Combine and Conquer: A Meta-Analysis on Data Shift and Out-of-Distribution Detection
Authors: Eduardo Dadalto Câmara Gomes
Florence Alberge
Pierre Duhamel
Pablo Piantanida
Venue: Accepted by TMLR ()

Abstract: This paper introduces a universal approach to seamlessly combine out-of-distribution (OOD) detection scores. These scores encompass a wide range of techniques that leverage the self-confidence of deep learning models and the anomalous behavior of features in the latent space. Not surprisingly, combining such a varied population using simple statistics proves inadequate. To overcome this challenge, we propose a quantile normalization to map these scores into p-values, effectively framing the problem into a multi-variate hypothesis test. Then, we combine these tests using established meta-analysis tools, resulting in a more effective detector with consolidated decision boundaries. Furthermore, we create a probabilistic interpretable criterion by mapping the final statistics into a distribution with known parameters. Through empirical investigation, we explore different types of shifts, each exerting varying degrees of impact on data. Our results demonstrate that our approach significantly improves overall robustness and performance across diverse OOD detection scenarios. Notably, our framework is easily extensible for future developments in detection scores and stands as the first to combine decision boundaries in this context. The code and artifacts associated with this work are publicly available\footnote{\url{https://github.com/edadaltocg/detectors}}.

Published in Transactions on Machine Learning Research (07/2024)
Combine and Conquer: A Meta-Analysis on Data Shift and Out-of-Distribution Detection
Eduardo Dadalto Florence Alberge", Pierre Duhamel Pablo Piantanida
'Laboratoire des Signaux et Systèmes (I22),, Université Paris-Saclay
"SATIE Laboratory, Université Paris-Saclay, ENS Paris-Saclay, CNRS
BILLS International Laboratory Learning Systems, Mila Quebec AI Institute
*CNRS, CentraleSupélec
Reviewed on OpenReview:
Abstract
This paper introduces a universal approach to scamlessly combine out-of-distribution (OOD) detection scores. These scores encompass wide range of techniques that leverage the self- confidence of deep learning models and the anomalous behavior of features in the latent space. Not surprisingly, combining such varied population using simple statistics proves inadequate. To overcome this challenge, we propose quantile normalization to map these scores into p-values, effectively framing the problem into a multi-variate hypothesis test. Then, we combine these tests using established meta-analysis tools, resulting in a more effec- live detector with consolidated decision boundaries. Furthermore, we create a probabilistic interpretable criterion by mapping the final statistics into a distribution with known param- eters. Through empirical investigation, we explore different types of shifts, each exerting varying degrees of impact on data. Our results demonstrate that approach significantly improves overall robustness and performance across diverse OOD detection scenarios. No- tably, our framework is easily extensible for future developments in detection scores and stands as the first to combine decision boundarics in this context. The code and artifacts associated with this work are publicly available
1 Introduction
Deploying AI systems in real-world applications is not without its challenges. Although these systems are evaluated static scenarios, they encounter a dynamic and evolving environment in practice. One of the most pressing issues is preventing and reacting to data distribution shift (Quionero-Candela et al. 2009). It occurs when the data distribution used to train AI model no longer matches the data required process in test time. It can happen gradually or suddenly and can be caused by various factors, e.g., changes in user behavior or degradation in operating conditions, which can have severe consequences in safety-critical applications (Amodei et al. 2016 such as autonomous vehicle control (Bojarski et al. 2016) and medical diagnosis (Subbaswamy Saria 2020). For instance, a predictive model of the Earth's 's temperature based on historical data may face challenges due to the evolving nature climate change. Historical patterns and trends may become less reliable indicators of future temperature changes, which could undermine the dangers of climate change unless mechanism detect such drifts in place.
Modern machine learning models can be difficult and expensive to adapt. Even though shifts in distributions can result in significant performance decline, in reality, distributions also undergo harmless shifts (Gemaque et al. 2020 As a result, practitioners should focus on discerning detrimental shifts that harm predictive performance from unimportant shifts that have little impact. This paper explores ways to improve the delection of performaance-dggradingg shifts by ensembling existing detectors an unsupervised manner. Each

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detector can be formalized test of equivalence of the source distribution (from which training data is sampled) and larget distribution (from which real-world data is sampled) through the lens of a predictive model. Our approach is motivated by the fact that different detection algorithms may make trivial mistakes in different parts the data space without any assumptions on the test data distribution (Birnbaum 1954).
The challenge to develop a widely applicable method for combining delectors to alleviate individual catastrophic mistakes.
Combine and Conquer draws inspiration from meta-analysis Glass 1976). which consolidates findings from various statistical hypothesis tests to derive unified estimate. To the best of our knowledge, this is the first work employing such methodologies in out-of-distribution (OOD) and data distribution shifts detection. Given that the underlying distributions (in-distribution and out distribution) are unknown and only partial information (derived from the training and validating samples) is available about one of the hypotheses, each score may carry relevant information for the decision. Each score induces a distinct probability mass transformation of the same data point, contingent upon the underlying hypothesis under test. Viewed in this light, this multi-score approach can be seen as means of enhancing the diversity of the scores by aggregating different scoring mechanisms, which is useful for testing different out-of-distribution scenarios. Consoquently, this increases the likelihood that at least one will successfully identify the correct hypothosis.
We summarize our contributions as follows:
1. We present simple and convenient ensembling algorithm for combining existing out-of-distribution data detectors, leading to better generalizability by incorporating effects that may not be apparent in individual detectors.
A probabilistic interpretable detection criterion is obtained by adjusting the final statistics to align with a distribution characterized by known parameters.
3. A framework to adapt any single example detector to window-based data shift detector.
We validate our contributions through a comprehensive empirical investigation encompassing classic OOD detection and introduce a benchmark on window-based data distribution shift detection.
2 Related Works
Window-based data shift detection. This line of work proposes methods for detecting shifts in data distribution using multiple samples. Lipton et al. (2018) presents a technique for detecting prior probability shifts. Rabanser et al.] (2019) studics two-sample tests with high dimensional inputs through dimensionality reduction techniques from the input space to a projected space. Cobb & Looveren
(2022) explores two sample conditional distributional shift detection based on maximum conditional mean discrepancies to segment relevant contexts in which data drift is diminishing. These studies, along with our own, demonstrate detection methods for detecting shifts in windowed data. For survey adapting models to these shifts, please refer
to Gama et al. 2014) and Lange et al. 2022).
Misclassification detection. Misclassification detection aims to reject in-distribution samples misclassified in test time with roots in rejection option (Chow 1957) and uncertainty quantification (Abdar et al. 2021 A natural baseline is the classification model's 's maximum softmax output (Hendrycks & Gimpel 20177 Geifman
& Yaniyy 2017). Other works (Granese et al., 2021), introduced a framework that considers the entire probability vector output to detect misclassifications. Gal & Ghahramani (2016); Lakshminarayaaaa et al.)
(2016) are popular approaches for estimating uncertainty from a Bayesian inference perspective. Even though this line of work focuses mainly on detecting problematic in-distribution samples while we focus on distributional drifts, our framework could be extended to it.
Out-of- distribution detection. OOD detection is also referred to in the literature as open-set recognition Geng et al. 2021). ono-class novelty detection (Pimentel et al.) (014), or semantic anomaly detection
WWng et al. 2020). Overall, methods are taxonomized into confidence- based Hein et al. (2019) Hendrycks
& Gimpel [0017 Liang et. al. (2018); Hsu et al. (2020): Liu et al. 2020 Hendrycks et al. 2002 Sun & Li
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2022
which rely the logits and softmax outputs; feature-based (Sastry & Oore
2020
Quintanilha al,
2019
Huang et al.
