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arxiv: 2301.05636 · v3 · submitted 2023-01-13 · 📊 stat.ME

Improving Power by Conditioning on Less in Post-selection Inference for Changepoints

Rachel Carrington , Paul Fearnhead This is my paper

Pith reviewed 2026-05-24 10:22 UTC · model grok-4.3

classification 📊 stat.ME
keywords post-selection inferencechangepoint detectionselective p-valuesMonte Carlo approximationpower improvementgenomic sequence analysisconditional inference
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The pith

Conditioning on less information yields more powerful valid p-values for changepoint detection.

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

The paper establishes that post-selection inference procedures for changepoints can be made more powerful by conditioning the p-value on a smaller subset of the selection information. This produces an ideal selective p-value whose exact computation is intractable, yet which can be approximated by Monte Carlo sampling through repeated perturbations of the observed data followed by re-running the changepoint detector. The resulting Monte Carlo p-values remain exactly valid for any number of samples drawn, and modest sample sizes already deliver noticeable gains in power. On human GC-content genomic sequences the procedure raises the count of declared significant changepoints from 17 to 27 relative to earlier conditioning choices.

Core claim

By deliberately conditioning on less of the information that determined which changepoints were selected, one obtains an ideal selective p-value that cannot be evaluated in closed form but admits an unbiased Monte Carlo estimator; this estimator is valid for any finite number of perturbations and empirically yields higher power while remaining easy to implement by re-applying any existing post-selection method to each perturbed data set.

What carries the argument

The Monte Carlo approximation to the ideal selective p-value, obtained by generating perturbations of the data set and re-applying the post-selection inference procedure to each one.

If this is right

  • The Monte Carlo p-values are exactly valid for any finite sample size.
  • Substantial power gains occur even with very small numbers of Monte Carlo perturbations.
  • Implementation requires only generating perturbations and re-running the existing post-selection routine on each.
  • Application to genomic GC-content data increases the number of significant changepoints from 17 to 27.

Where Pith is reading between the lines

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

  • The same reduction in conditioning information could be explored in other post-selection settings that currently condition on the full selection event.
  • Because validity holds for any sample size, users can trade computational budget directly against power without recalibrating thresholds.
  • If the underlying changepoint detector is computationally cheap, the Monte Carlo overhead remains modest even for genome-scale sequences.

Load-bearing premise

Perturbations of the original data set generate samples whose distribution matches the conditional law required by the ideal selective p-value.

What would settle it

In repeated simulations under the global null of no changepoints, the fraction of Monte Carlo p-values falling below a nominal level alpha exceeds alpha by more than binomial sampling variability.

Figures

Figures reproduced from arXiv: 2301.05636 by Paul Fearnhead, Rachel Carrington.

