__ Summary and Contributions__:
----- Update -----
I have read the author response as well as the other reviews. I agree with some of the concerns raised by the other reviewers, but also do not find them to be significant to question the overall value and insights in this paper.
I would still vote for accept, but lower my score to 7.
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This work further analyzes the recently observed issue that Deep Generative Models (DGMs) regularly assign higher likelihood to out-of-distribution (OOD) samples/outliers. Based on the phenomenon that typical sets (regions of largest probability mass where samples likely fall into) must not coincide with density level sets (high-density/likelihood regions) in high dimensions, a novel white noise test for outlier detection is proposed. This test shows a marked improvement in detection performance over previous tests *using the same models* on common benchmarks (CIFAR-10, SVHN, CelebA, TinyImagNet in-/out-of-distribution combinations), thereby suggesting that DGMs are not necessarily uncalibrated, but rather that existing likelihood-based test might be improperly formulated/applied. Additional experiments on separating the effects of low-level textures vs. high-level semantic features are presented, followed by a discussion of current benchmarks as well as evaluation practices and the introduction of new semantic-focused benchmarks. The paper concludes with a discussion on the intrinsic difficulty of density estimation in a finite-sample, high-dimensional setting that also highlights model inductive biases (e.g. of recent EBMs) that also suggests future work is needed to better understand the issue.

__ Strengths__:
- The problem of high-likelihood outliers is a relevant and important problem from a scientific (understanding DGMs) but also practical perspective (trustworthy, robust models in applications).
- The presented analysis is scientifically sound and presents new insights into this challenging problem, namely that the geometry of high-dimensional spaces must be taken into account for outlier testing and that the separation of low-level textures and high-level semantic features is crucial to the problem.
- The proposed white noise test generalizes and improves over previous tests for typicality.
- The suggestion and introduction of new semantic-oriented benchmarks extends and corrects shortcomings of the current testbed for OOD detection.

__ Weaknesses__:
- The analysis mainly focuses on model biases caused by learning objectives (e.g. likelihood or energy), but there are further inductive biases (e.g. network architecture) that very likely have an effect on the problem. This should be mentioned.
- Important works of the related deep anomaly detection line of research [7, 5, 2, 3, 6, 1, 4] should be included which consider other objectives than likelihood/energy based on reconstruction, one-class classification, and self-supervision.
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robustness and uncertainty. In NeurIPS, pages 15637–15648, 2019.
[4] K. H. Kim, S. Shim, Y. Lim, J. Jeon, J. Choi, B. Kim, and A. S. Yoon. RaPP: Novelty detection with
reconstruction along projection pathway. In ICLR, 2020.
[5] L. Ruff, R. A. Vandermeulen, N. Görnitz, L. Deecke, S. A. Siddiqui, A. Binder, E. Müller, and M. Kloft.
Deep one-class classification. In ICML, pages 4393–4402, 2018.
[6] L. Ruff, R. A. Vandermeulen, N. Görnitz, A. Binder, E. Müller, K.-R. Müller, and M. Kloft. Deep
semi-supervised anomaly detection. In ICLR, 2020.
[7] S. Zhai, Y. Cheng, W. Lu, and Z. Zhang. Deep structured energy based models for anomaly detection. In ICML, volume 48, pages 1100–1109, 2016.

__ Correctness__:
- The reasoning and theoretical arguments are technically sound.
- The empirical evaluation is scientifically rigorous.

__ Clarity__:
- The paper is well structured and written clearly. The overall exposition and presentation is excellent.
- The careful, scientific tone and wording of this work is pleasantly refreshing in these times of tendencies towards verbally inflated contributions.

__ Relation to Prior Work__:
Overall, this work is well-placed into the existing literature and includes very recent works. However, the related deep anomaly detection line of research [7, 5, 2, 3, 6, 1, 4] that considers other objectives such as reconstruction, one-class classification, and self-supervision should be included to bridge these related lines.

__ Reproducibility__: Yes

__ Additional Feedback__: - The paper title is very generic.

__ Summary and Contributions__:
This paper proposes a new outlier testing method that does not rely on likelihoods. Previously, the outlier testing methods based on the likelihood values of deep generative models have been shown to give inaccurate results. The proposed method argue that this is not exactly because the likelihood is not calibrated in the models but because the high likelihoods can be atypical in high-dimensional spaces. On this account, the authors present a new method relying on martingale-difference testing and white-noise testing.

__ Strengths__:
The paper is well-written and well-organized in general. The proposing method is a new line of research for outlier testing up to my knowledge, and the experimental results on popular image datasets seem promising.

__ Weaknesses__: The paper could have given a more detailed introduction regarding the statistical tests. In particular, since the data are not exactly temporal sequences but being treated as one in the testing, it would be nice if such logical gaps are discussed. Some of my questions are:
- How the typicality test is related to the IID condition and the weaker conditions (MD, WN)?
- If it were a latent variable model (LVM) endowing the latent space with IID Gaussians, checking such conditions seems plausible. But in case of auto-regressive models and considering its MD construction (R(x)), why do we expect the in-distribution sequence to be white noise and outliers to be not?