2021 Zhu et al.
2022
Colombo et. al.
2022:
Dong et al.) 2021; Song et al. 2022 Lin
et a..
2021 Djurisic et al. 2023 Lee et al.
2018 Ren et al.
2021
Sun et al. 2022 Darrin et al. 2023) which
explores latent representations: mixed fcature-logits (Sun ct al.
2021
Gomes ct al.
2022
Wang ct al. 2022
Dadalto et al. 2023; Djurisic et al.
2023): ; training, likelihood estimation and reconstrucl ion based (Schleg)
ct al. 2017 Vernekar et al.) 2019: Xiao et al. 2020 Ren al. 2019 Zhang ct al., 2021 Kirichenko et al.
2020) methods. We consider these methods to be complementary
our work as they focus on developing single discriminative OOD scores. The authors in Haroush et al. (2022) propose comparable approach
for OOD detection, framing it as statistical hypothesis testing issue. They aggregate p-values based on statistics obtained from various channels of a single convolutional network in a hierarchical manner. However, this approach is heavily reliant on the architect ure of convolutional neural networks and dimension reduction functions. It does not account for the correlation between the test statistics, as highlighted in Section 4.2 therein. Morcover, a recent benchmark (Zhang et al.) (202)) shows no evident winner detecting OOD data. In this paper, we introduce novel approach that involves combining multiple detectors to enhance performance and mitigate the risk of catastrophic failures when specific method fails to detect certain types of data.
3 Preliminaries and Methodology
This section discusses the methodology for detecting distribution shifts in high dimensional data streams inputted to deep neural networks. We define data stream in Section 3.1 we recall the various types of shifts in Section 3.2 and we formalize single sample and window-based detection in Section 3.3
3.1 Background
Let X' C Rd be continuous feature space, and let y [1....,C) C} denote the label space related to some task of interest. We denote by Pxy and yxy the underlying source and target probability density functions (pdf) associated with the distributions P and model x + y is trained on some training set D, = {(11, (1), ..... (æn, that, given an input æ E X, outputs prediction on y, i.e., f(x) = arg maxyey Ppix (y x). At test time, an unlabeled sequence of inputs or data stream is expected, sampled from the marginal target distribution dx
X' 3/ respectively. We assume that machine learning
Bn)) ~ pxy, which yields model
Definition 3.1 (Data stream). A data stream S is a finite or infinite sequence of not necessarily independent observations typically grouped into windows (i.e., sets WITh = Ij+m-13 gx) of same size m,
S = ==( (111 am--
U WTT j=1
(1)
3.2 Data-Shift
In real-world applications, data streams usually suffer from well-studied phenomenon known as data dis- tribution shift? (or data shift for short). Data shift occurs when the test data joint probability distribution differs from the distribution a model expects, i.e., Pxy v) # qxy v). Due to this mismatch, the model's response may suffer a drop in accuracy. Lot B E [0 be a mixture coofficient, we will write the true joint test pdf gxy as a mixture of pdís p and
yXY (z, v) = (1- 3) (x + (xx
(2)
Note that when B = 0, the test distribution matches the training distribution, i.e., there no shift. Con- versely, when 1, we have the largest shift between training and testing environments. In this work, we
Also referred to in the literature data distribution drift.
We assume that unknown and differs significantly from p. te. 1/2 Sxxx Im(=) - u(=)ldz
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focus detecting when any kind of data shift happens between the training and testing distributions and not estimating the mixing parameter 3 or the true pdls involved.
One could categorize different kinds of shifts that may happen by decomposing a joint pdf into
((, Q (xx
QX
(3)
posterior covariate
likelihood prior
Briefly, novelly drift, also referred to as concept evolution (Masud et al. 2011), is usually attributed to the presence of novel classes or concepts. As a result, the conditional distribution is not adapted anymore, i.e, P(Y]X) + QX).X). Naturally, (( Q(Y), and Q(X) are allowed to change. Covariate shift often happens because the input data comes from a different domain as g(X) changes, e.g., in image recognition, the drawing of concepts are introduced in testing, while the training features are natural pictures only. Finally, a prior shift or label shift usually occurs when the test label distribution is biased towards some classes, e.g., the majority of samples in testing come from a class in which the predictive model less proficient, causing overall accuracy degradation. All of these data shifts may have negative impacts on the model. Shifts that do not affect the classifier's performance are referred to virtual drifts, e-g., mild corruptions to images may result in ((X) drifting but without affecting Q(Y]X). However, this list is not exhaustive the principal objective of this paper does not lie in the precise categorization of diverse drift phenomena but rather in the establishment of a more robust detection framework for distinet scenarios.
Considering our objective of enhancing the overall reliability of Al systems real-world applications, our focus detecting any form of data drift that might result model deterioration without having access to labeled samples. Consequently, our primary emphasis will be on detecting data drift by examining the discriminative model, as the high dimensionality of the imput data poses significant challenges for generative modeling approaches. Through empirical validation, we demonstrate that our strategy can proficiently detect novelty and covariate shifts by measuring detection performance on provoked covariate shift and when introduced novel concepts.
3.3 Detection Framework
Predictions on production AI system can be made sample by sample or window by window in data stream. Both can be interpreted as a statistical hypothesis test.
3.3.1 Single Example Detection
On single example level (equivalent to OOD detection), let s (x, + R be a confidenco-wware score function that measures how adapted the input is to the model. A low score indicates the sample is untrustworthy, and a high value indicates otherwise. This score can be simply converted to a binary detector through a threshold 7 E IR, i.e., d(-) [s(+,)) S 7]- Finally, the role of the system (d, f) is only to keep a prediction if the imput sample x is not rejected by the detector d, i.c., if d(x) = O. This setup is identical to novelty, anomaly, or OOD detection. Formally, the null and alternative hypothesis writes:
Ho : (X, ~ pxy and HA (X, 8) ~ OXY.
(4)
We assume that the score functions are confidence oriented, i.c., greater values indicate more confidence in prediction. So, we frame the statistical hypothesis test as a left-tailed test (Lehmann & Romano 2005). Even though single-sample detection is adapted for anomaly detection, is not well adapted for detecting distribution shifts.
3.3.2 Multiple Examples Detection
In a window based detection scenario, we make the assumptions that 1.) there are multiple reference sam- ples available, 2.) the instance's class label are not available right after prediction, and 3.) the model is not updated. So, given reference window WT ~ Pxy with samples and test window WTT (=f...., z'm) ~ 9x with sample size m, our task is to determine whether they are both sampled from the source distribution or,
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equivalently, whether Pxy (x: y) equals 9xy (x',9) where y' =(x'). The mull and alternative hypothesis of the two-sample test of homogeneity writes:
Ho PXY (x, v) axP (x', 1') and HAT Pxy (x, y) # 9x8 (x',0).
(5)
In this case, the mull hypot that the two distributions are identical for all (æt u); the alternative is that they are not identical, which is a two-sided test. As testing this mull hypothesis on a continuous and high dimensional space is unfeasible, we will compute a univariate score on each sample of the windows. With a slight abuse of notation let 8 (W", ; (s(21, f),...; s(mm; f)} be multivariate prozy variable to derive unified large-scale window-based data shift detector. To compute the final window score, rely the
Kolmogorov-Smirnov MMsssy 1951 (wo-sample hypothesis lest over the proxy variable. The test statistic
writes:
KS(WF, WT)) = sup 6E3
Fm (T) (6)-
(6)
where Py and FTT the empirical cumulative distribution functions (ccdf) of the scores of each sample of the first and the second widows, respectively, defined in Equation 77). Finally, the KS statistic is compared to threshold y E IR to obtain the window-based binary detector D(.) [KS(:, WT) S 7).
