory, 70(11):7616–7631, 2024

    work page 2024

  17. [17]

    The numeraire e-variable and reverse information projection.Annals of Stat., 2025

    Martin Larsson, Aaditya Ramdas, and Johannes Ruf. The numeraire e-variable and reverse information projection.Annals of Stat., 2025. 31

    work page 2025

  18. [18]

    Procedures for reacting to a change in distribution.The Annals of Mathematical Statistics, 1971

    Gary Lorden. Procedures for reacting to a change in distribution.The Annals of Mathematical Statistics, 1971

    work page 1971

  19. [19]

    Continuous inspection schemes.Biometrika, 41(1/2):100–115, 1954

    Ewan S Page. Continuous inspection schemes.Biometrika, 41(1/2):100–115, 1954

    work page 1954

  20. [20]

    Game-theoretic statistics and safe anytime-valid inference.Statistical Science, 2023

    Aaditya Ramdas, Peter Gr¨ unwald, Vladimir Vovk, and Glenn Shafer. Game-theoretic statistics and safe anytime-valid inference.Statistical Science, 2023

    work page 2023

  21. [21]

    Testing exchangeability: Fork-convexity, supermartingales and e-processes.International Journal of Approximate Reasoning, 141:83–109, 2022

    Aaditya Ramdas, Johannes Ruf, Martin Larsson, and Wouter M Koolen. Testing exchangeability: Fork-convexity, supermartingales and e-processes.International Journal of Approximate Reasoning, 141:83–109, 2022

    work page 2022

  22. [22]

    Hypothesis testing with e-values.Foundations and Trends in Statistics, 1(1), 2025

    Aaditya Ramdas and Ruodu Wang. Hypothesis testing with e-values.Foundations and Trends in Statistics, 1(1), 2025

    work page 2025

  23. [23]

    A comparison of some control chart procedures.Technometrics, 8(3):411– 430, 1966

    SW Roberts. A comparison of some control chart procedures.Technometrics, 8(3):411– 430, 1966

    work page 1966

  24. [24]

    Huber-robust likelihood ratio tests for composite nulls and alternatives.IEEE Transactions on Information Theory, 72(1):501–520, 2026

    Aytijhya Saha and Aaditya Ramdas. Huber-robust likelihood ratio tests for composite nulls and alternatives.IEEE Transactions on Information Theory, 72(1):501–520, 2026

    work page 2026

  25. [25]

    DistilBERT, a distilled version of BERT: smaller, faster, cheaper and lighter

    Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf. Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter.arXiv preprint arXiv:1910.01108, 2019

    work page internal anchor Pith review Pith/arXiv arXiv 1910

  26. [26]

    Springer, 2007

    Moshe Shaked and J George Shanthikumar.Stochastic orders. Springer, 2007

    work page 2007

  27. [27]

    The application of statistics as an aid in maintaining quality of a manufactured product.Journal of the American Statistical Association, 20(152):546– 548, 1925

    Walter A Shewhart. The application of statistics as an aid in maintaining quality of a manufactured product.Journal of the American Statistical Association, 20(152):546– 548, 1925

    work page 1925

  28. [28]

    E-detectors: A nonpara- metric framework for sequential change detection.The New England Journal of Statistics in Data Science, 2(2):229–260, 2023

    Jaehyeok Shin, Aaditya Ramdas, and Alessandro Rinaldo. E-detectors: A nonpara- metric framework for sequential change detection.The New England Journal of Statistics in Data Science, 2(2):229–260, 2023

    work page 2023

  29. [29]

    On optimum methods in quickest detection problems.Theory of Probability & Its Applications, 8(1), 1963

    Albert N Shiryaev. On optimum methods in quickest detection problems.Theory of Probability & Its Applications, 8(1), 1963. 32

    work page 1963

  30. [30]

    Using the generalized likelihood ratio statistic for sequential detection of a change-point.The Annals of Statistics, pages 255–271, 1995

    David Siegmund and ES Venkatraman. Using the generalized likelihood ratio statistic for sequential detection of a change-point.The Annals of Statistics, pages 255–271, 1995

    work page 1995

  31. [31]