Figure 1
Figure 1. Figure 1: The p-value of Jewell et al. (2022) then fixes the ψ value so the conditional distribution of ϕ is uniform on the coloured line – i.e. all values that are consistent with detecting a change at τˆ for that value of ψ. The p-value is the probability of observing a more extreme value than that for the data – which is the proportion of the line that is red. By comparison, the p-value of (4) allows ψ to vary. I… view at source ↗
Figure 1
Figure 1. Figure 1: Comparison of the p-value of Jewell et al. (2022) (left-hand column) and the ideal p-value (right￾hand column) for the case of a univariate ψ parameter. We have used the probability inverse mapping to transform ϕ and ψ so that they are uniformly and independently distributed on [0, 1] under the prior. We view data sets as being a function of (ϕ, ψ), and the selection event – which corresponds to the inform… view at source ↗
Figure 2
Figure 2. Figure 2: QQ plots of p-value estimates, simulated under H0 with T = 1000, for different values of h and N. On each plot the ordered p-values obtained using different values of N (N = 1, 5, 10, 50) are plotted against theoretical quantiles from U(0, 1). In (a) and (b) the p-values are calculated as in Equation 6, where all ψ (j) ’s are simulated randomly. In (c) and (d), we take ψ (1) = U T Xobs. If p-values are val… view at source ↗
Figure 3
Figure 3. Figure 3: QQ plots of p-values for changepoints obtained using binary segmentation: h is the window size and δ the size of the change in the model from which we simulate. In (a), p-values from our method are plotted against theoretical quantiles from U(0, 1) for N = 1, 2, 5, 10, 20, 50. (b), (c) and (d) show QQ plots of p-values calculated using our method (with N = 2, 5, 10, 20, 50) against p-values from the method… view at source ↗
Figure 4
Figure 4. Figure 4: Rejection rates of H0 for binary segmentation, plotted against N. On each plot the three lines show the proportion of samples where the p-value was below 0.05, leading H0 to be rejected. Each line corresponds to a different size of change δ: green corresponds to δ = 3, blue to δ = 2, and red to δ = 1. 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 Number of samples Power h = 10 0.2 0.4 0.6 0.8 1.0 0 10 20 30 40 50 N… view at source ↗
Figure 5
Figure 5. Figure 5: Rejection rates of H0 for different values of h, δ, and N, when we simulate from a model with 4 changepoints, and apply binary segmentation with 4 changepoints. For each h, the process is run three times with changes of size δ = 1, 2, 3, which are shown on the plots as red, blue, and green lines respectively. Each line corresponds to a changepoint. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plots showing the number of true positives found in 1000 simulations using our method with N = 10 against the number found using the method of Jewell et al. (2022) (left) and MOSUM (right). A small amount of noise has been added so that points corresponding to different simulated data sets can be distinguished. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: QQ plots of p-values when σ 2 is estimated using median absolute deviation, rather than being assumed known. In each case T = 1000 and h = 10 and we estimate 1 changepoint using binary segmen￾tation and calculate the associated p-value. In the left-hand plot we simulate from H0 with no changes; in the right-hand plot we simulate from a model with a single change of size δ = 2 at τ = 500. amount of noise ha… view at source ↗
Figure 8
Figure 8. Figure 8: QQ plots of p-values obtained when simulating from a tν distribution with ν degrees of freedom. simulated from a tν distribution with ν = 5, 10 degrees of freedom. In panels (a) and (b) we simulate from H0 and find that the p-values approximately follow the expected distribution U(0, 1). In panel (c) we simulate data with a single change of size δ = 1; in this case the p-values are smaller than under H0, s… view at source ↗
Figure 9
Figure 9. Figure 9: Estimated changepoints in GC content data. Binary segmentation was used to estimate 38 changepoints, and we set h = 10. Each vertical line corresponds to an estimated changepoint; changepoints found to be significant at significance level α = 0.05 are shown in red, with others shown in grey. In the top panel, we used N = 1 (equivalent to the method of Jewell et al. (2022)) to calculate p-values; in the bot… view at source ↗
Figure 10
Figure 10. Figure 10: QQ plot of p-values for L0 segmentation; h is the window size and δ the size of the change in the model from which we simulate. In (a), p-values from our method are plotted against theoretical quantiles from U(0, 1) for N = 1, 2, 5, 10, 20, 50. (b), (c) and (d) show QQ plots of p-values calculated using our method (with N = 2, 5, 10, 20, 50) against p-values from the method of Jewell et al. (2022) (equiva… view at source ↗
Figure 11
Figure 11. Figure 11: Rejection rates of H0 for L0 segmentation. On each plot the three lines show the proportion of samples (of 1000 total) where the p-value was below 0.05, resulting in H0 being rejected, for different values of N. Each line corresponds to a different size of change δ: green corresponds to δ = 3, blue to δ = 2, and red to δ = 1. B Additional Simulations B.1 Results for L0-penalised and Wild Binary Segmentati… view at source ↗
Figure 12
Figure 12. Figure 12: Rejection rates of H0 for different values of h, δ, and N, when we simulate from a model with 4 changepoints, and apply wild binary segmentation with 4 changepoints. For each h, the process is run three times with changes of size δ = 1, 2, 3, which are shown on the plots as red, blue, and green lines respectively. Each line corresponds to a changepoint. 6 8 10 0 500 1000 1500 2000 Position GC content 6 8 … view at source ↗
Figure 13
Figure 13. Figure 13: Estimated changepoints in GC content data. L0 segmentation was used to estimate 38 changepoints, and we set h = 10. Each vertical line corresponds to an estimated changepoint; changepoints found to be significant at significance level α = 0.05 are shown in red, with others shown in grey. In the top panel, we used N = 1 (equivalent to the method of Jewell et al. (2022)) to calculate p-values. The middle an… view at source ↗
Figure 14
Figure 14. Figure 14: QQ plots of p-values obtained when simulating from a Laplace distribution under H0 and H1; s is the scale parameter. B.2 Correlation of p-values To investigate empirically whether p-values at distinct locations (i.e. non-overlapping regions of interest) are uncorrelated, we simulated from a model with T = 400 and three changepoints at τ = 100, 200, 300. Using binary segmentation, changepoints were detecte… view at source ↗
read the original abstract

Post-selection inference has recently been proposed as a way of quantifying uncertainty about detected changepoints. The idea is to run a changepoint detection algorithm, and then re-use the same data to perform a test for a change near each of the detected changes. By defining the p-value for the test appropriately, so that it is conditional on the information used to choose the test, this approach will produce valid p-values. We show how to improve the power of these procedures by conditioning on less information. This gives rise to an ideal selective p-value that is intractable but can be approximated by Monte Carlo. We show that for any Monte Carlo sample size, this procedure produces valid p-values, and empirically that noticeable increase in power is possible with only very modest Monte Carlo sample sizes. Our procedure is easy to implement given existing post-selection inference methods, as we just need to generate perturbations of the data set and re-apply the post-selection method to each of these. On genomic data consisting of human GC content, our procedure increases the number of significant changepoints that are detected from e.g. 17 to 27, when compared to existing methods.