4 Main Contribution: Arbitrary Scores Combination
This section explains in detail the core contribution of the paper: an algorithm to effectively combine arbitrary detection score functions from a diverse family of detectors. Section 4.1 discusses why basic statistics may fail to combine OOD detectors and motivates a more principled approach based on meta- analysis (Glass] 1976), statistical technique that combines the results of multiple studies to produce a single overall estimate.
The first step of our Combine and Conquer algorithm is to transform the individual scores into p-values through quantile normalization (Conover & Iman 1981 (ef. Section 4.2 Then, with multiple detectors, the p-values can be combined using a p-valuc combination method (cf. Section 4.3) Finally, we introduce an addit ional stalistical treatment, since the p-values of the multiple Lests over the same sample are often correlated (cf. Section 4.4
4.1
Simple Normalization for Score Aggregation May Fall Short
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Energy
Probabiltty Density
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Maha
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Probabliity
Probability
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Energy
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Maha
Ours
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Score
Transformed Score
Test Statistic
(a) Score's pdf (in-distribution).
(b) Quantile transformation's edf.
(c) Combined p-value edf.
Figure 1: Illustration of the three steps of the Combine and Conquer algorithm. This example shows three disparate score functions evaluated on in-distribution data. Our main experiments combine 14 scores.
Common approaches for combining different detection scores often revolve around calculating mean de Car- valho 2016) of the scores while incorporating certain assumptions. These assumplions typically entail con- siderations such as whether all scores should contribute equally to the final composite score, or a weighted sum should be computed, assigning greater importance select methods. Additionally, there's the question of whether outlier score values should be favored over others, and whether a more conservative or permissive approach should be adopted in score combination. For instance, using the product of available scores could
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yield low combined score if any individual scores are low, while selecting the minimum maximum value among all anomaly scores can influence the method to be more conservative permissive. While these combination methods are all viable, their effectiveness heavily relies on the distributional characteristics of the involved scores. Given that the choice of aggregation mothod hinges on the data's characteristics, it's pertinent to delve into the unique attributes of OOD detection scores.
One inherent limitation in OOD detection is the absence of access to sufficiently representative dataset of outlier data, which poses challenges for techniques like metaleorning Opitz & Maclin (1999) and other supervised ensembling methods that require data to train meta-model. Additionally, detection scores often exhibit distinct distribution shapes with varying moments, as illustrated in Figure la To mitigate some of these effects, several simple statistical approaches are commonly employed. One such method is normal standardization or z-score normalization, where each individual score random variable S; = s; (x f) is transformed into standard score Z, = (S; S,)/as,, where Zz represents the distance between the raw score and the population mcan in units of the standard deviation as.. While this approach corrects for the first two moments of the distributions, does not account for skewness, kurlosis, or multimodality. Another frequently used normalization technique is min-max scaling, which involves transforming scores to fall within the range of zero and one using statistics Z; = (S min S (max S; min S;). However, min-max scaling fails to address many other characteristics and does not provide control over the resulting distribution' 's moments, making the task of combining scores more challenging. To tackle this issue, we emphasize the importance of pre-processing the scores using quantile normalization instead.
4.2 Quantile Normalization: Managing Disparate Score Distributions
Each detector's s score r.v. S ===;; (X, follows very different distributions depending on the model's archi- tecture, the dataset it was trained on, and the score function 8;- To combine them effectively, we propose first to apply a quantile normalization (Bolstad et al., 2003), which exhibits interesting statistical properties (Gallón et al., 2013). Let S Ω + I. be a continuous univariate r.v. captured by a cumulative density function (cdf) (6) = Pr(S, $ 5) for i {I.....k} and E R. Its empirical edf F; R + [0,1] is defined by
1 E1 - Σ
(T) =
11S
(7)
which converges almost surely to the true edf for every 5 by the Dvoretzky Kiefer Wolfowitz Massart in- equality (Massart) [1990). We are going to estimate this function using a subsample of size of the training or validation set if available. The resulting r.v. is uniformly distributed in the interval [0, 1]. As result, for each detector i and sample æ, we can oblain a p-value:
pa(x) Pri, (S: S sa ()) Pr Pr S (x, Ho) v FT (c(
(8)
A decision is made by comparing the p-value to desired significance level If p < a, then the null hypothesis Ho is rejected, and the sample is considered OOD, Even though we derived everything for the single example delection case, this formulation can be exlended to the window-based scenario.
4.3 Combining Multiple P-Values
Our objective is to aggregate set of k 2 2 scores (or p-values) so that their synthesis exhibits better properties, such as improved robustness or detection performance, by consolidating each method's decision boundaries. Unfortunately, since q is unknown and is hard to estimate, designing an optimal test is unfeasible according to Neyman-Pearson's Pearson's Fundamental Lemma [Lehmann & Romano 2005). However, there are several possible empirical combination methods, such as Tippett (1931) min; P... Neyman & Pearson (1933) 2 E In(1-1)) - (p)), Wilkinson 1951] max; Pts Edgington (1972) 1/k >m P,, and Simes (1986) min; t for sorted p-values. We are going to explore in detail the Fisher's 's method (Fisher 1925 Mosteller & Fisher 1948) in the main manuscript, also referred to as the chi-squared method, and Stouffer's 's method (Stouffer et al. 1949) in the appendix Appendix A.1 as they exhibit good properties that will be explored in the following.
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If the p-values are the independent realizations of uniform distribution, ie., for in-distribution data, -2 -2E In P; ~ X2k follows chi-squared distribution with 2k degrees of freedom. Finally, for a test input at Fisher's detector score function can be defined as
sf(0, f)= = -2 Σ In (()
(9)
Fisher's test has interesting qualitative properties, such sensitivity to the smallest p-value, and it is generally more appropriate for combining positive-valued data (Heard & Subin-Delangee 2017) with matches the properties of most OOD scores.
4.4 Correcting for Correlated P-Values
It should be noted that Fisher's method depends the assumption of independence and uniform distribution of the p-values. However, the p-values for the same input sample are not independent. An in-distribution data point (one with high score) is likely to receive high scores across various useful score functions. Therefore, if one score function produces a high score for particular point, it's reasonable to expect that a second score function will also assign a high score to the same data point. Brown (1975) proposes correcting the Fisher statistics for correlation by modeling the r.v 8-()) using a scaled chi-squared distribution, i..,,
sF(1) ~ (K²), with == Var(S)))(2E(sr)) and k' = -=[[(()))) /
(10)
With this simple trick, approach more interpretable results, as know in advance the distribution followed by the in-distribution data under our combined score. As such, we can leverage calibrated confidence values given by the true edf and leverage more powerful single-sample statistical tests for window-based data shift detection.
