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REVIEW 4 major objections 5 minor 30 references

A method for classification of data with uncertainty using hypothesis testing

T0 review · 4 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Two hypothesis tests let a binary classifier abstain on ambiguous and out-of-distribution inputs using only training-data thresholds.

desk verdict A simple, clearly described abstention rule for binary classification that plausibly flags class overlap but makes an unsupported claim about OOD detection. read the letter →

arxiv 2502.08582 v2 pith:MZSIFF2C submitted 2025-02-12 cs.LG

Shoma Yokura , Akihisa Ichiki This is my paper · ORCID
classification cs.LG
keywords hypothesistestinguncertaintyquantificationbinaryclassificationrejectoptionout-of-distributiondetectionempiricaldistributionquantilethresholdpneumoniachestX-ray
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that a binary classifier can flag both kinds of dangerous inputs—data that falls in the overlap between the two classes and data that lies outside the training distribution—using only two hypothesis tests on the classifier's own score. For each class, the distribution of that score is estimated from the training data, and a central acceptance interval is cut at the $\alpha$-quantile and $(1-\alpha)$-quantile. A new input whose score falls inside both intervals is treated as ambiguous; a score outside both intervals is treated as out-of-distribution; a score inside exactly one interval is assigned to that class. The significance level $\alpha$ is set by the user, so no threshold optimization, resampling, or model modification is required. If the method holds up, high-risk applications such as clinical image diagnosis gain an explicit "I don't know" option with a tunable trade-off between coverage and accuracy.

What carries the argument

The machine is a pair of simultaneous hypothesis tests on the scalar score $g(x)$—the SVM discriminant value or the logit feeding a sigmoid—whose test statistic is the score itself. Each test's acceptance region is the central interval from the $\alpha$-quantile to the $(1-\alpha)$-quantile of that class's empirical training-score histogram. The whole argument runs on the position of the score relative to the two intervals: inside both means "both classes plausible", outside both means "neither class plausible", and these two zones are exactly the method's two kinds of uncertainty.

What would settle it

Run the method on a genuine out-of-distribution dataset whose scores mostly fall inside the training score range, for example images from a new domain that the network still scores near typical training scores; whenever the method confidently assigns those inputs to one of the two classes instead of flagging them uncertain, the claim that it detects out-of-distribution data is contradicted.

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Extended reading notes

Core claim

The central claim is that the uncertainty of a binary prediction can be read directly from the position of one scalar feature, $g(x)$, relative to two per-class empirical null distributions. For class $i$, the null hypothesis $H^i_0$ says $g(x)$ follows $F_i$, the class-$i$ distribution, and the acceptance region is the central interval between the empirical $\alpha$-quantile and $(1-\alpha)$-quantile of the training scores for that class. The decision rule is: accept both hypotheses, and the point is in the ambiguous overlap; reject both, and the point lies outside both training distributions and is reported as out-of-distribution; accept exactly one, and the point is classified as that class. The experiments reproduce the expected trade-off: a larger $\alpha$ narrows the acceptance intervals and, on the pneumonia dataset, raises coverage while lowering accuracy on the inputs that receive labels.

Load-bearing premise

The load-bearing premise is that each class's empirical training-score distribution is a valid null distribution for hypothesis tests on future inputs, so a new score that falls outside both central intervals can be read as out-of-distribution rather than as the result of calibration shift or a feature-distribution shift.

Editorial extensions

If this is right

  • Any classifier that emits a scalar score can gain an abstention rule by computing two per-class training quantiles; no resampling, architecture change, or threshold search is needed.
  • Raising $\alpha$ narrows the acceptance intervals; in the reported pneumonia experiment this raised coverage from 83.97% at $\alpha=1\%$ to 93.91% at $\alpha=5\%$ while lowering accuracy from 99.24% to 97.39%.
  • A rejection is interpretable: the method states whether the score was plausible for both classes, plausible for neither, or plausible for exactly one.
  • Applying the method to X-ray screening gives a clinical decision support in which the model declines to answer when the evidence is ambiguous, matching the "I don't know" behavior medical AI has been missing.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same recipe should transfer to any scalar confidence score, including softmax probability, distance to a decision boundary, or reconstruction error, so the abstention rule could be grafted onto models that were never designed for uncertainty.
  • Because the OOD verdict is defined purely by the training-score range, inputs that are out-of-distribution in reality but happen to score inside that range are invisible to the method; a dedicated OOD benchmark would reveal how large this blind spot is.
  • The empirical nulls ignore miscalibration; pairing the quantile intervals with calibration (for example temperature scaling of the logits) could make $\alpha$ an honest false-positive control rather than an empirical quantile statement.
  • A one-vs-rest extension of the two-test rule would give a multi-class abstention rule with the same "inside everything / outside everything" logic, but the paper only demonstrates the binary case.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper proposes a post-hoc abstention rule for binary classification. For a model that outputs a scalar feature g(x) (an SVM decision value or a logit), the method computes the empirical distribution of g over each class in the training set and defines a central acceptance interval for each class from its quantiles at the chosen significance level α. A new input is classified as class 1 or class 2 if its test statistic falls inside exactly one acceptance interval; it is declared uncertain if it falls inside both intervals (treated as class-overlap ambiguity) or outside both intervals (treated as out-of-distribution). The method is illustrated on a two-spiral SVM benchmark and on a DenseNet-121 pneumonia chest X-ray classification task, with coverage and accuracy reported for several significance levels. The authors emphasize that the method requires no resampling, no model modification, and no threshold optimization.

