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Michele Caprio

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10 papers

2 author rows

TMLR Journal 2026 Journal Article

Self-Supervised Laplace Approximation for Bayesian Uncertainty Quantification

Julian Rodemann
Alexander Marquard
Thomas Augustin
Michele Caprio

Approximate Bayesian inference typically revolves around computing the posterior parameter distribution. In practice, however, the main object of interest is often a model’s predictions rather than its parameters. In this work, we propose to bypass the parameter posterior and focus directly on approximating the posterior predictive distribution. We achieve this by drawing inspiration from self-training within self-supervised and semi-supervised learning. Essentially, we quantify a Bayesian model's predictive uncertainty by refitting on self-predicted data. The idea is strikingly simple: If a model assigns high likelihood to self-predicted data, these predictions are of low uncertainty, and vice versa. This yields a deterministic, sampling-free approximation of the posterior predictive. The modular structure of our Self-Supervised Laplace Approximation (SSLA) further allows us to plug in different prior specifications, enabling classical Bayesian sensitivity (w.r.t. prior choice) analysis. In order to bypass expensive refitting, we further introduce an approximate version of SSLA, called ASSLA. We study (A)SSLA both theoretically and empirically in regression models ranging from Bayesian linear models to Bayesian neural networks. Across a wide array of regression tasks with simulated and real-world datasets, our methods outperform classical Laplace approximations in predictive calibration while remaining computationally efficient.

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ISIPTA Conference 2025 Conference Paper

Conformal prediction regions are imprecise highest density regions

Michele Caprio
Yusuf Sale
Eyke H\"ullermeier

Recently, Cella and Martin proved how, under an assumption called <em>consonance</em>, a credal set (i. e. a closed and convex set of probabilities) can be derived from the conformal transducer associated with transductive conformal prediction. We show that the Imprecise Highest Density Region (IHDR) associated with such a credal set corresponds to the classical Conformal Prediction Region. In proving this result, we establish a new relationship between Conformal Prediction and Imprecise Probability (IP) theories, via the IP concept of a cloud. A byproduct of our presentation is the discovery that consonant plausibility functions are monoid homomorphisms, a new algebraic property of an IP tool.

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TMLR Journal 2025 Journal Article

Conformalized Credal Regions for Classification with Ambiguous Ground Truth

Michele Caprio
David Stutz
Shuo Li
Arnaud Doucet

An open question in Imprecise Probabilistic Machine Learning is how to empirically derive a credal region (i.e., a closed and convex family of probabilities on the output space) from the available data, without any prior knowledge or assumption. In classification problems, credal regions are a tool that is able to provide provable guarantees under realistic assumptions by characterizing the uncertainty about the distribution of the labels. Building on previous work, we show that credal regions can be directly constructed using conformal methods. This allows us to provide a novel extension of classical conformal prediction to problems with ambiguous ground truth, that is, when the exact labels for given inputs are not exactly known. The resulting construction enjoys desirable practical and theoretical properties: (i) conformal coverage guarantees, (ii) smaller prediction sets (compared to classical conformal prediction regions) and (iii) disentanglement of uncertainty sources (epistemic, aleatoric). We empirically verify our findings on both synthetic and real datasets.

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NeurIPS Conference 2025 Conference Paper

Integral Imprecise Probability Metrics

Siu Lun (Alan) Chau
Michele Caprio
Krikamol Muandet

Quantifying differences between probability distributions is fundamental to statistics and machine learning, primarily for comparing statistical uncertainty. In contrast, epistemic uncertainty---due to incomplete knowledge---requires richer representations than those offered by classical probability. Imprecise probability (IP) theory offers such models, capturing ambiguity and partial belief. This has driven growing interest in imprecise probabilistic machine learning (IPML), where inference and decision-making rely on broader uncertainty models---highlighting the need for metrics beyond classical probability. This work introduces the Integral Imprecise Probability Metric (IIPM) framework, a Choquet integral-based generalisation of classical Integral Probability Metric to the setting of capacities---a broad class of IP models encompassing many existing ones, including lower probabilities, probability intervals, belief functions, and more. Theoretically, we establish conditions under which IIPM serves as a valid metric and metrises a form of weak convergence of capacities. Practically, IIPM not only enables comparison across different IP models but also supports the quantification of epistemic uncertainty~(EU) within a single IP model. In particular, by comparing an IP model with its conjugate, IIPM gives rise to a new class of epistemic uncertainty measures---Maximum Mean Imprecision (MMI) ---which satisfy key axiomatic properties proposed in the Uncertainty Quantification literature. We validate MMI through selective classification experiments, demonstrating strong empirical performance against established EU measures, and outperforming them when classical methods struggle to scale to a large number of classes. Our work advances both theory and practice in Imprecise Probabilistic Machine Learning, offering a principled framework for comparing and quantifying epistemic uncertainty under imprecision.