Remark 1. Commonly, a score's binary detection threshold & is set based on certain quantile of the score's value on an in-distribution validation set. Usually, this value set to have 95% of entities correctly classified. By combining p- p-values with Fisher's method and correcting for correlation with Brown's method, we have that the detection threshold given by y == F
Remark 2. Given that Brown's method involves only linear scaling, il does not lead to any reranking of the scores. Consequently, any evaluation metric used for detection (e.g., AUROC) computed with this method will yield results identical to those obtained with the original Fisher's 's method statistic. Nevertheless, the benefit of employing Brown's s correction lies in the calibrution of scores based on a known underlying probability data distribution, as depicted in Figure 1c This calibration enhances the interpretabiitty of the results.
Algorithm summarizes the offline steps of Combine and Conquer. Finally, at test time, the aggregated binary detection function for an input sample a writes for given TPR desired performance a [0, 1]:
s = [1 shift detected, no shift detected.
d(z) =1 1 Fae(1) -2 > In (s((,))
(11)
5 Experimental Setup
In this section, present and detail the experimental setup from conceptual point of view. For all our main experiments, set as in-distribution dataset ImageNet- 1K [=ILSVRC2012: Deng al. 2009) on ResNet 2021 models. ImageNet is large-scale image for training and 50,000 annolated images for testing. It contains 1,000 different categories, one per image. Our experiments encompass a full-spectrum setting on i.) classic OOD via single example detection (Section 5.1). ii.) novelty shift via. independent window-based detection (Section 5.2; Par. 1), iii.) covariate shift via independent window-based detection (Section 5.2 Par. 2), and iv.) sequential shift detection via sequential window-based detection (Section 5.3
(He et al. 2016) and Vision Transformers (Dosovitskiy et al. classification dataset containing 14 million annotated images
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Algorithm 1 Offline preparation algorithm for combining multiple detectors for OOD detection.
Require: Classifier f. in-distribution held-out data set D. {.i,.... Tr and k 2 detection score functions denoted by 81....8k.
S + Orxk for 2, E Dr do
b Initialize empty r x matrix > Fill the matrix with in-distribution scores
for j E (1....,R) k) do S + sz(z)) end for end for
for j {1, [1....,k) do F2() + Ei- 1|S. .
b Define the empirical cdfs to compute p-values
end for
> The following steps are for the Fisher- Brown method. They can be casily adapted to other methods for E (1....,r} do
Pj x 2 In (S:
end for
/ - 1/r 1/r
i= Pp: o² 1 Ei= (P: 11)
c c c 1-p2)
W' 2
return Fl...., Fr,cK'
5.1 Classic
Detection
OOD detection benchmarks are created by appending in the same set in-distribution data (from the testing dataset) and novelty from different datasets (Hendrycks & Gimpel 2017). We evaluate the performance of the detectors by mixing the 50,000 testing samples from ImageNet with the curated datasets from Bitterwolf et al. (2023) that contain a clean subset without any semantic overlap with ImageNet of far-OOD datasets:
SSB-Eas Easy [Vaze al.) 2022) (farthest novelty from ImageNet- 21K ranked by the total path distance between their nodes in the semantic trees of the two datasets), OpenImage-O O (OI-O) (Wang et al.
2022
(images
belonging to novel classes from the OpenImago-v3 dataset (Krasin et al. 2017 Places (Zhou al.
2017
(images of 365 natural scenes categories, e.g., patio, courlyard, swamp, etc.), iNaturalist (Horn el al.
2017
(samples with concepts from 110 plant classes different from ImageNet- 1K ones), and Textures [Cimpoi
ct al. 2014) (collection of textural pattern images observed in nature); and the near-OOD datasets: SSB-
Hard
Vaze el al. 2022) (closes novelty from Net-e ranked by the total path distance between their
nodes in the semantic trees of the two datasets), Species (Hendrycks et al., (022) (plant species sourced from
(Horn al.] (017) not belonging to ImageNett-IK or ImageNet-21K), and finally NLNCO (Bitterwolf al.
2023
(images from 64 categories manually curated from several popular datasets).
For the evaluation metrics, we consider the popular Area Under the Receiver Operating Characteristic curve (AUROC), which measures how well the OOD score distinguishes between in-distribution and out-of- distribution data in a wwreshold-indeeeedeeee manner (higher is better). The ROC curve is constructed by plotting the true positive rate (TPR) against the false positive rate (FPR) at various threshold values. More rigorously, the AUROC corresponds to the probability that randomly drawn in-distribution sample has a higher score than a randomly drawn OOD sample for a confidence-based score.
For the baselines, we consider the following post-hoc detection methods (14 in total): MSP (Hendrycks & Gimpel 2017), Energy (Liu et al.) 2020). Mahalanobis or Maha for short (Lee et al. 2018). Igcood Goomes et al., 2022), MaxCos (Techapanurak el al.) 2020). ReAct (Sun et al.] 2021), ODIN (Lingg el al.
2018), DICE Sun & Li )022), VIM (Wang et al. 2022). KL-M (Hendrycks et al. 2022). Doctor (Granese et al. p RMD Ren al.
2021 KNN (Sun et al. 2022), GradN (Huang et al. 2021). We followed
the hyperparameter selection procedure suggested in the original papers when needed. New methods can
be easily integrated into our universal framework and should improve the robustness and, potentially, the
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performance of the group detector. In Section 6 delve into the empirical findings and examine whether optimal subset of detectors exists that enhances detection performance.
5.2 Independent Window-Basee Detection
Novelty shift. To simulate a novelty shift at test time, we fabricate fully ID windows and corrupted windows formed by a mixture of ID and OOD data from the OpenImage-O (OI-O) (Wang el al. 2022)
dataset with mixing parameter 3 as defined Equation (2). As a result, the model will encounter windows that contain novelty from natural images. The objective of the detectors is to classify each test window being corrupted not in order to secure the predictor. To do so, each test window is compared to fixed reference window of size and with window sizes [W] E f 1000). We use the KS two sample test described in Section 3.3 &s window-based (est statistics. Evaluation metrics and baselines are the same as described in Section 5. Figure 2 shows the Fisher's test statistic computed on windows with different mixture amounts and sizes. Figure 2a shows the distribution of the test statistics for different mixture values from & (fully ID window) ) to = (fully OOD window). Figure 2b displays how the distribution of the lest statistic changes from flatter to peaky
1000 extracted from clean validation set. We ran experiments with E [0.
increase the window size in the simulations (better seen in color). Finally, Figure 2c demonstrates how the detection performance is affected by window size increase and mixture coefficient. As expected, note an AUROC of 0.5 for the case with 3 = 0. With window size as low as 8, we can perfectly distinguish fully corrupted windows from normal ones. Similar qualitative behavior is observed for all detectors.
15
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[w]) B
B
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010, 0
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0.4
010, 1
100,
0.6
2
200,
0.6
1.0
0.2 0.4
0.6
0.8
0.0
0.2
0.4
0.6
0.8
1.0
100
1000
Test Statistic
Test Statistic
Window Size
(a) From ID to OOD window.
(b) Flat to peaky windows.
(c)
Detection performance.
Figure 2: Test statistic distributional behavior and detection performance as a function of the novelty shift intensity and window size. Experiments ran for Fisher's method on a ResNet-50.
Covariate shift. To simulate covariate shift at test time, we ran experiments with the ImageNet-R (IN-R) (Hendryeks et al.) 2021 datasct. This dataset contains images from different domains than natural objects for 200 TD classes of ImageNet. The shifted do- mains are, for instance, cartoons, graffiti, origami, Similarly to the novelty drift setup described in the previ- ous paragraph, we suppose that the windows arrive inde- pendently. We use the same reference window to compute metrics and vary the mix parameter and window size in the same way. Figure X is analogous to Figure 2 and shows the behavior of the combined p-values for detecting covariate shift in windows of a data stream. We draw similar quali- lalive auservalions from it. Table [ display the accuracy of each model studied on the new domain. We can see that the drift severe without masking only the classes present on IN-R, with top-1 accuracy of around 1% only. However, we compute the top accuracy only the 200 classes by
paintings, plastic objects, tattoos, etc..
