Score Combining for Contrastive OOD Detection

Edward T. Reehorst; Philip Schniter

REVIEW 5 major objections 5 minor 58 references

Score Combining for Contrastive OOD Detection

T0 review · 5 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A generalized likelihood-ratio test derived from a negative-means model of z-scored inlier scores outperforms heuristic score-combining rules for contrastive out-of-distribution detection.

desk verdict Clean GLRT-based score combining for contrastive OOD detection with an honest writeup, but the claim of beating Fisher/Stouffer rests on sub-0.01 AUROC gaps with no error bars. read the letter →

arxiv 2501.12204 v1 pith:24DSEF2T submitted 2025-01-21 cs.LG

Edward T. Reehorst , Philip Schniter This is my paper · ORCID

classification cs.LG MSC 62F0362H15

keywords out-of-distributiondetectioncontrastivelearninghypothesistestinggeneralizedlikelihoodratiotestscorecombiningnegative-meansmodelconformalp-valuesanomaly

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

Out-of-distribution (OOD) detection asks whether a test sample comes from a known inlier distribution or from something new. The leading contrastive detectors, CSI and SupCSI, compute many intermediate inlier scores and merge them with hand-crafted rules. This paper proposes replacing those rules with a formal hypothesis test: each score is converted to a z-value (the inverse-normal transform of its empirical percentile), an inlier is modeled as $\mathcal{N}(0, I)$ on those z-values, and an OOD sample as having every z-coordinate mean at most $-\epsilon$; the generalized likelihood ratio test (GLRT) of this negative-means model becomes the combining rule. On dataset-versus-dataset experiments with CIFAR-10, SVHN, LSUN, ImageNet, and CIFAR-100, and leave-one-class-out experiments on CIFAR-10, the GLRT combination improves the area under the ROC curve and the detection rate at a fixed false-alarm rate over the original CSI/SupCSI rules and over Fisher, Bonferroni, Simes, Benjamini-Hochberg, Stouffer, and ALR combinations. If the claim holds, it replaces heuristic weight tuning with a principled, extensible way to fuse any number of inlier scores.

What carries the argument

The load-bearing object is the negative-means (NM) hypothesis-testing model and its generalized likelihood ratio test statistic. The model (equation 15) says that inlier z-values are independent standard normals while OOD z-values are normals with every coordinate mean at most $-\epsilon$; the GLRT statistic $t_{\mathrm{GLRT}}(z) = (\tfrac{1}{2} z^{-} - z)^T z^{-}$ with $z^{-}_l = \min\{z_l, -\epsilon\}$ is the log-likelihood ratio after replacing the unknown alternative mean by its constrained maximizer. The machinery also includes the empirical CDF transform $\hat{z}_l(x) = \Phi^{-1}(\hat{F}_l(s_l(x)))$ that converts arbitrary score distributions to z-values, and a conformal p-value threshold (Algorithm 3) that certifies a user-specified false-alarm rate up to a chosen failure probability using a validation set.

What would settle it

Measure the off-diagonal correlations of the empirical z-value vectors on a held-out inlier set; if a GLRT built with the estimated inlier covariance (Appendix A.2) does not beat or match the identity-covariance GLRT on the same novelty sets, the independence assumption is doing essential work rather than being a harmless device.

Watch

Extended reading notes

Core claim

The paper's central claim is that score combining for contrastive OOD detection should be posed as a negative-means hypothesis test on z-values rather than as a heuristic sum, max, or weighted average. Under the null hypothesis $H_0 : Z \sim \mathcal{N}(0, I)$, each empirical z-value is standard normal; under the alternative $H_1 : Z \sim \mathcal{N}(\mu, I)$ with $\mu_l \le -\epsilon$ for every $l$, an OOD sample is, on average, worse on every score. Solving this composite testing problem with a GLRT yields the statistic $t_{\mathrm{GLRT}}(z) = (\tfrac{1}{2} z^- - z)^T z^-$, where $z^-_l = \min\{z_l, -\epsilon\}$. The paper claims that thresholding this statistic outperforms the heuristic combining rules of CSI and SupCSI on the tested image benchmarks, and also outperforms Fisher, Bonferroni, Simes/BH, Stouffer, and ALR p-value combiners.

