I examine the technology of machine learning from the perspective of rhetoric, which is simply the art of persuasion. Rather than being a neutral and "objective" way to build "world models" from data, machine learning is (I argue) inherently rhetorical. I explore some of its rhetorical features, and examine one pervasive business model where machine learning is widely used, "manipulation as a service."

The machine learning community progresses (in part) by improving the "accuracy" of its systems. The EU AI Act explicitly refers to "accuracy" as part of its compliance measures for high-risk AI systems. Are we talking about the same thing? This work presents "accuracy" as a case-study for differing requirements of social worlds, the technological machine learning community and the legal community. While competition on accuracy contributes to technological development, machine learning scholars simultaneously recognize accuracy's shortcomings regarding the usefulness and effectiveness of machine learning systems. The legal counterpart embraces the vagueness of "accuracy," leaving interpretative flexibility for technological and societal changes. At the same time, accuracy is a core element of compliance within the EU AI Act. We elaborate on five main tensions, (a) nature of accuracy, (b) notion of performance, (c) scope of validity, (d) ends, and (e) statisticalness, to show that the two communities project disparate, and sometimes contradictory, expectations on accuracy. Both legal and technical communities lack precise understanding of "accuracy" beyond the contextual boundaries of their community. The resulting frictions, , based on the empirical or normative understanding of accuracy, are symptoms of an unresolved (and unresolvable) debate on what accuracy is. We constructively use the frictions to recommend baselines and interventional studies in standardization, and demand for tools to extend the validity of accuracy measurements.

Machine Learning research, including work promoting fair or equitable algorithms, often relies on the concept of a data-generating probability distribution. The standard presumption is that since data points are 'sampled from' such a distribution, one can learn from observed data about this distribution and, thus, predict future data points which are also drawn from it. We argue, however, that such true probability distributions do not exist and that the rhetoric around them is harmful in social settings. We show that alternative frameworks focusing directly on relevant populations rather than abstract distributions are available and leave classical learning theory almost unchanged. Furthermore, we argue that the assumption of true probabilities or data-generating distributions can be misleading and obscure both the choices made and the goals pursued in machine learning practice. Based on these considerations, we suggest avoiding the assumption of data-generating probability distributions in the social world.

A key result in statistics is the data processing inequality, originally proved by Blackwell (1951) and later refined by DeGroot (1962) in terms of statistical uncertainty. It states that the Bayes risk of a statistical experiment obtained by stochastically modifying another experiment cannot be lower than the Bayes risk of the original experiment, regardless of the loss function or prior chosen. In machine learning, this result underlies applications such as the information bottleneck principle and some feature learning techniques. However, machine learning problems are constrained learning problems: the model class used does not include all measurable functions. We present a simple counterexample showing that the classical data processing inequality fails to hold in such a setting. Hence, we formulate a generalized data processing inequality, requiring the constrained Bayes risk of a joint distribution (with respect to a loss function and a constrained hypothesis class) to lower bound the constrained Bayes risk on the stochastically modified distribution, regardless of the choice of distribution. We show this inequality to be equivalent to a set containment condition on a specific function set induced by the loss and model class, called the superprediction set. Finally, we derive sufficient conditions for this containment.

Fueled by discussions around "trustworthiness" and algorithmic fairness, calibration of predictive systems has regained scholars attention. The vanilla definition and understanding of calibration is, simply put, on all days on which the rain probability has been predicted to be p, the actual frequency of rain days was p. However, the increased attention has led to an immense variety of new notions of "calibration." Some of the notions are incomparable, serve different purposes, or imply each other. In this work, we provide two accounts which motivate calibration: self-realization of forecasted properties and precise estimation of incurred losses of the decision makers relying on forecasts. We substantiate the former via the reflection principle and the latter by actuarial fairness. For both accounts we formulate prototypical definitions via properties Γ of outcome distributions, e.g., the mean or median. The prototypical definition for self-realization, which we call Γ-calibration, is equivalent to a certain type of swap regret under certain conditions. These implications are strongly connected to the omniprediction learning paradigm. The prototypical definition for precise loss estimation is a modification of decision calibration adopted from Zhao et al. [73]. For binary outcome sets both prototypical definitions coincide under appropriate choices of reference properties. For higher-dimensional outcome sets, both prototypical definitions can be subsumed by a natural extension of the binary definition, called distribution calibration with respect to a property. We conclude by commenting on the role of groupings in both accounts of calibration often used to obtain multicalibration. In sum, this work provides a semantic map of calibration in order to navigate a fragmented terrain of notions and definitions.