Table 1: Top-1 accuracies in percentage.
Model
Train
Val.
IN-R
IN-R (m)
RN-50
87.5
76.1
1.33
36.2
RN-101
90.0
77.4
1.67
39.3
RN-152
90.2
78.3
0.67
41.4
ViT-S-16
88.0
81.4
1.33
46.0
ViT-B-16
90.5
84.5
3.33
56.8
VIT-L-16
92.3
85.8
1.67
64.3

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Published in Transactions on Machine Learning Research (07/2024)
(m)asking the other 800, we observe an amclioration in performance. In our experiments, we simulate the more realistic and challenging scenario by supposing this mask is unavailable.
5.3 Sequential Drift Detection
Unlike the independent window-based detection setting introduced in Section 5.2 in this setup, we im- plement sliding window of size 64 with stride of one, so that the resulting windows contain over- lapping data samples. We assume that the samples arrive sequentially and that the labels are unavail- able to compute the true accuracy of the model on the current or past test windows. The objective is to measure how well the moving average of the detection score will correlate with the moving accuracy of the model. By having a high correlation with accuracy, a machine learning practitioner can use the values of the score as an indicator the system is suffering from any degrading data distribution shift.
To simulate a progressive sequential drift in a data strem, we ran experiments with the corrupted ImageNet (IN-C) (Hendrycks & Di- etterich 2019) dataset. This dataset contains corrupted versions of ImageNet test data by introducing image pre-processing functions, such as adding noise, changing brightness, pixelating, compressing, etc. The intensity of the drift increases over time from intensily 0 (training warmup set and part of the validation set without corrup- tions) to 5. Figure 3 illustrates the monitoring pipeline with the moving accuracy on the left y-axis and the score's moving average on the right, y-axis. The score's moving average can effectively follow the accuracy (hidden variable). The dashed lines are the timestamps in which the drift intensity progressively increascs.
1.0
Moving Average
25
0.8
550
Aczuracy
0.6
-75
0.4
100
125
02-
Moving Accuracy
150
Time
Figure 3: Data stream monitoring with correlation 0.98
6 Results and Discussion
Out-of-distribution detection. Table 2 displays the experimental result on classic OOD detection for a ResNet-50 model on the setup described in Section 5.1 Fisher's method achieves state-of-ther the-art results on average AUROC, surpassing the previous SOTA by 1.4% (MaxCos). Also, the other six standard p-value combination strategics also achieve great results, validating our proposed meta-framowork of Section 4 Sim- ilar FPR and her archilectures tables are available in the Appendix A Apart from achieving overall great performance capabilities, the most compelling observed property is the robustness compared to individual detection metrics. Figure 4 shows the ranking per dataset and on average for selected methods. We can ob- serve that even though several detectors achieve top-1 performance in a few cases, there are several datasets in which they underperform, sometimes catastrophically. This not true for the group methods, which can effectively combine the existing detectors to obtain a final score that successfully combines the multiple decision regions. For instance, Combine and Conquer with Fisher/ fisher/Brown BBrown keeps top-4 performance in all cases on the ResNet-50 ImageNet benchmark and Stouffer /Halterr///tuuggggg is top-5 in all cases.
Standard approaches for scores combination. Table 3 shows the performance of simple normalization statistics (min-max scaling and standard normalization) combined with simple score combination methods (mean, min, max). Note that the quantile normalization range bounded, which helps the interpretability of the scores model monitoring pipcline. The range for min-max scaling not [0, 1] in this case bocause the statistics are computed on a held-out validation set. We also observe better performance overall for the Combine and Conquer methodology.
Independent window-based detection. Figure 5 displays results on novelty shift detection. Figure 5a shows the detectors' performance with the window size, showcasing a small edge in performance for Vim, Fisher's, and Stouffer's methods. Figure 5b displays the impact of the mixture parameter. Figure 5c shows that model size mildly impacts detection performance, with registered improvements for ResNet-152 and ResNet-101 over ResNet-50 on Fisher's method. The confidence interval bounds are computed over 10 different seeds and quite narrow for all methods. Similar observations drawn in the covariate shift
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Table Numerical results regarding AUROC (values in percentage) comparing p-value combination methods against literature for a ResNet-50 model trained on ImageNet. The left-hand side shows results on out-of- distribution detection, and the right-hand hand side shows results on novelty (O1-O) and covariate (IN-R) shift detection with [W]
and 3 1. We recall the basic combination methods (with their equations): Edgington (1/k Ei=>>=1 Pt), Pearson (2 o In(1 - p,)), Simes (min; fp, for sorted p-values), Tippet (min; Pt), and Wilkinson (max; P:).
Detection SSB-E
Data Shift Detection
Method
Avg.
NINCO
Spec.
OL-O
Fisher/ /Browu
89.8
75.8
84.3
88.7
91.0
93.0
93.1
95.9
96.4
94.3 (0.2)
95.7 (0.4)
Stouffer Hartung
89.6
75.5
84.6
89.0
90.9
92.8
92.7
95.8
95.5
92.8 (0.2)
95.5 (0.4)
Edgington
95.5
(0.2)
95.3 (0.3)
Pearson
89.2
74.6
84.9
89.4
90.9
92.4
91.8
95.5
91.1
92.2 (0.3)
93.9 (0.4)
Sinnes
89.2
75.0
83.0
87.6
89.5
92.3
93.1
95.7
97.0
83.6 (0.5)
86.6 (0.7)
Tippet
88.5
74.8
80.9
86.7
87.3
91.7
93.5
95.9
97.2
82.0 (1.0)
81.5 (0.7)
Wilkinson
86.5
68.7
83.3
89.0
88.
89.5
86.3
93.6
93.1
71.2 (1.8)
77.4 (0.9)
MaxCos
88.4
69.6
82.7
88.2
89.9
92.2
89.7
96.1
98.4
92.2 (0.3)
95.5 (0.4)
ReAct
87.4
75.0
82.3
90.4
96.6
(0.3)
91.5 (0.4)
ODIN
85.4
72.9
80.3
83.9
87.7
88.8 8
90.0
91.4
88.3
92.2 (0.5)
93.6 (0.4)
DICE
(0.3)
90.1 (0.4)
Energy
85.0
72.1
79.6
83,1
87 :2
88.7
90.0
90.7
88. 1
91.9 (0.3)
93.4 (0.4)
Igenod
84.7
71.4
80.1
83.0
88.8
88.0
88.8
90.2
87.6
91.0 (0.3)
93.3 (0.3)
VIM
84.3
66.4
78.9
80.7
89.3
90.3
83.7
87.9
97.5
92.2 (0.5)
95.4 (0.4)
KL-M
84.3
73.9
80.7
86,
87.3
85.7
85.2
90.0
85.3
86.9 (0.6)
91.4 (0.9)
Doctor
RMD
83.5
78.2
82.7
87.7
82.9
84.9
81.3
87.6
82.7
89.9 (0.3)
93.1 (0.6)
MSP
86.1
(0.5)
89.0 (0.4)
KNN
83.4
64.3
79.6
83.3
88.0
87.2
83.0
84.1
97.6
84.6 (0.5)
80.2 (0.8)
GradN
82.6
63.3
74.4
83.1
76.2
84.4
91.1
96.0
92.5
49.7 (1.0)
67.4 (1.2)
Maha
69.6
55.3
65.7
70.6
73.9
60.0
72.7
88.4
71.2 (1.8
77.6
Fisher
3
Stouffer
Ranking
MaxCos
ReAct
10
ODIN
VIM
15
RMD
Avg.