Load-bearing premise

The derivation assumes the z-values of an inlier are independent standard normals; the paper says this independence does not hold, so if the scores are strongly correlated, the GLRT is no longer a likelihood-ratio test for the true distribution.

Editorial extensions

If this is right

New inlier scores can be added to a contrastive detector by appending their empirical z-values, since the GLRT is agnostic to how the scores were produced; glrt-SupCSI+ combines 24 scores and improves dataset-vs-dataset AUROC over glrt-SupCSI.
The GLRT removes the need to hand-tune CSI's weighting coefficients $\lambda^{\mathrm{con}}_j$ and $\lambda^{\mathrm{shift}}_j$, because every score enters through its own z-value.
A validation set of size $v$ gives a threshold such that, with probability at least $1-\delta$, the false-alarm rate is at most $\alpha$, via the beta-distributed conformal p-value described in Section 3.3.
A positive offset $\epsilon$ (the paper uses 0.25) improves AUROC over the $\epsilon = 0$ cone problem.
The benefit of adding extra base scores is not guaranteed: glrt-SupCSI+ underperforms glrt-SupCSI on leave-one-class-out CIFAR-10, so base-score selection remains part of the design.

Reading between the lines

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

Editorial inference: the average AUROC gap between the GLRT and the best classical combiners (Fisher, Stouffer) is well under a percentage point, while Bonferroni, Simes/BH, and ALR are much further behind, which suggests the GLRT's practical value is stability across datasets rather than a large leap over a well-chosen classical merger.
Editorial inference: because the independence assumption in (15) is false, a natural stress test is to compare the GLRT against a covariance-aware GLRT on benchmarks with strongly correlated scores; the paper's appendix finds the estimated-covariance variant underperforms, hinting that the identity covariance acts as a helpful inductive bias.
Editorial inference: the conformal false-alarm machinery of Section 3.3 does not depend on the GLRT statistic itself, so any score combiner could inherit the same finite-sample threshold guarantee by reusing Algorithm 3.
Editorial inference: since the pipeline only sees marginal z-values, it could fuse heterogeneous evidence (classifier logits, nearest-neighbor distances, density estimates) in other one-class and anomaly-detection tasks, not just contrastive image representations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take on 2501.12204. The paper does something real: it replaces the heuristic score-averaging rules in CSI/SupCSI with a GLRT derived from a negative-means model over empirical z-values, and it adds a conformal calibration step that carries a finite-sample false-alarm guarantee. The derivation of t_GLRT (eq. 20) is correct, the writing is clear, and the honesty is refreshing: Section 3.1 states plainly that the independence assumption in (15) is not expected to hold, Appendix A.2 actually tests a covariance-generalized variant, and the Limitations paragraph admits that adding scores can hurt, which their own SupCSI+ results show.

What is genuinely new: applying the convex-cone GLRT of Wei et al. (2019) to combine self-supervised OOD scores, with the epsilon > 0 extension and the empirical-CDF z-transform. Modest but real. The conformal false-alarm calibration in Section 3.3 is the cleanest part of the paper and is directly reusable.

Soft spots, in proportion. The headline claim, that GLRT beats Fisher and Stouffer, is not established by the evidence. Tables 6-9 and 13-14 show average AUROC gaps of 0.0002-0.009 and DR gaps of a point or less, apparently from a single run per condition: no error bars, no seeds, no paired tests. In Table 8 (DR at 5% FAR with CIFAR-10 inliers), Stouffer actually wins. At that resolution the honest summary is 'competitive with, occasionally marginally better than, Fisher/Stouffer.' Second, epsilon = 0.25 is chosen using one of the evaluation tasks (Figure 4, CIFAR-100 vs CIFAR-10). The paper discloses this and fixes epsilon for all tasks, but the theoretically motivated epsilon = 0 was nonetheless abandoned on the basis of that single plot; a nested selection would be cleaner. Third, the independence assumption is false, but the stress-test note overweights it: Appendix A.2 shows identity covariance beats an estimated covariance, so the misspecification does not impeach the empirical comparison, it only voids the theoretical optimality claim. And no code is released, which sharpens the missing error bars.