We can use structural causal models (SCMs) to help us evaluate the consequences of actions given data. SCMs identify actions with structural interventions. A careful decision maker may wonder whether this identification is justified. We seek such a justification. We begin with decision models, which map actions to distributions over outcomes but avoid additional causal assumptions. We then examine assumptions that could justify causal interventions, with a focus on symmetry. First, we introduce conditionally independent and identical responses (CIIR), a generalisation of the IID assumption to decision models. CIIR justifies identifying actions with interventions, but is often an implausible assumption. We consider an alternative: precedent is the assumption that “what I can do has been done before, and its consequences observed,” and is generally more plausible than CIIR. We show that precedent together with independence of causal mechanisms (ICM) and an observed conditional independence can justify identifying actions with causal interventions. ICM has been proposed as an alternative foundation for causal modelling, but this work suggests that it may in fact justify the interventional interpretation of causal models.

We introduce a notion of distance between supervised learning problems, which we call the Risk distance. This distance, inspired by optimal transport, facilitates stability results; one can quantify how seriously issues like sampling bias, noise, limited data, and approximations might change a given problem by bounding how much these modifications can move the problem under the Risk distance. With the distance established, we explore the geometry of the resulting space of supervised learning problems, providing explicit geodesics and proving that the set of classification problems is dense in a larger class of problems. We also provide two variants of the Risk distance: one that incorporates specified weights on a problem's predictors, and one that is more sensitive to the contours of a problem's risk landscape.

Corruption is notoriously widespread in data collection. Despite extensive research, the existing literature on corruption predominantly focuses on specific settings and learning scenarios, lacking a unified view. There is still a limited understanding of how to effectively model and mitigate corruption in machine learning problems. In this work, we develop a general theory of corruption from an information-theoretic perspective - with Markov kernels as a foundational mathematical tool. We generalize the definition of corruption beyond the concept of distributional shift: corruption includes all modifications of a learning problem, including changes in model class and loss function. We will focus here on changes in probability distributions. First, we construct a provably exhaustive framework for pairwise Markovian corruptions. The framework not only allows us to study corruption types based on their input space, but also serves to unify prior works on specific corruption models and establish a consistent nomenclature. Second, we systematically analyze the consequences of corruption on learning tasks by comparing Bayes risks in the clean and corrupted scenarios. This examination sheds light on complexities arising from joint and dependent corruptions on both labels and attributes. Notably, while label corruptions affect only the loss function, more intricate cases involving attribute corruptions extend the influence beyond the loss to affect the hypothesis class. Third, building upon these results, we investigate mitigations for various corruption types. We expand the existing loss-correction results for label corruption, and identify the necessity to generalize the classical corruption-corrected learning framework to a new paradigm with weaker requirements. Within the latter setting, we provide a negative result for loss correction in the attribute and the joint corruption case.

The standard approach to causal modelling especially in social and health sciences is the potential outcomes framework due to Neyman and Rubin. In this framework, observations are thought to be drawn from a distribution over variables of interest, and the goal is to identify parameters of this distribution. Even though the stated goal is often to inform decision making on some target population, there is no straightforward way to include these target populations in the framework. Instead of modelling the relationship between the observed sample and the target population, the inductive assumptions in this framework take the form of abstract sampling and independence assumptions. In this paper, we develop a version of this framework that construes causal inference as treatment-wise predictions for finite populations where all assumptions are testable in retrospect; this means that one can not only test predictions themselves (without any fundamental problem) but also investigate sources of error when they fail. Due to close connections to the original framework, established methods can still be be analysed under the new framework.