Significance. If the method's claims were fully supported, it would be an attractive addition to the selective-classification toolbox: it is simple, transparent, requires only training-set feature quantiles, and its threshold is interpretable as a significance level. The visual histogram-based diagnostics in Figures 2 and 5 are genuinely useful, and applying a weighted loss to handle class imbalance in the X-ray experiment is a sensible methodological choice. However, the central out-of-distribution-detection claim currently rests on no held-out OOD evaluation, and the only independent experiment does not compare against existing abstention methods. As a result, the significance of the paper is conditional on substantial additional validation rather than established by the current evidence.

major comments (4)
  1. [Abstract; §4.4–4.5] The central claim that the method "is capable of detecting ... out-of-distribution data" is not tested anywhere in the manuscript. The chest X-ray experiment uses a test set containing only 390 pneumonia and 234 normal images, which are drawn from the same two classes as the training data. Table 1 reports coverage, accuracy, recall, precision, specificity, and F1-score on those in-distribution test images, but no out-of-distribution image set is used and no metric for OOD detection (for example, AUROC for the "outside both intervals" decision) is reported. An OOD test set or a synthetic distribution shift is required before this claim can be evaluated.
  2. [§3, Figs. 2–3] The spiral experiment does not provide independent evidence for OOD detection because all data in Figure 1 are used as training data and the visualization is performed on the same input grid. Type II uncertainty is defined, by construction, as the test statistic lying outside both per-class central intervals, i.e., in the intersection of the rejection regions I1 and I2; therefore the observation that the squares in Figures 3(b)–3(d) appear outside the training envelope is a restatement of the decision rule rather than a detection result. A held-out region of the input space, a separately generated spiral, or a separate OOD dataset would be needed to test the claim.
  3. [§2.1–2.3, Eq. (4)] The method relies on an unstated identifiability assumption: that OOD inputs map to feature values t outside the central intervals of both per-class empirical training distributions. The null distribution Fi is approximated by the empirical distribution of training feature values, but this only describes where in-distribution training features lie; it does not imply anything about where OOD features will land. Because the scalar logit g(x) is not a sufficient statistic for the in-distribution versus OOD distinction, there is no guarantee that OOD inputs will fall outside the training quantile intervals. The authors should either state and empirically justify this assumption with an explicit OOD experiment, or remove the OOD-detection claim from the abstract and title.
  4. [§4.4, Table 1] The experimental results are reported without baselines. A confidence-thresholded softmax classifier, an MC-dropout variance threshold, or a conformal-prediction abstention rule would provide a minimal comparison for the coverage–accuracy trade-off, and all reported metrics are single-run values with no error bars or seed information. As it stands, Table 1 shows that the method abstains on some fraction of the test set, but it does not show that this abstention is more effective than simply thresholding the model's own confidence.
minor comments (5)
  1. [Throughout] There are several typos that should be corrected: "modefication" in the Introduction, "Applicatioin" in the Section 4 heading, "croses" in the Figure 1 caption, and "the the acceptance regions" in the description of experiment (iii) in Section 3.
  2. [§4.5] The statement in the Discussion that "coverage does not necessarily decrease as the significance level increases" is confusing because increasing α shrinks both central intervals, which affects the two uncertainty types in opposite directions; the authors should explain which effect dominates in the X-ray data.
  3. [§2.3] Equation (4) writes the empirical distribution using a Dirac delta, but the text later refers to "the interval between the 2.5% and 97.5% points"; it would be clearer to define the empirical CDF as a step function and then define the quantile interval explicitly.
  4. [References] There are duplicate entries for Arulananth et al. (2024), and the Hinton and van Camp reference is missing page numbers; these should be cleaned up.
  5. [§4.2/4.4] No information is given about the number of training runs or random seeds, and no code or data availability statement is provided, which limits reproducibility.

Circularity Check

2 steps flagged · score 6.0 of 10

OOD and ambiguity 'detection' reduces to the quantile-interval definition; no independent OOD test is performed.

  1. self definitional [Section 2.1, Section 2.3 (Eq. 4), abstract]
    "The two classes are deemed uncertain if both types of hypothesis testing can be rejected, i.e., if the value of the test statistic is an element of I1 and I2. Additionally, if neither type of hypothesis testing can be rejected, i.e., if the value of the test statistic is an element of I c 1 and I c 2, then the two classes are considered uncertain as well."

    This defines the two uncertainty classes purely by membership in the acceptance/rejection regions of per-class tests, and Eq. (4) sets those regions from the empirical training feature histograms. The abstract then claims the method 'is capable of detecting ... out-of-distribution data,' i.e., it predicts OOD by checking t ∈ I1 ∩ I2, the very condition used to define OOD. The X-ray experiment has no OOD test set, and the spiral experiment uses all data as training, so nothing falsifies the defining equivalence. The OOD claim is a restatement of the definition plus an unstated assumption that OOD inputs map outside the training quantile intervals.