SSB-H
NINCO
Spedles
SSB-E
OL0
Places
iNat
Textures
Figure 4: Ranking in terms of AUROC for few selected methods for the ResNet-50 model. Note that the two displayed methods to combining tests obtain top-5 ranking in every dataset, while state-of-the-art individual detectors vary significantly in performance.
results displayed in Figure 10 except for the network scale impact, where we obtained more or less the same results for all sizes. On the right-hand side of Table 2 we showed that for both shifts, we demonstratod improved performance by combining p-values, especially with Fisher's method. We also observe from the table that the novelty shift benchmark is slightly easier than the covariate shift benchmark, which is probably biased because most OOD detectors were developed for the novel class scenario. Additional results are available in the Appendix
Results in a sequential stream. Table the average results for the ImageNet-CC dataset, including kinds of covariate drifts. We can observe that the most performing methods are the scores function based on the softmax and logit outputs and that Fisher's method is on par with top-performing methods. We emphasize that, even though MSP and Doctor work well in this benchmark, they demonstrated poor
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Table 3: Comparative performance in terms of average AUROC for the OOD detection benchmark.
Simple Combination
Ours
Normaliz.
Range
Mean
Min
Max
Stouffer
Fisher
Min-Max
-00, 00)
89.1
87.9
85.6
Standard Quantile
00, 00)
89.3
87.4
86.7
10. 1]
89.3
88.5
86.5
89.6
89.8
1.0
0.9
AUROC
Stouffer
Fisher
0.8
Fisher
Stouffer
AUROC
0.9
VIM
0.7
VIM
0.8
0.7
RMD
ResNet50
0.6
MSP
0.6
0.7
ResNat101
Maha
0.5
Maha
ResNat152
100
0.4
0.6
0.8
1.0
100
Window Size
B
Window Size
(a) [W] impact with 3 0.8.
(b) 8 impact with [W] 10.
(c) Model size impact (3 = 0.8).
Figure 5: Independent data shift (Openlmage-O) O) detection performance ResNet-50 model (ImageNet).
performance on other benchmarks, notably on Table 2
This supports our claim that combining scores is the most effective approach for improving robustness and performance in general data shift detection.
Table 4: Average Pearson's correlation coefficient with the hidden accuracy with one standard deviation in parenthesis for top and bottom performing detection methods across 19 different corruptions on the sequential data shift detection scenario on a ResNet-50 model.
Fisher
Doctor
MSP
Igeood
KNN
RMD
GradN
Maha
Avg.
0.96 (0.03) 0.96 (0.03) 0.96 (0.03)
0.95 (0.03)
0.92 (0.07)
0.92 (0.03)
0.91 (0.07)
0.81 (0.21)
On the distillation of the best subset of detectors. We provide a supervised study to showcase the potential impact of finding an optimal subset of detectors. We computed the performance of all possible subsets of j < k methods, and we report our results in Figure G We found out that 1.) surprisingly, removing the worse detector from the pool does not necossarily increase performance; 2.) increasing the size of the subset improves probable denection on average and on worst performance; 3.) best subset selection benefits harder to find OOD samples; and 4.) not surprisingly, the best combination for the easy benchmark may be very different from the best subset on the harder one. We also list the best subset of four methods on average performance: {GradN, ReAct, MaxCos, RMD), on casy dataset (SSB-Easy) (DICE, MaxCos, KL-M, VIM), and on hard dataset (SSB-Hard)) (MSP, GradN, ReAct, RMD]. Their AUROC and relative gain w.r.t all methods combined together are equal to 91.4 (+188%), 92.0 (+1.1)%, and 79.7 (+4.9%), respectively. These observations support the main claim of the paper that in a data-free scenario with specialized methods, combining all of them should greatly improve the safety of the underlying system.
Limitations. Our study acknowledges that there is no one-size-fits-al detector or a universally superior combination method, finding supported by previous research (Heard & Rubin- Rubin-Delanchy 2017; Fang et al, 2022). This recognition underlines the inherent complexity of real-word world ML applications. Additionally, we recognize that the empirical cumulative distribution function may be susceptible to estimation errors, and the effectiveness of individual detector score functions can influence the performance of the aggregated score. Even though this work stands to improve the reliability of OOD and data-shift detection that allows for safer deployment of machine learning models, there's a risk associated with becoming too confident in the ability of these detection mechanisms. If machine learning practitioners become overly reliant on the OOD detector, they may deploy these models into domains or situations where the detection mechanism fails to
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0.929
0.80
0.96
Nal
0.900
Species
0.875
0.75
0.90
Places
AUROO
0.85
01-0
0.850
SSBE
0.70
Textures
NINCO
0.775
0.85
0.70
Avg.
12
4
8
12
16
Number of Methods
Number Methods
Number Methods
(a) SSB-Fasy.
(b) SSB-Hard.
Average for all datasets with 95% CT.
Figure 6: Evaluation of all possible subsets of detectors the OOD detection benchmark. The dashed red line indicates the performance combining all detectors.
identify OOD data accurately. It is also important to note that although investigation primarily focused computer vision applications, similar techniques be applied to diverse scenarios and application domains.
Future directions. Several avenues for future research remain open for exploration. One promising direction involves investigating the performance patterns of detectors across various types of drifts to fa- cilitate subset selection, ultimately improving detection accuracy. However, this may necessitate validation on held-out labeled data or domain expertise accurately reflect the prior importance of the p-values. Additionally, proposed algorithm could be integrated into incremental and online lcarning algorithms, enhancing their adaptability to evolving data streams and offering exciting opportunities for advancing ma- chine learning applications. Furthermore, instance-dependent, yielding different detector weights for different instances, given our demonstration that various scoring strategies are effective for different types data inputs.
intriguing future direction entails designing a method that is
7 Summary and Concluding Remarks
This paper presents a versatile and efficient method for combining detectors effectively handle shifts in data distributions. By transforming diverse scores into p-values and leveraging meta-analysis techniques, we have illustrated the creation of unified decision boundaries that mitigate the risk of catastrophic failures seen with individual detectors. Our use of Fisher's method, adjusted for correlated p-values, demonstrates strong interprelability as a detection criterion. Through meticulous empirical validation, we' ve confirmed the effectiveness our approach in both single-instance out-of-distribuute detection and window-base data distribution shift detection, achieving notable robustness and detection performance across diverse domains. Looking forward, our framework establishes solid groundwork for enhancing the safety of AI systems.
Acknowledgments
This work has been supported by the project PSPC AIDA: 2019-PSPC-09 funded by BPI-France and granted access to the HPC/AI resources of IDRIS under the allocation 2023 AD011012803R2 made by GENCT.
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A
Appendix
A.1
Combining Multiple P-Values with Stouffer's method
The Stouffer et al. 1949 test statistics for combining p-values is given by:
ss(.) Σ + (p,(.) ==1
(12)
where 1- 1 is the probil, i.e., (a) = erl (2ar 1); where erf is the Gauss error function. If the
p-values are independent, ss(.) ((O,)), where N(,, 0²) is the normal distribution with mean j and standard deviation a.
