NeurIPS 2020

### Review 1

Summary and Contributions: The manuscript focuses on detecting change points and presents a novel method for computing p-values estimated from CPs detected through dynamic programming. The p-values are computed under a selective inference framework, producing valid p-values.

Strengths: The theoretical presentation is good and the DP algorithm is a computationally feasible algorithm as demonstrated in the empirical evaluation when applied to real world datasets. The algorithm is novel and solves a widely applicable problem. This work is very relevant to the NurIPS community due to the generality of the problem and practicality of the DP solution.

Weaknesses: The experiment section is very short, leaving all details to the appendix. Interpretation of the presented results is therefore very difficult. It would be great if some space was found for some minimal information about the experimental setup, which would increase clarity significantly. As it stands, the real world experiments are especially difficult to interpret.

Correctness: The claims are well explained and the method is solid. The empirical methodology is also good.

Clarity: The paper is easy to follow, however the empirical results are under detailed. A little bit more detail would go a long way to improving the clarity of the manuscript.

Relation to Prior Work: The manuscript is well placed within existing literature.

Reproducibility: Yes

Additional Feedback: This was a great read and a great method. Are you planning on releasing an implementation? Given the generality of the problem and the excellent results I'd imagine an implementation would be widely used.

### Review 2

Strengths: This work presents an extensive and thorough treatment of the problem of estimating accurate p-values for change points found by optimal partitioning. While many methods for change point detection exists, to the best of my knowledge few have considered estimating the p-values of the detected change points. This is relevant in practice as it allows to evaluate the significance of a detected change point. Moreover the theoretical work could be relevant to the broader ML community.

Weaknesses: In comparison to the depth of the theoretical treatment, the numerical experiments are rather brief. Without the additional details in the supplementary material it is difficult to evaluate the numerical results fully. While I understand the space limitation, the experimental section could have been given more attention. For instance, Table 1 shows the results for two "bioinformatics related datasets", but no context is provided about their source, descriptive statistics, or true number of change points. Thus in its current form Table 1 simply shows numbers that are larger for the proposed method, which in itself isn't very informative.

Correctness: Overall the presented methodology appears to be correct, although I have not checked all proofs in full detail. With regards to Algorithm 1, the authors mention in Sec. 3.3 that the number of possible CP vectors increases exponentially. Subsequently a reduced set of possible CP vectors is introduced, which certainly would reduce the runtime of Algorithm 1. However, unless I've missed it, it is not quantified *how much* smaller this set is. Does the complexity of Algorithm 1 still grow exponentially but at a smaller rate? A relatively minor note is that the authors write in the caption of Figure 7 that the runtime is almost linear. Judging by the means of the boxplots however it seems that the runtime is more likely to be quadratic. It is also not clear if this runtime includes the change point detection step as well, or whether it is only the runtime for computing the p-values.

Clarity: The paper is overall well written and clearly structured. However, the theoretical description is fairly dense, with quite a lot of different notation that needs to be introduced to present the method. This can make the description occasionally difficult to follow. I believe some of the notation can be clarified or streamlined to alleviate this problem. For example, while change point configurations are generally described with lowercase bold $\tau$, eq (9) introduces uppercase bold T^{opt} as a change point configuration (instead of bold lowercase $\tau^{opt}$). Similarly, Sec. 3.2 introduces lowercase bold $t$ for change point vectors, instead of bold $\tau$. This might lead to unnecessary confusion. Another approach that may help clarify the many different quantities that are introduced is to provide a description in language. For instance, (6) could be described as "the probability that the absolute difference in mean for a random sequence is equal or greater to that of the observed sequence, given the same set of change points, ..." It would be good if the authors mention early on that their method is for univariate time series. While this fact can be extracted from the definition of x^{obs}, it would be good to clarify.

Relation to Prior Work: The related work on estimating p-values in change point detection is discussed properly, as is the related work on selective inference. With regards to the dynamic programming algorithm and the computational tricks in Lemma's 1-3 to speed it up, it seems that this paper might not be the first to introduce these for change point detection (see [1-3], among others). It would be good if this is clarified in case these overlap with those in earlier works. One stream of related work that is not discussed in this manuscript is that on Bayesian online change point detection (i.e. [4, 5] and subsequent works). It could be argued that existing Bayesian methods already give posterior probabilities for the existence of a change point at every time step. While I appreciate that this work takes a hypothesis-testing approach, it seems that it would be appropriate to discuss the differences between the two streams of research. [1]: Rigaill - Pruned dynamic programming for optimal multiple change-point detection, 2010. [2]: Killick et al - Optimal detection of changepoints with a linear computation cost (2011) [3]: Maidstone et al. - On optimal multiple changepoint algorithms for large data (2017) [4]: Adams & Mackay - Bayesian Online Change Point Detection (2007) [5]: Fearnhead & Liu - On-line inference for multiple changepoint problems (2007)