__ Correctness__: The claims and methods seem correct. The experiments, however, only cover the image datasets; the claims would be more strengthened if datasets from other domains are tested.

__ Clarity__: The paper is clearly written in general, but there are several parts (mentioned in the weaknesses) that can be improved.

__ Relation to Prior Work__: Yes. The prior work is discussed in the Sec. 5 and the comparative experiments have been conducted and reported.

__ Reproducibility__: Yes

__ Additional Feedback__:

__ Summary and Contributions__: The main contributions of this paper are:
(1) It proposes a residual autocorrelation test for outlier detection and empirically reports that its performance surpasses those of likelihood-based methods on natural images
(2) The authors show that the standard image datasets can be successfully distinguished using a simple linear model and proposes a new semantic-oriented evaluation where capturing high-level semantics is more important for OOD detection
(3) It confirms that density estimation in OOD regions is hard for both deep generative models and energy-based models and suggests that current OOD evaluation protocols be revised or be developed on a problem-dependent basis.

__ Strengths__: The proposed residual autocorrelation test is empirical powerful for standard image datasets. Moreover, it reveals that low textual information is often enough to solve the task, casting doubt on using the benchmark for DGM-based OOD detection methods.
Also, the paper presents a new semantic-oriented OOD evaluation where textual or background difference is minimized and reveals that DGMs are useful for detecting different semantics when combined with the proposed white noise test.
The empirical analyses show the difficulty of developing a universally effective OOD test, suggesting that future OOD tests be better configured in problem-dependent ways.

__ Weaknesses__: One main question is: what are the potential differences between the proposed measure and the standard typicality test? For example, typicality is defined in terms of likelihoods where the white noise is defined as a lack of autocorrelations. What are the implications of this discrepancy for OOD detection? For instance, typicality requires a stronger IID assumption but that does not necessarily mean that typicality test cannot work in practice. (The concentration result for white noise test also relies on the IID assumption) What are the results if we apply typicality test on the residuals or on the transformed latent variables of DGMs? (As far as I know, Salisnick, et al. 2019 have not studied this matter in detail) It would be desirable for authors to discuss these matters in paper in more detail.
In addition, though the white noise test is empirically promising, it relies on the assumption that DGMs will successfully remove autocorrelation structures from their residuals for inlier samples. However, the authors do provide some empirical evidence that there autocorrelation is scarce between the residuals of deep AR models in Appendix A.

__ Correctness__: The claims and methods are reasonable and sound.

__ Clarity__: The paper is clear overall.

__ Relation to Prior Work__: While many prior works have explored the use of likelihood-based methods incorporating DGMs for OOD detection, this work is distinctive for that it proposes to test for autocorrelation structures in the residuals of DGMs instead of relying on likelihoods.
In addition, it introduces a semantic-oriented evaluation where capturing higher-level semantics is more important in contrast to the standard image datasets which can be distinguished even with very low-level information.

__ Reproducibility__: No

__ Additional Feedback__: Typos
- Line 251, "as measured" -> "as measured by"

__ Summary and Contributions__: The paper proposes a statistical test for detecting outliers, which performs much better than simple log likelihood test. The authors also explain why the log likelihood test can fail, and point to problems with designing a universal outlier detection test.

__ Strengths__: The method is well motivated and clearly explained. The empirical tests are thorough. The paper is a valuable contribution to an actively researched area, and the method presented can have many important practical and scientific applications.

__ Weaknesses__: Line 95: "While IID sequences are automatically MD and WN" is slightly inaccurate, as IID sequence may have E[x] <> 0 or Var[x] <> 1.
Table 1 should report sample sizes used and confidence intervals for the AUROC values (see e.g. https://ncss-wpengine.netdna-ssl.com/wp-content/themes/ncss/pdf/Procedures/PASS/Confidence_Intervals_for_the_Area_Under_an_ROC_Curve.pdf).

__ Correctness__: Overall yes. I wish the Authors commented more on the fact that their method depends on a deep autoregressive model (e.g. PixelCNN). How much would the results change if they used a different model?

__ Clarity__: Yes, apart from some minor problems with style, e.g. "Samples with a lower value of this statistics is considered outlier" or "The test statistics, having the form of a Bayes factor, and can also be viewed that comparing two competing hypotheses (pmodel and pgeneric) without assuming either is true [19]" on page 5.
I like the fact that the Authors take the trouble to motivate their work and make the problem easy to understand.
Conference and journal papers should not be cited as arXiv abstracts (e.g. Kingma and Welling, 2014 or Hendrycks, Mazeika and Dietterich, 2018). Titles of papers should be formatted properly ("Bayes" not "bayes", etc).

__ Relation to Prior Work__: Yes.

__ Reproducibility__: Yes

__ Additional Feedback__: I would try using the CIFAR-10.1 as an outlier set against CIFAR-10 as the inlier set.
There is also the thorny question of the method's dependence on the autoregressive model used. How much do the results depend on the model used?
=== EDIT AFTER FEEDBACK ===
The Authors have addressed some of my concerns explicitly, and hopefully other manuscript quality problems will be resolved in the camera-ready copy. I keep the score unchanged.