A.2 Correcting for correlated p-values with Hartung's method
Hartung 1999)
method aims to correct Stouffer's s test for correlated p-values. The group statistics write:
E w (P:(-)
SH (Hw, p)
N),
(13)
(a p) Eimn wi? + p (Ix wi
Ho
with p real-valued parameter and Σi= wi # O. Hartung showed that an unbiased estimator of p based on Pj under Ho is given by:
27
0 è=1-E
(p,)
Σ + (p.) k
1
(14)
Assuming equal weights, repeated similar experiment the one of Figure ! replacing the chi-squared with a standard normal to see how well the correction works. We can observe in Figure that the corrected statistic indeed approximates standard normal distribution. Unlike Brown's method, Hartung's method corrects the statistics directly instead of correcting the parameters of the underlying distribution.
1.0
0.8
Probability
0.6
0.4
Stouffer's method
0.2
Standard Normal Hartung's correction
0.0
-4
-2
2
Test Statistic
Figure 7: Stouffer's method corrected for correlated p-values with Hartung's method to obtain standard normal distribution when evaluated on in-distribution data (null hypothesis), also obtaining interpretable results.
19

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A.3
Additional Plots
8
6
1.0
15
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[e]]
8
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0.0
Density
Density
100,0 0
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010,
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0.2
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0.0
0.2
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0.6
0.8
1.0
100
1000
Test Statistic
Test Statistic
Window Size
(a) From ID to OOD window.
(b) ID vs. OOD windows.
(c) Detection performance.
Figure Test statistic behavior and detection performance in function of the covariate shift intensity and window size. Experiments ran on ResNet-50.
1.0
15
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[W]] 8
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100,
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0.4 Test Statistic
0.6
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0.8
1.0
10
100
1000
Test Statistic
Window Size
(a) From ID to OOD window.
(b) ID vs. OOD windows.
(c) Detection performance.
Figure 9: Test statistic behavior and detection performance in function the covariate shift intensity and window size. Experiments ran on a ViT-L-16.
1.0
1.0
1.0
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Fisher
Stouffer
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Stouffer
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ResNet50
MSP
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Maha
ResNet152
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B
Window Size
(a) [W] impact with = 0.8.
(b) B impact with W 10.
(c) Model size impact (B = 0.8).
Figure 10: Covariate shift (ImageNet-R) detection performance on ResNet-50 model (ImageNet).
20

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Published in Transactions on Machine Learning Research (07 (274))
1.0
AUROC
0.8
0.9
Fisher
Stouffer
0.8
0.9
Fisher Stouffer
AUROC
0.9
VIM
0.7
VIM
0.8
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RMD
MSP
0.6
MSP
0.7
V-T-L-16
Mihh
0.5
VIT B-16
10
100
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0.8
1.0
100
Window Size
B
Window Size
(a) [W] impact with 0.8.
(b) 8 impact with [| 10.
(c) Model size impact (B3 = 0.8).
Figure 11: Covariate shift (ImageNet-R) detection performance ViT-I-16 model (ImageNet).
0.98
0.94
1.000
INat
0.97
0.975
Species
0.92
AUROC
0.96
AUROC
AUROC
0.925
01-0
Places
0.95
0.800
Textures
0.88
0.875
NINCO
0.86
0.850
Avg.
4
8
10
12
14
4
8
10
12
14
10
15
Number
Methods
Number
(a) SSB-Easy.
(b) SSB-Hard.
Average for all dalasets with 95% CL.
Figure 12: Evaluation of all possible subsets of detectors the OOD detection benchmark for ViT-L-16 model. The dashed red line indicates the performance combining all detectors.
A.4 Additional Tables
Table Numerical results in terms of AUROC (values in percentage) comparing p-value combination methods against literature for ViT-L-16 model trained on ImageNet.
Method
Avg.
SSB-H
NINCO
Spec.
SSB-E
O1-O
Places
iNat.
Text.
Maba
96.8
92.7
94.8
96.6
97.4
98.6
96. 9
99.8
97.6
VIM
96.6
92.1
93.9
95.6
97.77
98.5
96.7
99.7
98.2
RMD
92.4
94.8
96.2
96.3
97.9
95.7
99.5
95.6
Fisher/Brown
96.1 1
91.8
93.4
91.6
97.3
98.0 0
96.8
99.5
97.1
Vovk
93.4
99.5
Simes
96.0
91.7
93.4
94.6
97.1
98.0
97.0
99.5
97.0
HaaufferHHartung
91.5
93.3
94.4
97.3
97.9
96.7
99.4
97.1
ReAct
95.9
93.9
91.7
96.9
96.6
97.
91.1
99.5
96.3
Edgington
95.7
90.9
92.8
93.9
97.1
97.7
96.8
99.2
97.1
Energy
93.2
Tippet
95.5
90.9
92.3
91.6
96. 4
97
96.9
99.3
96.
Pearson
90.4
99.0
MaxL.
95.5
91.2
92.6
93.2
97.0
97.6
96.1
99.3
96.8
ODIN
95.5
91.2
92.6
93.2
97.0
97.
96.1
99.3
96.8
Igeood
95.4
90.8
92.6
93.2
97.1
97.6
96.0
99.2
96.7
MaxCos
94.9
89.7
91.2
92.9
97.0
96.9
96.2
98.2
97.1
GradN
KNN
93.4
85.4
89.2
91.9
96.3
96.1
94.3
97.6
96.4
Doctor
90.3
93.7
MSP
92.5
88.2
89.5
91.3
98.5
94.0
92.4
98.0
93.0
KL-M
92.1
85.1
89.0
90.6
93.5
91.2
92.5
98.0
93.7
Wilkinson
91.2
81.6
85.0
87.1
94.2
94.7
96.3
95.7
95.2
DICE
76.3
60.2
63.6
67.0
79.8
80.8
94.3
81.9
82.5
21


Reviews:
[{"review_id": "b1C1yPyHkC", "reviewer": "Reviewer_FtKf", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Overall, this is a simple and well-described method, using statistics to combine existing detectors in a way that results in well-behaved (nicely varying between 0 and 1) OOD/domain-shift score functions, that outperform any individual detector, and most combinations of fewer detectors as well.\n\nMost reassuringly, the variance in performance decreases substantially as more methods are used, and this performance level tends towards the higher available for each dataset, as shown in figs 4 and 6.\n\nThe text mentions that some distribution shifts may not have an impact on model performance, but I didn't see anything that measures how much this method might improve (or worsen) detection of these cases where it might not be desired.  For example, if a model is trained on imagenet-1k, but then applied to a subset of only 10 (in-distribution) classes, will this detect a window-based distribution shift?  This may or may not be desired depending on the application.\n\nIn addition, I would have liked even more on the effectiveness of the approach as different detectors are included or omitted from the system.  p.11 presents a good analysis of this, showing the performance of all possible selections.  But I still would have been in explorations such as, for example, if one selects the top 5 methods according to one subset of measurement datasets (or synthetically corrupted val set), how well does that work compared to selecting all of them, for the held-out subset of datasets?  Likewise, I'd be curious to see how well combining all but one or two of the best overall methods (maxcos) compares to using just that one (or two) method(s)?\n\n\nMore minor questions/comments:\n\n- The text mentions several combination methods, including a few simple types of average, but I didn't initially see these compared because they are referred to only by the names from 4.3 in table 2.  A reminder of which is which (i.e. using the same defs as in 4.3) would be good to include in the table row or caption.\n\n- Is there any binning or smoothing of p-values?\n\n- eq 6:  lowercase $w$ is not defined.  what is the sup_$w$ computed over?  (it appears this is defined later on in eq 7 as the empirical cdf threshold)\n\n\n- Fig 5c:  Model size having only a small effect is a nice behavior.  How much of this is from the ensembling inherent in the score combination method?  What if fewer (or only one) shift detection method are used --- does this difference get any larger?  If so, this would be another benefit of combining methods (less dependence on model size/choice).\n\n\n- what is an example of virtual drift that doesn't affect the model, related to the decompositions in eq (3), as was described for three types affecting shifts?\n\n- could use a little more explicit explanation of why p values are modeled as uniform RVs.  This is by construction of the p values, I believe.\n\n- similarly, could elaborate more on non-independence, though this is pretty clear:  an in-distrib (high score) point is likely to be scored high for any useful score function, so if one function has high score, a second function is likely too as well\n\n- eq.2:  this linear mixture is not used in this method, only in evaluations.  