Who gets value: anyone working on OOD score ensembles or p-value/z-value combining. The GLRT formulation is worth engaging and the calibration trick is directly usable. I would not desk-reject on the empirics, but a referee should demand multiple seeds, error bars, and a proper evaluation of epsilon. Recommendation: yes, send to peer review. It is an honestly written methods paper with a correct derivation and a useful calibration tool; the empirical overreach is fixable with compute and a more careful writeup.

Referee Report

5 major / 5 minor

Summary. The paper proposes to replace the heuristic score-combining rules used by the CSI and SupCSI contrastive OOD detectors (Eqs. (6) and (8)) with a generalized likelihood ratio test (GLRT) derived from a negative-means model on z-values obtained from an empirical CDF transform of the base scores (Eqs. (15) and (20)). The authors also give a conformal-prediction procedure that, using a validation set, yields a false-alarm-rate guarantee (Algorithms 2 and 3). Experiments on dataset-vs-dataset tasks with CIFAR-10, SVHN, LSUN, ImageNet, and CIFAR-100, and on leave-one-class-out tasks with CIFAR-10, claim that the GLRT combination outperforms the original CSI/SupCSI heuristics and the classical Fisher, Bonferroni, Simes/BH, Stouffer, and ALR combiners.

Significance. The statistical derivation of the GLRT statistic is sound and the proposed framework is a principled alternative to ad-hoc ensembling of contrastive scores. The false-alarm-rate guarantee in Section 3.3 is a useful contribution, as is the appendix's explicit examination of the independence assumption through a general-covariance variant. However, the central empirical claim of outperformance is currently not statistically established: the reported gains over Fisher and Stouffer are very small in AUROC and detection rate, the experiments appear to be single-run, and the single tuned parameter epsilon is selected using one of the evaluation tasks. If the empirical claim could be supported by repeated-seed results with proper selection of epsilon, the paper would be a genuinely useful contribution to OOD detection methodology.

major comments (5)

[Section 4.2, Figure 4] The free parameter epsilon in Eq. (15) is set to 0.25 by tuning on the CIFAR-10-inliers vs CIFAR-100-novelties task, which is one of the evaluation tasks reported in Tables 2, 4, 6, and 8. This leaks OOD information into the method and makes the reported improvements over eps=0 and over the classical combiners hard to interpret. The authors should either choose epsilon on a held-out validation split that does not use any evaluation OOD dataset, or report the main comparisons for a range of epsilon values, including epsilon=0, to show that the conclusions are not artifacts of this choice.
[Tables 6-9 and 13-14] No error bars, confidence intervals, number of seeds, or paired significance tests are reported. The average AUROC differences between GLRT and Fisher/Stouffer are as small as 0.0002-0.009 (e.g., Table 6 avg 0.9677 vs 0.9670 for Fisher and Stouffer; Table 7 avg 0.9928 vs 0.9926/0.9916), and the DR differences are 0.1-1.0 percentage points. Given the stochasticity of contrastive training, these differences are within plausible run-to-run noise. The central claim that the GLRT outperforms Fisher and Stouffer therefore needs repeated-seed experiments with variance reporting or paired tests before it can be accepted.
[Section 3.1, Eq. (15)] The authors explicitly state that the independence assumption in Eq. (15) is not expected to hold in practice. The GLRT derivation and its Neyman-Pearson/minimax optimality rely on this assumption, while the actual z-values are derived from correlated scores. Appendix A.2 partially addresses this by substituting a sample covariance, but it only compares identity versus sample covariance on the dataset-vs-dataset AUROC tables and finds that identity performs better. This does not resolve the misspecification concern for correlated non-Gaussian z-values. The authors should either temper the optimality language to describe the GLRT as a heuristic score combination that is principled only under the NM model, or provide additional diagnostics on the actual distribution of the z-vectors.
[Section 4.3, Tables 8-9] The claim that the GLRT outperforms all classical combiners is not uniformly supported by the tables. In Table 8, Stouffer achieves an average DR of 86.3 compared to 86.2 for GLRT, and in Tables 8-9 GLRT and Fisher tie on average. The text in Section 4.3 acknowledges this, but the abstract and conclusion state the outperformance claim without this qualification. The claims should be restated as 'on average over AUROC' and should acknowledge that the advantage over Fisher/Stouffer is not consistent across all metrics and tasks.
[Section 3.3, Eq. (24)] The false-alarm-rate guarantee is stated as holding 'with probability at least 1-delta' over the random validation set, and the procedure in Algorithm 2 searches for a threshold a that controls the beta upper tail. This is a valid conformal-style argument, but the guarantee is average over the validation set under the assumption that Xval is drawn i.i.d. under H0. In the experiments, no results using this finite-sample guarantee are reported; all DR-versus-FAR results instead use a threshold computed directly on the test set. The paper should either include experiments demonstrating the conformal procedure, or explicitly state that the guarantee is a methodological contribution not evaluated empirically.