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 manuscript proposes an improvement to post-selection inference for changepoint detection by conditioning on less information than standard selective p-values. This yields an ideal (intractable) selective p-value that is approximated via Monte Carlo by generating B perturbations of the observed data, re-running the full changepoint procedure on each, and constructing a rank-based p-value from the resulting test statistics. The central claims are that the Monte Carlo procedure produces exactly valid selective p-values for any finite B and that modest B suffices for noticeable power gains, as illustrated on human GC-content genomic data where the number of detected significant changepoints rises from 17 to 27.

Significance. If the validity argument holds, the approach would be a practical and low-cost extension of existing post-selection methods for changepoints, requiring only data perturbations and re-application of the base procedure. The empirical demonstration of power improvement with small B is a concrete strength, and the method's modularity (building directly on prior selective-inference code) aids reproducibility. However, the significance is tempered by the need to confirm that the perturbation mechanism exactly reproduces the conditional null distribution given selection.

major comments (2)
  1. [Monte Carlo approximation procedure (likely §3–4)] The validity claim for any Monte Carlo size B rests on the perturbed replicates being exchangeable with the observed statistic under the null conditional on the selection event. The manuscript implements this via data perturbations followed by re-running the changepoint algorithm, but does not explicitly verify or prove that the chosen perturbation distribution coincides with the law of the data under the null given selection (particularly when the selection event is discrete). If this match fails, the rank statistic is no longer uniform and type-I error control is lost even though the algebraic form of the p-value is preserved.
  2. [Validity theorem / proof of Monte Carlo validity] The abstract and introduction assert that the procedure 'produces valid p-values' for any B. The supporting argument should be located in the section deriving the selective p-value; if it relies on an implicit assumption that perturbations are drawn from the exact conditional distribution, this needs to be stated as a theorem with the precise conditions on the perturbation kernel.
minor comments (2)
  1. [Implementation details] Clarify the exact form of the perturbation distribution (e.g., additive Gaussian noise with what variance?) and whether it is the same for all candidate changepoint locations.
  2. [Empirical results / genomic data example] In the genomic application, report the specific changepoint detection algorithm, the definition of the selection event, and the value of B used to obtain the increase from 17 to 27 significant changepoints.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the need to make the validity argument for the Monte Carlo procedure more explicit. We address each major comment below and will revise the manuscript to strengthen the presentation of the supporting theorem.

read point-by-point responses

  1. Referee: The validity claim for any Monte Carlo size B rests on the perturbed replicates being exchangeable with the observed statistic under the null conditional on the selection event. The manuscript implements this via data perturbations followed by re-running the changepoint algorithm, but does not explicitly verify or prove that the chosen perturbation distribution coincides with the law of the data under the null given selection (particularly when the selection event is discrete). If this match fails, the rank statistic is no longer uniform and type-I error control is lost even though the algebraic form of the p-value is preserved.

    Authors: We agree that an explicit statement of the conditions is warranted. The perturbation mechanism is constructed so that, conditional on the selection event, the observed statistic and the B perturbed statistics are exchangeable under the null (by drawing perturbations from the same conditional law that defines the selective p-value). This ensures the rank-based p-value is exactly uniform for any finite B, analogous to a Monte Carlo test. However, the current text leaves the precise matching of the perturbation kernel to the conditional distribution implicit, especially for discrete selection events. We will add a formal theorem in the section deriving the Monte Carlo approximation that states the required conditions on the perturbation kernel for exchangeability to hold. revision: yes

  2. Referee: The abstract and introduction assert that the procedure 'produces valid p-values' for any B. The supporting argument should be located in the section deriving the selective p-value; if it relies on an implicit assumption that perturbations are drawn from the exact conditional distribution, this needs to be stated as a theorem with the precise conditions on the perturbation kernel.

    Authors: The validity claim for any B follows directly from the exchangeability of the observed and perturbed statistics under the conditional null, which is ensured by the choice of perturbation kernel matching the law of the data given selection. We will revise the manuscript to include an explicit theorem (with the precise conditions on the kernel) in the section on the selective p-value derivation, rather than leaving the argument implicit. This will make the finite-B validity self-contained and address the concern about discrete selection events. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper's derivation extends prior post-selection inference by conditioning on less information to obtain an ideal selective p-value, then approximates it via Monte Carlo perturbations of the data followed by re-application of the changepoint procedure. Validity for any Monte Carlo sample size B is claimed via a rank-based construction that does not reduce to a fitted parameter or self-definition by construction. No load-bearing self-citations, ansatzes smuggled via citation, or uniqueness theorems imported from the authors' prior work are exhibited in the provided text that would make the central result equivalent to its inputs. The perturbations are external to the fitting process, and the method remains falsifiable via the conditional distribution argument.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The approach builds on existing post-selection inference without introducing new free parameters or entities; it modifies the conditioning in the p-value calculation using data perturbations.

axioms (1)
  • domain assumption Standard assumptions of the changepoint model and the fixed detection algorithm allow for valid selective inference.
    Invoked implicitly when describing the post-selection procedure and its extension.

pith-pipeline@v0.9.0 · 5732 in / 1167 out tokens · 28860 ms · 2026-05-24T10:22:51.237966+00:00 · methodology

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Reference graph

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