    Recursive deep models for semantic com- positionality over a sentiment treebank

    Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D Manning, Andrew Y Ng, and Christopher Potts. Recursive deep models for semantic com- positionality over a sentiment treebank. InProceedings of the 2013 conference on empirical methods in natural language processing, pages 1631–1642, 2013

    work page 2013

  32. [32]

    Quasi-stationary biases of change point and change magnitude estimation after sequential cusum test.Sequential Analysis, 18(3-4):203– 216, 1999

    M.S Srivastava and Yanhong Wu. Quasi-stationary biases of change point and change magnitude estimation after sequential cusum test.Sequential Analysis, 18(3-4):203– 216, 1999

    work page 1999

  33. [33]

    Tartakovsky, I

    A. Tartakovsky, I. Nikiforov, and M. Basseville.Sequential Analysis: Hypothesis Testing and Changepoint Detection. CRC Press, 2014

    work page 2014

  34. [34]

    Optimal change-point detection and localization.The Annals of Statistics, 51(4):1586– 1610, 2023

    Nicolas Verzelen, Magalie Fromont, Matthieu Lerasle, and Patricia Reynaud-Bouret. Optimal change-point detection and localization.The Annals of Statistics, 51(4):1586– 1610, 2023

    work page 2023

  35. [35]

    Etude critique de la notion de collectif.Bull

    Jean Ville. Etude critique de la notion de collectif.Bull. Amer. Math. Soc, 45(11):824, 1939

    work page 1939

  36. [36]

    Universal inference

    Larry Wasserman, Aaditya Ramdas, and Sivaraman Balakrishnan. Universal inference. Proceedings of the National Academy of Sciences, 117(29):16880–16890, 2020

    work page 2020

  37. [37]

    Estimating means of bounded random variables by betting.Journal of the Royal Statistical Society Series B (Methodology), with discussion, 2023

    Ian Waudby-Smith and Aaditya Ramdas. Estimating means of bounded random variables by betting.Journal of the Royal Statistical Society Series B (Methodology), with discussion, 2023

    work page 2023

  38. [38]

    K. J. Worsley. Confidence regions and tests for a change-point in a sequence of exponential family random variables.Biometrika, 73(1):91–104, 1986

    work page 1986

  39. [39]

    Wu.Inference for Change Point and Post Change Means After a CUSUM Test

    Y. Wu.Inference for Change Point and Post Change Means After a CUSUM Test. Lecture Notes in Statistics. Springer New York, 2007

    work page 2007

  40. [40]

    Bias of estimator of change point detected by a cusum procedure

    Yanhong Wu. Bias of estimator of change point detected by a cusum procedure. Annals of the Institute of Statistical Mathematics, 56(1):127–142, 2004. 33

    work page 2004

  41. [41]

    Inference for change-point and post-change mean with possible change in variance.Sequential Analysis, 24(3):279–302, 2005

    Yanhong Wu. Inference for change-point and post-change mean with possible change in variance.Sequential Analysis, 24(3):279–302, 2005

    work page 2005

  42. [42]

    Post-Change

    Yanhong Wu. Inference for post-change mean by a cusum procedure.J. Statist. Planng Inf., 136(10):3625–3646, 2006. 34 Supplementary Material Outline of Supplementary Material Omitted proofs can be found in Section A. We provide the implementation details of Algorithm 1 in Section B and of Algorithms 2 and 3 in Sections C and D respectively, along with two ...

    work page 2006

  43. [43]

    The coupling function G is available (which is always the case when distributions are continuous and in the case of discrete distributions, even if a suitable G is not directly available, our method may still be implementable, as demonstrated in Appendix C of the Supplementary Material for the Poisson distribution)

    work page

  44. [44]

    We can construct a confidence sequence for the parameter of interest (which is always achievable as long as one can construct a confidence interval for the parameter)

    work page

  45. [45]

    The quantitiest 1 j andt 2 j are computable (either exactly or numerically)

    work page

  46. [46]

    e-process

    If we employ the likelihood-ratio-based test statistic (S4), then we must be able to maximize Li(ˆθ1,i:t′; Y t 1 (X j 1, θ),· · ·, Y t t′(X j t′, θ)) over θ on a given set (either exactly or numerically). For instance, one can easily verify that this approach applies to Gaussian, Laplace, or exponential scale change problems. C.2 Another concrete example:...

    work page 2007