minor comments (5)

[Abstract and Section 1] The name 'Benjamini-Hochwald' appears in the abstract and contribution list; this should be 'Benjamini-Hochberg'.
[Algorithm 3] In Algorithm 3, the line 'return “OOD” if bq(x) ≤ a' is preceded by 'white guaranteeing a false-alarm rate'; this should read 'while guaranteeing'.
[Section 5] The conclusion refers to 'the non-negative means problem (15)', but Eq. (15) is the negative-means problem with means ≤ -ϵ; the wording should be corrected for consistency with Section 3.1.
[Figure 2] Figure 2 plots the proposed test statistic for m=1, but the paper does not mention in the caption or text that this is the univariate case; a brief note would avoid confusion.
[Table 1] The row for glrt-SupCSI+ marks both 'Heuristic' and 'GLRT' columns as applicable; it would be clearer to state explicitly that the base scores themselves are computed with the SupCSI/OC-SVM/Mahalanobis methods while the combining is GLRT-based.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: GLRT statistic is a standard derivation from an explicit model, comparisons are external benchmarks, and the disclosed ε selection is a hyper-parameter choice, not a fitted prediction.

full rationale

The derivation is self-contained and the empirical claims are not equivalent to their inputs. The empirical z-values in (21) are probability-integral transforms of the inlier score CDF, so their marginal null distribution is standard normal by construction; the paper explicitly acknowledges that the independence assumption in (15) is false and probes the misspecification in Appendix A.2 by substituting the sample covariance. The GLRT statistic (20) is obtained by a standard closed-form likelihood-ratio computation from the stated negative-means model (15), with no benchmark result or target AUROC used as an input. The comparisons against CSI/SupCSI and against Fisher, Bonferroni, Simes/BH, Stouffer, and ALR use identical base scores and are external empirical benchmarks, so outperformance is not implied by the model construction. The paper contains no load-bearing self-citations: the only optimality citation (Wei et al., 2019) is external and covers the ε=0 special case, while the ε=0.25 choice is a disclosed hyper-parameter selection made on one evaluation pair; that choice could affect one table row but is not a fitted outcome renamed as a prediction. The acknowledged limitations (which base scores to combine, and dependence on transformation sets) are scope restrictions rather than hidden circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The proposed method rests on standard statistical results (GLRT, conformal prediction) plus several domain assumptions. One free parameter (epsilon) is tuned on a subset of the evaluation tasks. No invented entities are introduced.

free parameters (1)

epsilon (NM model parameter) = 0.25
Chosen by examining AUROC vs epsilon in Figure 4 for the CIFAR-10 inlier / CIFAR-100 novelty task, then fixed for all other tasks. This uses OOD data to set a parameter of the detector.