  2. self definitional [Section 3 (spiral benchmark, Figs. 2-3)]
    "The purpose of this experiment is to visualize class regions. Therefore, all the data in Figure 1 are used as training data. ... A comparison of these results reveals that the proposed method effectively distinguishes between data while preserving the classification of data whose basis is ambiguous."

    Ambiguity is not measured against an external label; type I uncertainty is defined in Section 2.1 as the feature lying in both acceptance regions (the central quantile intervals of the two empirical training distributions). Marking the overlap region near the decision boundary as 'uncertain' therefore restates the construction. Because all data are training data, the figure cannot show that the method detects genuinely ambiguous inputs better than the definition already does.

full rationale

The paper's threshold rule is self-contained: thresholds are α and 1−α quantiles of per-class training feature histograms, and the X-ray experiment reports an internal coverage/accuracy tradeoff. That part is not circular. The circularity lies in the two headline capabilities. Section 2.1 defines type I uncertainty (ambiguity) as the feature lying in both central acceptance regions and type II uncertainty (OOD) as the feature lying outside both rejection regions; Section 2.3 makes these regions from the empirical training distributions (Eq. 4). The abstract then presents detection of ambiguous and OOD data as a property of the method. For ambiguity, the spiral 'benchmark' uses all data as training and simply colors the overlap of the two quantile intervals, which is the definition. For OOD, no OOD data are tested at all (the X-ray test set contains only pneumonia positive/negative), so the claim rests on the unstated assumption that OOD inputs map to feature values outside the training intervals. The observed X-ray metrics are real evidence for the familiar coverage-confidence tradeoff, but they do not test the OOD claim. One or two definitional reductions affect the central claims, so a score of 6 (partial circularity) is appropriate.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The method rests on treating training-logit histograms as true null distributions and on a decision rule that is asserted rather than derived. The only free parameter that the authors explicitly vary is α, but the SVM kernel parameter and DenseNet training details are unreported and also shape the entire pipeline. No new entities are introduced.

free parameters (3)
  • Significance level α = 1.0%, 2.5%, 5.0%
    Chosen by hand in Section 4.4; it directly determines the quantile thresholds and thus the coverage/accuracy tradeoff. The paper frames this as context-dependent rather than optimized, but it is still a free parameter.
  • SVM RBF kernel parameter γ
    Used in Section 3 but its value is not reported; it fully determines the training feature distribution and therefore all thresholds.
  • DenseNet training hyperparameters = learning rate 1e-3; epochs, batch size, seed unreported
    Training procedure affects g(x) and hence the empirical quantiles; unreported details make reproduction impossible.
assumptions (3)
  • domain assumption For each class, the feature values g(x) follow a fixed probability distribution F_i that can be approximated by the empirical distribution of training features (Eq. 4).
    Section 2.3; this is the foundation of the hypothesis tests. It ignores distribution shift, model miscalibration, and the fact that g(x) depends on the trained model.
  • ad hoc to paper A level-α central interval of the empirical distribution functions as a valid acceptance region for a hypothesis test on new data.
    Sections 2.1 and 4.4; the paper assumes the α-quantile interval has the usual hypothesis-test interpretation, but no proof of size or power is provided and the null distribution is itself estimated.
  • ad hoc to paper If a test score falls in both acceptance intervals or in both rejection regions, the correct decision is to abstain.
    Section 2.1; this decision rule is asserted, not derived from a loss function or error-cost model.

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Cite this review

Pith. "Pith review of A method for classification of data with uncertainty using hypothesis testing." pith.science (2026). https://pith.science/paper/MZSIFF2C

@misc{pith2026250208582,
  author       = {Pith},
  title        = {Pith review of: A method for classification of data with uncertainty using hypothesis testing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZSIFF2C}},
  note         = {Machine review of arXiv:2502.08582}
}
abstract

Binary classification is a task that involves the classification of data into one of two distinct classes. It is widely utilized in various fields. However, conventional classifiers tend to make overconfident predictions for data that belong to overlapping regions of the two class distributions or for data outside the distributions (out-of-distribution data). Therefore, conventional classifiers should not be applied in high-risk fields where classification results can have significant consequences. In order to address this issue, it is necessary to quantify uncertainty and adopt decision-making approaches that take it into account. Many methods have been proposed for this purpose; however, implementing these methods often requires performing resampling, improving the structure or performance of models, and optimizing the thresholds of classifiers. We propose a new decision-making approach using two types of hypothesis testing. This method is capable of detecting ambiguous data that belong to the overlapping regions of two class distributions, as well as out-of-distribution data that are not included in the training data distribution. In addition, we quantify uncertainty using the empirical distribution of feature values derived from the training data obtained through the trained model. The classification threshold is determined by the $\alpha$-quantile and ($1-\alpha$)-quantile, where the significance level $\alpha$ is set according to each specific situation.

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