publications

2024

1. UAI

   Label-wise Aleatoric and Epistemic Uncertainty Quantification

   Yusuf Sale, Paul Hofman, Timo Löhr, and 3 more authors

   In The 40th Conference on Uncertainty in Artificial Intelligence, 2024

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   We present a novel approach to uncertainty quantification in classification tasks based on label-wise decomposition of uncertainty measures. This label- wise perspective allows uncertainty to be quantified at the individual class level, thereby improving cost-sensitive decision-making and helping under- stand the sources of uncertainty. Furthermore, it allows to define total, aleatoric, and epistemic un- certainty on the basis of non-categorical measures such as variance, going beyond common entropy- based measures. In particular, variance-based measures address some of the limitations associated with established methods that have recently been discussed in the literature. We show that our pro- posed measures adhere to a number of desirable properties. Through empirical evaluation on a variety of benchmark data sets – including applications in the medical domain where accurate uncertainty quantification is crucial – we establish the effectiveness of label-wise uncertainty quantification.

   @inproceedings{salelabel,
     title = {{Label-wise Aleatoric and Epistemic Uncertainty Quantification}},
     author = {Sale, Yusuf and Hofman, Paul and L{\"o}hr, Timo and Wimmer, Lisa and Nagler, Thomas and H{\"u}llermeier, Eyke},
     booktitle = {The 40th Conference on Uncertainty in Artificial Intelligence},
     year = {2024},
   }

2. ICML

   Second-Order Uncertainty Quantification: A Distance-Based Approach

   Yusuf Sale, Viktor Bengs, Michele Caprio, and 1 more author

   In Forty-first International Conference on Machine Learning, 2024

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   In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in ma- chine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive un- certainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

   @inproceedings{sale2024secondorder,
     title = {Second-Order Uncertainty Quantification: A Distance-Based Approach},
     author = {Sale, Yusuf and Bengs, Viktor and Caprio, Michele and H{\"u}llermeier, Eyke},
     booktitle = {Forty-first International Conference on Machine Learning},
     year = {2024},
   }

3. SPIGM

   Quantifying Aleatoric and Epistemic Uncertainty: A Credal Approach

   Paul Hofman, Yusuf Sale, and Eyke Hüllermeier

   In ICML 2024 Workshop on Structured Probabilistic Inference & Generative Modeling, 2024

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   Uncertainty representation and quantification are paramount in machine learning, especially in safety-critical applications. In this paper, we pro- pose a novel framework for the quantification of aleatoric and epistemic uncertainty based on the notion of credal sets, i.e., sets of probability distributions. Thus, we assume a learner that produces (second-order) predictions in the form of sets of probability distributions on outcomes. Practically, such an approach can be realized by means of ensemble learning: Given an ensemble of learners, credal sets are generated by including sufficiently plausible predictors, where plausibility is mea- sured in terms of (relative) likelihood. We provide a formal justification for the framework and introduce new measures of epistemic and aleatoric un- certainty as concrete instantiations. We evaluate these measures both theoretically, by analyzing desirable axiomatic properties, and empirically, by comparing them in terms of performance and effectiveness to existing measures of uncertainty in an experimental study.

   @inproceedings{hofman2024quantifying,
     title = {Quantifying Aleatoric and Epistemic Uncertainty: A Credal Approach},
     author = {Hofman, Paul and Sale, Yusuf and H{\"u}llermeier, Eyke},
     booktitle = {ICML 2024 Workshop on Structured Probabilistic Inference {\&} Generative Modeling},
     year = {2024},
   }

4. arXiv

   Explaining Bayesian Optimization by Shapley Values Facilitates Human-AI Collaboration