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ISIPTA Conference 2025 Conference Paper

Optimal transport for $ε$-contaminated credal sets

Michele Caprio

We present generalized versions of Monge’s and Kantorovich’s optimal transport problems with the probabilities being transported replaced by lower probabilities. We show that, when the lower probabilities are the lower envelopes of $\epsilon$-contaminated sets, then our version of Monge’s, and a restricted version of our Kantorovich’s problems, coincide with their respective classical versions. We also give sufficient conditions for the existence of our version of Kantorovich’s optimal plan, and for the two problems to be equivalent. As a byproduct, we show that for $\epsilon$-contaminations the lower probability versions of Monge’s and Kantorovich’s optimal transport problems need not coincide. The applications of our results to Machine Learning and Artificial Intelligence are also discussed.

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IROS Conference 2025 Conference Paper

TAR: Teacher-Aligned Representations via Contrastive Learning for Quadrupedal Locomotion

Amr Mousa
Neil Karavis
Michele Caprio
Wei Pan 0004
Richard Allmendinger 0001

Quadrupedal locomotion via Reinforcement Learning (RL) is commonly addressed using the teacher-student paradigm, where a privileged teacher guides a proprioceptive student policy. However, key challenges such as representation misalignment between privileged teacher and proprioceptive-only student, covariate shift due to behavioral cloning, and lack of deployable adaptation; lead to poor generalization in real-world scenarios. We propose Teacher-Aligned Representations via Contrastive Learning (TAR), a framework that leverages privileged information with self-supervised contrastive learning to bridge this gap. By aligning representations to a privileged teacher in simulation via contrastive objectives, our student policy learns structured latent spaces and exhibits robust generalization to Out-of-Distribution (OOD) scenarios, surpassing the fully privileged “Teacher”. Results showed accelerated training by 2× compared to state-of-the-art baselines to achieve peak performance. OOD scenarios showed better generalization by 40% on average compared to existing methods. Moreover, TAR transitions seamlessly into learning during deployment without requiring privileged states, setting a new benchmark in sample-efficient, adaptive locomotion and enabling continual fine-tuning in real-world scenarios. Open-source code and videos are available at https://amrmousa.com/TARLoco/.

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TMLR Journal 2024 Journal Article

Credal Bayesian Deep Learning

Michele Caprio
Souradeep Dutta
Kuk Jin Jang
Vivian Lin
Radoslav Ivanov
Oleg Sokolsky
Insup Lee

Uncertainty quantification and robustness to distribution shifts are important goals in machine learning and artificial intelligence. Although Bayesian Neural Networks (BNNs) allow for uncertainty in the predictions to be assessed, different sources of uncertainty are indistinguishable. We present Credal Bayesian Deep Learning (CBDL). Heuristically, CBDL allows to train an (uncountably) infinite ensemble of BNNs, using only finitely many elements. This is possible thanks to prior and likelihood finitely generated credal sets (FGCSs), a concept from the imprecise probability literature. Intuitively, convex combinations of a finite collection of prior-likelihood pairs are able to represent infinitely many such pairs. After training, CBDL outputs a set of posteriors on the parameters of the neural network. At inference time, such posterior set is used to derive a set of predictive distributions that is in turn utilized to distinguish between aleatoric and epistemic uncertainties, and to quantify them. The predictive set also produces either (i) a collection of outputs enjoying desirable probabilistic guarantees, or (ii) the single output that is deemed the best, that is, the one having the highest predictive lower probability -- another imprecise-probabilistic concept. CBDL is more robust than single BNNs to prior and likelihood misspecification, and to distribution shift. We show that CBDL is better at quantifying and disentangling different types of uncertainties than single BNNs and ensemble of BNNs. In addition, we apply CBDL to two case studies to demonstrate its downstream tasks capabilities: one, for motion prediction in autonomous driving scenarios, and two, to model blood glucose and insulin dynamics for artificial pancreas control. We show that CBDL performs better when compared to an ensemble of BNNs baseline.

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NeurIPS Conference 2024 Conference Paper

Credal Learning Theory

Michele Caprio
Maryam Sultana
Eleni G. Elia
Fabio Cuzzolin

Statistical learning theory is the foundation of machine learning, providing theoretical bounds for the risk of models learned from a (single) training set, assumed to issue from an unknown probability distribution. In actual deployment, however, the data distribution may (and often does) vary, causing domain adaptation/generalization issues. In this paper we lay the foundations for a `credal' theory of learning, using convex sets of probabilities (credal sets) to model the variability in the data-generating distribution. Such credal sets, we argue, may be inferred from a finite sample of training sets. Bounds are derived for the case of finite hypotheses spaces (both assuming realizability or not), as well as infinite model spaces, which directly generalize classical results.

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ICML Conference 2024 Conference Paper

Second-Order Uncertainty Quantification: A Distance-Based Approach

Yusuf Sale
Viktor Bengs
Michele Caprio
Eyke Hüllermeier

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine 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 uncertainty 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.

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UAI Conference 2023 Conference Paper

Is the volume of a credal set a good measure for epistemic uncertainty?

Yusuf Sale
Michele Caprio
Eyke Hüllermeier

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 measure 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.

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