assumptions (5)

domain assumption Under H0, the z-values Z_l = Phi^{-1}(F_l(s_l(X))) are i.i.d. standard normal.
The empirical z-value construction assumes the training-set CDF estimates the true CDF and that scores are independent across l. Section 3.1 explicitly states independence is not expected to hold in practice, so this is a simplifying modeling assumption.
domain assumption Under H1, each z-value has mean at most -epsilon, i.e., all component means are shifted negative.
This negative-means model (15b) is the core hypothesis tested by the GLRT. It assumes OOD samples depress all base scores simultaneously, which may not hold for all novelty types.
domain assumption The empirical CDF bF_l provides a consistent estimate of the true null CDF, so bz_l is approximately standard normal under H0.
Algorithm 1 uses training-data empirical CDFs to convert scores to z-values; this requires the training distribution to match the test inlier distribution.
standard math The conformal p-value calibration in Algorithms 2-3 provides the stated false-alarm-rate bound.
Follows from Vovk (2012) results on conformal p-values and the beta distribution of the false-alarm rate given a random validation set.
ad hoc to paper epsilon=0.25 is an appropriate fixed value for all tasks.
The value is selected from one task (Figure 4) and applied everywhere, with no theoretical justification for why a single positive epsilon should work across diverse datasets.

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Cite this review

Pith. "Pith review of Score Combining for Contrastive OOD Detection." pith.science (2026). https://pith.science/paper/24DSEF2T

@misc{pith2026250112204,
  author       = {Pith},
  title        = {Pith review of: Score Combining for Contrastive OOD Detection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24DSEF2T}},
  note         = {Machine review of arXiv:2501.12204}
}

read the original abstract

In out-of-distribution (OOD) detection, one is asked to classify whether a test sample comes from a known inlier distribution or not. We focus on the case where the inlier distribution is defined by a training dataset and there exists no additional knowledge about the novelties that one is likely to encounter. This problem is also referred to as novelty detection, one-class classification, and unsupervised anomaly detection. The current literature suggests that contrastive learning techniques are state-of-the-art for OOD detection. We aim to improve on those techniques by combining/ensembling their scores using the framework of null hypothesis testing and, in particular, a novel generalized likelihood ratio test (GLRT). We demonstrate that our proposed GLRT-based technique outperforms the state-of-the-art CSI and SupCSI techniques from Tack et al. 2020 in dataset-vs-dataset experiments with CIFAR-10, SVHN, LSUN, ImageNet, and CIFAR-100, as well as leave-one-class-out experiments with CIFAR-10. We also demonstrate that our GLRT outperforms the score-combining methods of Fisher, Bonferroni, Simes, Benjamini-Hochwald, and Stouffer in our application.

Figures

Figures reproduced from arXiv: 2501.12204 by the authors.

**Figure 1.** Left: Histogram of CSI’s “norm” score {s1(xi)} n i=1 (top) and “cos” score {s2(xi)} n i=1 (bottom) for CIFAR-10 inliers X cifar10 test . Center: Histograms of the corresponding empirical z-values {zb1(s1(xi))} n i=1 and {zb2(s2(xi))} n i=1 with zb1(·) and zb2(·) constructed using CIFAR-10 inliers X cifar10 train . Right: Scatter plot of zb1(s1(xi)) (horizontal axis) versus zb2(s2(xi)) (vertical axis) for CIFAR-10 in… view at source ↗

**Figure 2.** Left: Plot of tGLRT(z) versus z ∈ R for ϵ ∈ {0, 0.1, 0.2, 0.3}, along with tstouffer(z) = z and tfisher(z) = ln Φ(z). Right: a plot of the same test statistics after shifting and scaling t(z) so that Pr{t(z) ≤ 0 | H0} = 0.1 and t ′ (z0) = 1 for z0 such that t(z0) = 0. Algorithm 1 The proposed GLRT-based OOD detector for a given threshold τ Require: test sample x, test threshold τ , training samples {xi} n i=1, score… view at source ↗

**Figure 3.** Examples of the beta pdf from (24) and thresholds [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: AUROC of glrt-SupCSI+ vs. GLRT parameter [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]

**Figure 5.** Figure 5: AUROC of eigen-score teig,k(z) versus eigenvalue λk for k = 1, . . . , m on two m = 24 score experiments from Section 4: CIFAR-10 inliers with SVHN novelties (left) and SVHN novelties with CIFAR-10 inliers (right). Next, we perform an eigen-analysis of the log-GLRT sco…

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