The course is organised in two parts: Part I: Foundations of Imprecise Probabilities (Lectures 2–6) and Part II: Learning with Imprecise Probabilities (Lectures 7–13). Dates are listed on the schedule. Slides are uploaded before each class; until then, the corresponding slides from the 2025/26 offering give an impression of the material.

  • Lecture 1: Introduction
    tl;dr: The fundamentals of statistical machine learning, how uncertainty arises through data, models and environments, and why the limits of today's methods are rooted in probability theory.
    Slides: TBA

    Topics:

    • Course overview, objectives, and logistics
    • Foundations of statistical machine learning
    • The roles of uncertainty in learning and decision-making
    • Limitations and pitfalls of conventional learning paradigms
    • Motivation for imprecise probabilistic machine learning

    Suggested readings: TBA

  • Lecture 2: From Precise to Imprecise Probabilities
    tl;dr: What uncertainty is and how probability is used to model it: interpretations of probability, the limits of precise probability models, and an overview of imprecise probabilistic models.
    Slides: TBA

    Topics:

    • Understanding uncertainty and probability
    • Interpretations of probability
    • Limitations of precise probability models (e.g., Ellsberg’s paradox)
    • Motivation for imprecise probabilities
    • Overview of imprecise probabilistic models: lower and upper previsions, sets of desirable gambles, credal sets, and related representations

    Suggested readings:

  • Lecture 3: Gambles, Sets of Desirable Gambles, and Lower Prevision
    tl;dr: The behavioural foundations of imprecise probability: gambles, sets of desirable gambles, lower and linear previsions, coherence and avoiding sure loss, and natural extension.
    Slides: TBA

    Topics:

    • Gambles and their interpretation
    • Sets of desirable gambles
    • Lower prevision, linear prevision, and lower probabilities
    • Coherence and avoiding sure loss
    • Natural extension and coherent inference

    Suggested readings: TBA

  • Lecture 4: Possibility Theory
    tl;dr: Possibility theory as a close relative of probability: possibility distributions, possibility and necessity measures, their connection to imprecise probability, and applications in machine learning.
    Slides: TBA

    Topics:

    • Historical background and motivation
    • Possibility distributions
    • Possibility and necessity measures
    • Connections to imprecise probability
    • Statistical properties and inference
    • Applications in machine learning

    Suggested readings:

  • Lecture 5: Belief Function Theory
    tl;dr: Belief function theory, which generalises both probability and possibility: belief and plausibility measures, Dempster's rule of combination, connections to imprecise probability, and applications in machine learning.
    Slides: TBA

    Topics:

    • Historical background and motivation
    • Belief functions and their interpretations
    • Belief and plausibility measures
    • Dempster’s rule of combination
    • Connections to imprecise probability
    • Applications in machine learning

    Suggested readings:

  • Lecture 6: Credal Sets
    tl;dr: Convex sets of probability measures as a unifying framework: from Bayesianism to quasi- and infra-Bayesianism, representations of credal sets, their connections to other imprecise models, and learning and reasoning with them.
    Slides: TBA

    Topics:

    • Motivation: decision-making under uncertainty
    • From Bayesianism to quasi-Bayesianism and infra-Bayesianism
    • Convex sets of probability measures and their representations
    • Connections to other imprecise probabilistic models, including lower previsions and sets of desirable gambles
    • Learning and reasoning with credal sets, including notions of independence and conditional independence

    Suggested readings:

  • Lecture 7: Learning and Decision Making I
    tl;dr: Updating and decision making when probabilities are uncertain or incomplete: conditioning rules, decision criteria such as maximality, E-admissibility, Gamma-maximin and interval dominance, dynamic consistency, and robustness.
    Slides: TBA

    Topics:

    • Updating and conditioning rules for imprecise probabilities
    • Decision making under imprecision
    • Decision criteria: maximality, E-admissibility, Γ-maximin, Γ-maximax, interval dominance
    • Dynamic consistency and sequential decision-making
    • Robustness, dilation, and the value of information

    Suggested readings:

  • Lecture 8: Learning and Decision Making II
    tl;dr: Classification and regression under imprecision: credal predictors and set-valued predictions, learning objectives and risk, calibration and evaluation of imprecise predictions, and selective prediction and abstention.
    Slides: TBA

    Topics:

    • Classification and regression under imprecision
    • Credal predictors and set-valued predictions
    • Learning objectives, loss functions, and risk
    • Calibration and evaluation of imprecise predictions
    • Selective prediction, abstention, and indeterminate decisions

    Suggested readings:

  • Lecture 9: Uncertainty Quantification
    tl;dr: Why uncertainty quantification matters, how to distinguish and measure aleatoric and epistemic uncertainty, axiomatic characterisations of uncertainty measures, and evaluation and calibration of uncertainty estimates.
    Slides: TBA

    Topics:

    • Quantifying aleatoric and epistemic uncertainty
    • Axiomatic characterization of uncertainty measures
    • Uncertainty quantification for structured outputs beyond token-level predictions
    • Evaluation and calibration of uncertainty estimates
    • Applications in machine learning

    Suggested readings:

  • Lecture 10: Conformal Prediction and Calibration
    tl;dr: Conformal prediction as a distribution-free framework for set-valued prediction with finite-sample coverage guarantees, its imprecise-probabilistic interpretation, and calibration of precise and imprecise probabilistic predictions.
    Slides: TBA

    Topics:

    • From probabilistic to set-valued prediction
    • Conformal prediction and finite-sample coverage guarantees
    • Conformal prediction from an imprecise-probabilistic perspective
    • Calibration of precise and imprecise probabilistic predictions
    • Validity, efficiency, and informativeness of predictive sets
    • Applications in machine learning

    Suggested readings:

  • Lecture 11: Epistemic Causality and Partial Identification
    tl;dr: Causal ambiguity and identifiability, partial identification and causal bounds, and how imprecise probabilities represent causal knowledge for robust inference and decision-making under causal uncertainty.
    Slides: TBA

    Topics:

    • Simpson’s paradox and causal ambiguity
    • Causal assumptions and identifiability
    • Partial identification and causal bounds
    • Imprecise probabilities for representing causal knowledge
    • Robust inference and decision-making under causal uncertainty

    Suggested readings: TBA

  • Lecture 12: Advanced Topics I — Sequential Decision-Making, Safety, and Alignment
    tl;dr: Imprecise Markov decision processes and robust reinforcement learning, exploration and abstention under epistemic uncertainty, AI safety under distribution shift, and preference and reward uncertainty in AI alignment.
    Slides: TBA

    Topics:

    • Sequential decision-making under imprecise uncertainty
    • Imprecise Markov decision processes and robust reinforcement learning
    • Exploration, abstention, and decision-making under epistemic uncertainty
    • AI safety under distribution shift and model uncertainty
    • Preference uncertainty, reward uncertainty, and AI alignment
    • Multi-agent decision-making with imprecise beliefs

    Suggested readings: TBA

  • Lecture 13: Advanced Topics II — Beyond the Observed World
    tl;dr: Compositional generalisation, extrapolation under partial knowledge, open-world learning, novelty and ignorance, scientific discovery under epistemic uncertainty, and metacognition about the limits of knowledge.
    Slides: TBA

    Topics:

    • Compositional generalisation and reasoning about unseen compositions
    • Extrapolation under partial knowledge and underspecification
    • Open-world and open-ended learning
    • Novelty, ignorance, and the unknown
    • Scientific discovery and hypothesis generation under epistemic uncertainty
    • Metacognition and reasoning about the limits of knowledge

    Suggested readings: TBA