   Julian Rodemann, Federico Croppi, Philipp Arens, and 7 more authors

   arXiv preprint arXiv:2403.04629, 2024

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   Bayesian optimization (BO) with Gaussian processes (GP) has become an indispensable algorithm for black box optimization problems. Not without a dash of irony, BO is often considered a black box itself, lacking ways to provide reasons as to why certain parameters are proposed to be evaluated. This is particularly relevant in human-in-the-loop applications of BO, such as in robotics. We address this issue by proposing ShapleyBO, a framework for interpreting BO’s proposals by game-theoretic Shapley values.They quantify each parameter’s contribution to BO’s acquisition function. Exploiting the linearity of Shapley values, we are further able to identify how strongly each parameter drives BO’s exploration and exploitation for additive acquisition functions like the confidence bound. We also show that ShapleyBO can disentangle the contributions to exploration into those that explore aleatoric and epistemic uncertainty. Moreover, our method gives rise to a ShapleyBO-assisted human machine interface (HMI), allowing users to interfere with BO in case proposals do not align with human reasoning. We demonstrate this HMI’s benefits for the use case of personalizing wearable robotic devices (assistive back exosuits) by human-in-the-loop BO. Results suggest human-BO teams with access to ShapleyBO can achieve lower regret than teams without.

   @article{rodemann2024explaining,
     title = {Explaining Bayesian Optimization by Shapley Values Facilitates Human-AI Collaboration},
     author = {Rodemann, Julian and Croppi, Federico and Arens, Philipp and Sale, Yusuf and Herbinger, Julia and Bischl, Bernd and H{\"u}llermeier, Eyke and Augustin, Thomas and Walsh, Conor J and Casalicchio, Giuseppe},
     journal = {arXiv preprint arXiv:2403.04629},
     year = {2024},
   }

5. arXiv

   Quantifying Aleatoric and Epistemic Uncertainty with Proper Scoring Rules

   Paul Hofman, Yusuf Sale, and Eyke Hüllermeier

   arXiv preprint arXiv:2404.12215, 2024

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   Uncertainty representation and quantification are paramount in machine learning and constitute an important prerequisite for safety-critical applications. In this paper, we propose novel measures for the quantification of aleatoric and epistemic uncertainty based on proper scoring rules, which are loss functions with the meaningful property that they incentivize the learner to predict ground-truth (conditional) probabilities. We assume two common representations of (epistemic) uncertainty, namely, in terms of a credal set, i.e. a set of probability distributions, or a second-order distribution, i.e., a distribution over probability distributions. Our framework establishes a natural bridge between these representations. We provide a formal justification of our approach and introduce new measures of epistemic and aleatoric uncertainty as concrete instantiations.

   @article{hofman2024quantifyinh,
     title = {Quantifying Aleatoric and Epistemic Uncertainty with Proper Scoring Rules},
     author = {Hofman, Paul and Sale, Yusuf and H{\"u}llermeier, Eyke},
     journal = {arXiv preprint arXiv:2404.12215},
     year = {2024},
   }

2023

1. UAI

   Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?

   Lisa Wimmer, Yusuf Sale, Paul Hofman, and 2 more authors

   In The 39th Conference on Uncertainty in Artificial Intelligence, 2023

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   The quantification of aleatoric and epistemic un- certainty in terms of conditional entropy and mutual information, respectively, has recently become quite common in machine learning. While the properties of these measures, which are rooted in in- formation theory, seem appealing at first glance, we identify various incoherencies that call their appropriateness into question. In addition to the measures themselves, we critically discuss the idea of an additive decomposition of total uncertainty into its aleatoric and epistemic constituents. Experiments across different computer vision tasks support our theoretical findings and raise concerns about current practice in uncertainty quantification.

   @inproceedings{wimmer2023quantifying,
     title = {{Quantifying Aleatoric and Epistemic Uncertainty in Machine Learning: Are Conditional Entropy and Mutual Information Appropriate Measures?}},
     author = {Wimmer, Lisa and Sale, Yusuf and Hofman, Paul and Bischl, Bernd and H{\"u}llermeier, Eyke},
     booktitle = {The 39th Conference on Uncertainty in Artificial Intelligence},
     pages = {2282--2292},
     year = {2023},
     organization = {PMLR},
   }

2. UAI

   Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?