Reproducibility: Yes

Additional Feedback: Minor comments and questions: - The caption of Figure 1 states that BinSeg p-values fail to detect some true CPs (C, D) due to the lack of power. Shouldn't this be "(C, D, F)"? - In Figure 4 & 5, $\Delta u$ is not explained, and the appendix is needed to make sense of the results in these figures. - The authors may want to consider referring the reader to some review papers on change point detection for background information, such as Aminikhanghahi & Cook (2017), Truong et al (2020), and Van den Burg & Williams (2020). Extremely minor comments: - In (5) and several other equations and definitions $\Sigma$ is not bold, whereas it is bold when introduced. - Footnotes typically follow punctuation - Caption Figure 2: programing -> programming - Line 100: "of a test statistic" - Line 173: fix "and apply to parametric case" - Line 174: "which" -> "that" - Line 198: "which makes ... can be substantially decreased" (unclear sentence) - Line 241: "almost *all* of them" Comments on broader impact statement: The authors give no discussion of potential negative impact of their work. They do note that the p-value is useful in the healthcare domain "for making medical decisions". Personally, I would be quite concerned if my physician made medical decisions based on p-values, and considering the many reports on the replication crisis and the role of p-values therein, this may not be the best example. In fact, it may actually be better suited as an example of the potentially negative effects of using the p-values obtained through this work.

### Review 3

Summary and Contributions: The paper proposes a method for more reliable statistical evaluation of change point detection, by computing an exact p-value (and hence allowing to search for the best change point vector). The exact p-value is based on the use of 'selective inference'. The authors propose a variant that does not result in over-conditioning, and to combat the computational effort require they propose a dynamic programming based approach. The resulting method is evaluated on artificial and real data and shown to outperform other CP detection methods, including an earlier CP method based on selective inference.

Strengths: + theoretically well founded + well structured and written paper with a mostly clear explanation, though dense at times + clever dynamic programming variant for computing the optimal CP vectors for a z + experiments demonstrate the meaningful results obtained, compared to state of the art methods

Weaknesses: - dense writing - computation time is reported, but only on itself, not in comparison to the other methods

Correctness: yes, having checked as much of the main paper as I could

Clarity: yes, well structured and written

Relation to Prior Work: yes, good discussion of related work and comparison

Reproducibility: Yes

### Review 4

Summary and Contributions: The authors introduce a selective inference approach to assess the p-value of a detected changepoint. The p-value is conditioned on the number of random line segments having the same optimal CP locations. which is made possible by incorporating several computational tricks.

Strengths: The paper is very well written and extremely accessible. The method is novel and clearly described. The results on simulated and real world data are convincing. The broader impact is given since the proposed methodology advances the field by increasing the statistical reliability of detected changepoints.

Weaknesses: One very minor comment is the low number of real world data sets used to validate the method. I am very aware of the problem that real world data sets to validate changepoint detection approaches are hard to find. I found the problem to be that usually, the location and existence of ground truth changepoints are debatable. Allow me to share a selection of some data sets that I have found to be useful in the past (sometimes appropriate transformations have to be applied, e.g. log-diff): Transatlantic internet traffic has daily and weekly changes in mean (https://ieeexplore.ieee.org/abstract/document/1716452 2006) Crest toothpaste market share increases due to a change in management and marketing (Section 1.1.4 https://onlinelibrary.wiley.com/doi/pdf/10.1002/9781118032978#page=281) The seasonally adjusted monthly US civilian unemployment rate changes due to business cycles (Montgomery, Forecasting the US unemployment rate https://www.tandfonline.com/doi/abs/10.1080/01621459.1998.10473696) The public drunkenness intakes in Minneapolis has two ground truth changepoints when effective political (counter-)interventions took place (Meidinger, Applied Time Series Analysis for the Social Sciences, 1980) The yearly global average temperature changes due to climate change (https://data.giss.nasa.gov/gistemp/) I also suggest to add one more internal comparison partner to the experiments that derives the significance of changepoints with a Bonferroni correction OptSeg-BF, as in Figure 1 (naive). Although correcting by the number of detected changepoints is reasonable, the authors mention the problem of double-dipping with this approach. Therefore, an equally justified way of correcting would incorporate that the segmentation method already searched a larger space of hypothesis before returning the changepoint candidates. A stricter Bonferroni correction can therefore be justified, e.g. by dividing by the all possible configurations of changepoint locations (alpha/(n choose k)) or by the number of possible change point locations (alpha/n). Doing that, the Bonfferroni correction would be a bit more conservative. In Figure 1, 0.05/1192052400 = 4e-11 or 0.05/100 = 0.0005 -- the second approach giving the desired results. Adding on that, maybe you could briefly justify why double-dipping is not an issue for the unknown K case, where you return segmentations for all K where the p-value for each changepoint in the kth case is unaffected by the other cases of k.

Correctness: Yes

Clarity: Yes

Relation to Prior Work: Yes

Reproducibility: Yes