might make more sense to define this in the evaluation methods procedure instead, as mentioning it earlier sets the stage for trying to recover the individual components $p, v, \\beta$ which is not actually the method described\n\n- also note: I did not check the citations of all the referenced theorems or lemmas for correctness of application, but did consider whether the conclusions they are used to support seemed reasonable according to the authors' descriptions, and found these descriptions clear.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper presents a method to combine out-of-distribution or domain-shift detectors, using techniques from \"classical\" statistics.  Scores for each detector are calibrated using quantile p-values, then combined using $\\chi^2$ or other combination method.  These are also monotonically adjusted for correlative dependence, with a distribution scaling.  Numerous experiments using imagenet demonstrate the effectiveness of the method and its robustness.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "None required.  I mentioned a few in my review above that I feel would strengthen the paper.", "broader_impact_concerns": "This work stands to improve reliability of OOD and domain-shift detection, which can enable safer application of models.  However, over-confidence in an OOD detection ability may also facilitate wider deployments of classifiers, including into domains where the OOD detection itself fails.  This could be discussed in the Limitations section.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "B1C0gLcQ3H", "reviewer": "Reviewer_gsyy", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The main proposal of this paper, i.e. combining multiple individual metrics for out-of-distribution / data shift detection together, is straightforward and clear, and the arguments that are provided for doing so are sensible. The statistical framework based on meta-analysis that is proposed is also well-motivated and reasonably clearly described. I was not able to find any major mistakes in either the proposed framework or the statistical foundations for it. The empirical results seem to convincingly indicate that the proposed approach for combining individual metrics performs better and more consistently than each of the respective individual metrics.\n \nI need to indicate that I am not an expert in out-of-distribution / data-shift detection, so it is difficult for me to properly judge the scientific contribution of the paper in this regard. I am also not an expert in statistical meta-analysis, so it is also hard for me to judge the quality/standard of the paper in this respect.\n \nI found several issues that I think should be addressed before this paper could be accepted for publication.\n \n[1] I think there is some confusion / mix up with regards to the terms *concept drift*, *covariate shift* and *prior shift*. Concept drift is defined as a change in the conditional distribution of Y given X (and with no change in the input distribution X). In subsection 3.3, it is stated that the instances’ class labels (i.e., the Y’s) are not available for the detection of data shifts. It seems to follow from this that it becomes impossible to detect concept drifts, yet the authors still claim to do this (e.g., subsection 5.2). It seems to me that what is described in subsection 5.2 as being a covariate shift, is actually more akin to a prior shift.\n \n[2] I found section 5 hard to read and understand. A reason for this is that I am unfamiliar with most of the datasets / benchmarks that are used, while at the moment it seems the paper is written in such a way that knowledge of these is assumed. I would encourage the authors to make changes to this section so that it is also possible for readers unfamiliar with the used datasets to follow it.\n \n[3] In subsection 4.1 several conceptual arguments are made for why combining different individual metrics in standard or naïve ways is not optimal. However, I could not find an empirical evaluation of these standard approaches. I think it would be important to include these standard approaches in the empirical comparisons, to back up the conceptual arguments from subsection 4.1.\n \nMinor issues:\n- several issues with formatting of references (e.g., top of last paragraph p2)\n- inconsistent use of capitalisation (e.g., title of subsections 4.3 and 4.4)", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes that for detecting out-of-distribution samples as well as for detecting data shifts, it is beneficial to combine multiple different individual metrics. However, the paper argues that combining such different metrics is not trivial as standard approaches for doing so have issues. Instead, the paper proposes to use the statistical approach of meta-analysis to build a framework for combining individual metrics in a principled way. In a series of experiments, it is convincingly demonstrated that the proposed combination of metrics performs better than each of the respective individual metrics.", "claims_and_evidence_yn": "No", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "The main changes I encourage the authors to make is addressing the three main issues (labeled [1], [2] and [3]) that are raised under “Strengths and Weaknesses”.", "broader_impact_concerns": "No concerns in this regard.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Gx74OUop5P", "reviewer": "Reviewer_fkys", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The paper tackles an important OOD detection problem, which is of great relevance to real world applications. \n- The paper is well written in general. \n- The proposed method is technically sound. \n- The empirical evaluation is thorough. \n- I particularly like the experiments and discussion on window based detection, which is often neglected in typical OOD detection papers. \n\nWeaknesses/Questions:\n- As a crucial step, quantile normalization needs to be performed with training/validation data. Typically, both of these dataset do not contain any OOD samples. Moreover, the model could be heavily overfit to the training data. Are there any risks in doing quantile normalization using such datasets? Would this lead to false positives? \n- How is the significant level selected?\n- Any theoretical guarantee  on the proposed detection function (equation 11)? \n- For window-based detection, typically window size can be adjusted. I wonder if there is a way to select an optimal window size?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces a method to improve out-of-distribution (OOD) detection by integrating various detection scores that reflect the self-confidence of deep learning models and anomalous behavior in latent space. Traditional combination techniques using simple statistics are insufficient for this task. The solution involves quantile normalization of detection scores into p-values, effectively turning the problem into a multi-variate hypothesis test. This approach allows for the use of meta-analysis tools to create a unified, effective OOD detector with solid decision boundaries. Empirical studies show significant enhancements in robustness and performance across different OOD scenarios, demonstrating the framework's extensibility and marking a first in combining decision boundaries for OOD detection in this manner.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "It would be great if the authors can address the questions and weaknesses in the above section.", "broader_impact_concerns": "No concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This paper presents a method for combining multiple out of distribution detection methods / scores in order to obtain a more robust combined score.  The authors argue that very simple methods for combining scores does not work well, so instead they develop a more elaborate strategy using p-values in the context of a statistical hypothesis test.

The reviewers all voted to accept the paper (two leaning accept and one accept. The reviewers found the paper well motivated, the methodological contribution mathematically sound and well justified, and they found the empirical results convincing.  One reviewer remarked that additional ablations would strengthen the work ("I would have liked even more on the effectiveness of the approach as different detectors are included or omitted from the system").  Although the reviewers have expressed that additional experiments / ablations would make the work stronger, they found the existing empirical work convincing and thorough enough.  Therefore, we leave it up to the authors to determine whether they would like to add these, but won't require revisions for acceptance.

One reviewer found that, although the authors addressed their concerns in the author response, they didn't find the response particularly helpful or insightful. They cited this as a reason to 'lean accept' rather than outright accept.
Decision: Accept (poster)