   Yusuf Sale, Michele Caprio, and Eyke Hüllermeier

   In The 39th Conference on Uncertainty in Artificial Intelligence, 2023

   Abs Bib PDF

   Adequate uncertainty representation and quantification have become imperative in various scientific disciplines, especially in machine learning and artificial intelligence. As an alternative to representing uncertainty via one single probability measure, we consider credal sets (convex sets of probability measures). The geometric representation of credal sets as d-dimensional polytopes implies a geometric intuition about (epistemic) uncertainty. In this paper, we show that the volume of the geometric representation of a credal set is a meaningful mea- sure of epistemic uncertainty in the case of binary classification, but less so for multi-class classification. Our theoretical findings highlight the crucial role of specifying and employing uncertainty measures in machine learning in an appropriate way, and for being aware of possible pitfalls.

   @inproceedings{sale2023volume,
     title = {Is the Volume of a Credal Set a Good Measure for Epistemic Uncertainty?},
     author = {Sale, Yusuf and Caprio, Michele and H{\"u}llermeier, Eyke},
     booktitle = {The 39th Conference on Uncertainty in Artificial Intelligence},
     pages = {1795--1804},
     year = {2023},
     organization = {PMLR},
   }

3. COPA

   Conformal Prediction with Partially Labeled Data

   Alireza Javanmardi, Yusuf Sale, Paul Hofman, and 1 more author

   In Conformal and Probabilistic Prediction with Applications, 2023

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   While the predictions produced by conformal prediction are set-valued, the data used for training and calibration is supposed to be precise. In the setting of superset learning or learning from partial labels, a variant of weakly supervised learning, it is exactly the other way around: training data is possibly imprecise (set-valued), but the model induced from this data yields precise predictions. In this paper, we combine the two settings by making conformal prediction amenable to set-valued training data. We propose a generalization of the conformal prediction procedure that can be applied to set-valued training and calibration data. We prove the validity of the proposed method and present experimental studies in which it compares favorably to natural baselines.

   @inproceedings{javanmardi2023conformal,
     title = {Conformal Prediction with Partially Labeled Data},
     author = {Javanmardi, Alireza and Sale, Yusuf and Hofman, Paul and H{\"u}llermeier, Eyke},
     booktitle = {Conformal and Probabilistic Prediction with Applications},
     pages = {251--266},
     year = {2023},
     organization = {PMLR},
   }

4. Epi UAI

   A Novel Bayes’ Theorem for Upper Probabilities

   Michele Caprio, Yusuf Sale, Eyke Hüllermeier, and 1 more author

   In International Workshop on Epistemic Uncertainty in Artificial Intelligence, 2023

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   In their seminal 1990 paper, Wasserman and Kadane establish an upper bound for the Bayes’ posterior probability of a measurable set, when the prior lies in a class of probability measures and the likelihood is precise. They also give a sufficient condition for such upper bound to hold with equality. In this paper, we introduce a generalization of their result by additionally addressing uncertainty related to the likelihood. We give an upper bound for the posterior probability when both the prior and the likelihood belong to a set of probabilities. Furthermore, we give a sufficient condition for this upper bound to become an equality. This result is interesting on its own, and has the potential of being applied to various fields of engineering (e.g. model predictive control), machine learning, and artificial intelligence.

   @inproceedings{caprio2023novel,
     title = {A Novel Bayes’ Theorem for Upper Probabilities},
     author = {Caprio, Michele and Sale, Yusuf and H{\"u}llermeier, Eyke and Lee, Insup},
     booktitle = {International Workshop on Epistemic Uncertainty in Artificial Intelligence},
     pages = {1--12},
     year = {2023},
     organization = {Springer},
   }

5. arXiv

   Second-Order Uncertainty Quantification: Variance-Based Measures

   Yusuf Sale, Paul Hofman, Lisa Wimmer, and 2 more authors

   arXiv preprint arXiv:2401.00276, 2023

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   Uncertainty quantification is a critical aspect of machine learning models, providing important insights into the reliability of predictions and aiding the decision-making process in real-world applications. This paper proposes a novel way to use variance-based measures to quantify uncertainty on the basis of second-order distributions in classification problems. A distinctive feature of the measures is the ability to reason about uncertainties on a class-based level, which is useful in situations where nuanced decision-making is required. Recalling some properties from the literature, we highlight that the variance-based measures satisfy important (axiomatic) properties. In addition to this axiomatic approach, we present empirical results showing the measures to be effective and competitive to commonly used entropy-based measures.

   @article{sale2023second,
     title = {Second-Order Uncertainty Quantification: Variance-Based Measures},
     author = {Sale, Yusuf and Hofman, Paul and Wimmer, Lisa and H{\"u}llermeier, Eyke and Nagler, Thomas},
     journal = {arXiv preprint arXiv:2401.00276},
     year = {2023},
   }

2022

1. THESIS

   G-Framework in Statistics

   Yusuf Sale

   2022

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   In order to achieve reliable results via statistical methodology, one important goal is to account for potential uncertainty. Shige Peng introduced an un- certainty counterpart of Kolmogorov’s probabilistic setting: the G-Framework. While this framework is well-known in mathematical finance, work within the G-Framework in statistics is limited. This thesis motivates nonlinear expectations for decision-making under uncertainty in dynamic and non-dynamic situations. Switching the viewpoint from probability spaces to expectation spaces, we discuss the theoretical foundations of the G-Framework, emphasizing comprehensibility. We motivate nonlinear expectations for subsequent application in statistics by notions that emerged in various academic communities and are like- wise concerned with decision-making under uncertainty: Choquet expectations express probabilistic uncertainty from the viewpoint of non-additive measures and g-expectations, which represent a nonlinear class of expectations based on backward stochastic differential equations (BSDE). For explicit understanding, we provide the required foundations of stochastic calculus in a self-contained form. The applicability of the G-Framework in statistics is particularly evident from the respective Law of Large Numbers and Central Limit Theorem. To emphasize the applicability, this thesis motivates a notion of sublinear regression.

   @article{sale2022g,
     title = {G-Framework in Statistics},
     author = {Sale, Yusuf},
     year = {2022},
   }

2019

1. THESIS

   Robuste Minimax-Tests und eine Robustifizierung des Bayes-Faktors

   Yusuf Sale

   2019

   Abs Bib PDF

   Die robuste Statistik befasst sich mit statistischen Verfahren, die mögliche Abweichungen von zugrundeliegenden Modellannahmen berücksichtigen. Insbesondere stellt man fest, dass der Likelihood-Quotienten-Test im Allgemeinen nicht robust gegenüber Abweichungen von zugrundeliegenden Verteilungsannahmen ist. Dahingehend wird in der vorliegenden Arbeit eine robuste Version des Likelihood-Quotienten-Tests mathematisch über Umgebungsmodelle und Least Favorable Pair motiviert und vorgestellt. Dabei stellt man fest, dass der gestutzte Likelihood-Quotienten-Tests über wünschenswerte minimax Eigenschaften besitzt, weshalb man in diesem Kontext auch von robusten Minimax-Tests spricht. Diese Minimax-Resultate werden anhand des Huber-Strassen-Theorems auf bialternierende Kapazitäten in polnischen Räumen generalisiert. Das Huber-Strassen-Theorem und eben- so die damit verbundene weitreichende Bedeutung für die robuste Statistik werden in der vorliegenden Arbeit thematisiert. Neben mathematischer Ausführlichkeit in der Beweisführung und Veranschaulichung methodologischer Konstrukte, wird zudem versucht die praktische Relevanz solcher robusten Verfahren zu veranschaulichen. Um die Methodik robuster Verfahren ebenso in den bayesianischen Rahmen zu übertragen, beschäftigt sich die Arbeit weiterhin mit dem Bayes-Faktor, der eine Alternative zu klassischen Hypothesentests darstellt. Anhand der erzielten Ergebnisse wird eine robuste Version des Bayes-Faktors vorgeschlagen und ebenso verdeutlicht, wieso eine Robustifizierung sich als besonders schwierig herausstellt.

   @article{sale2019robuste,
     title = {Robuste Minimax-Tests und eine Robustifizierung des Bayes-Faktors},
     author = {Sale, Yusuf},
     year = {2019},
   }