What does it mean to say that, for example, the probability for rain tomorrow is between 20% and 30%? The theory for the evaluation of precise probabilistic forecasts is well-developed and is grounded in the key concepts of proper scoring rules and calibration. For the case of imprecise probabilistic forecasts (sets of probabilities), such theory is still lacking. In this work, we therefore generalize proper scoring rules and calibration to the imprecise case. We develop these concepts as relative to data models and decision problems. As a consequence, the imprecision is embedded in a clear context. We establish a close link to the paradigm of (group) distributional robustness and in doing so provide new insights for it. We argue that proper scoring rules and calibration serve two distinct goals, which are aligned in the precise case, but intriguingly are not necessarily aligned in the imprecise case. The concept of decision-theoretic entropy plays a key role for both goals. Finally, we demonstrate the theoretical insights in machine learning practice, in particular we illustrate subtle pitfalls relating to the choice of loss function in distributional robustness.

The most common approach to causal modelling is the potential outcomes framework due to Neyman and Rubin. In this framework, outcomes of counterfactual treatments are assumed to be well-defined. This metaphysical assumption is often thought to be problematic yet indispensable. The conventional approach relies not only on counterfactuals but also on abstract notions of distributions and assumptions of independence that are not directly testable. In this paper, we construe causal inference as treatment-wise predictions for finite populations where all assumptions are testable; this means that one can not only test predictions themselves (without any fundamental problem) but also investigate sources of error when they fail. The new framework highlights the model-dependence of causal claims as well as the difference between statistical and scientific inference.

We argue that insurance can act as an analogon for the social situatedness of machine learning systems, hence allowing machine learning scholars to take insights from the rich and interdisciplinary insurance literature. Tracing the interaction of uncertainty, fairness and responsibility in insurance provides a fresh perspective on fairness in machine learning. We link insurance fairness conceptions to their machine learning relatives, and use this bridge to problematize fairness as calibration. In this process, we bring to the forefront two themes that have been largely overlooked in the machine learning literature: responsibility and aggregate-individual tensions.

Machine learning is about forecasting. Forecasts, however, obtain their usefulness only through their evaluation. Machine learning has traditionally focused on types of losses and their corresponding regret. Currently, the machine learning community regained interest in calibration. In this work, we show the conceptual equivalence of calibration and regret in evaluating forecasts. We frame the evaluation problem as a game between a forecaster, a gambler and nature. Putting intuitive restrictions on gambler and forecaster, calibration and regret naturally fall out of the framework. In addition, this game links evaluation of forecasts to randomness of outcomes. Random outcomes with respect to forecasts are equivalent to good forecasts with respect to outcomes. We call those dual aspects, calibration and regret, predictiveness and randomness, the four facets of forecast felicity.

Supervised learning has gone beyond the expected risk minimization framework. Central to most of these developments is the introduction of more general aggregation functions for losses incurred by the learner. In this paper, we turn towards online learning under expert advice. Via easily justified assumptions we characterize a set of reasonable loss aggregation functions as quasi-sums. Based upon this insight, we suggest a variant of the Aggregating Algorithm tailored to these more general aggregation functions. This variant inherits most of the nice theoretical properties of the AA, such as recovery of Bayes' updating and a time-independent bound on quasi-sum regret. Finally, we argue that generalized aggregations express the attitude of the learner towards losses.

Machine learning typically presupposes classical probability theory which implies that aggregation is built upon expectation. There are now multiple reasons to motivate looking at richer alternatives to classical probability theory as a mathematical foundation for machine learning. We systematically examine a powerful and rich class of alternative aggregation functionals, known variously as spectral risk measures, Choquet integrals or Lorentz norms. We present a range of characterization results, and demonstrate what makes this spectral family so special. In doing so we arrive at a natural stratification of all coherent risk measures in terms of the upper probabilities that they induce by exploiting results from the theory of rearrangement invariant Banach spaces. We empirically demonstrate how this new approach to uncertainty helps tackling practical machine learning problems.

We introduce two new classes of measures of information for statistical experiments which generalise and subsume $\phi$-divergences, integral probability metrics, $\mathfrak{N}$-distances (MMD), and $(f,\Gamma)$ divergences between two or more distributions. This enables us to derive a simple geometrical relationship between measures of information and the Bayes risk of a statistical decision problem, thus extending the variational $\phi$-divergence representation to multiple distributions in an entirely symmetric manner. The new families of divergence are closed under the action of Markov operators which yields an information processing equality which is a refinement and generalisation of the classical data processing inequality. This equality gives insight into the significance of the choice of the hypothesis class in classical risk minimization.

Mixable loss functions are of fundamental importance in the context of prediction with expert advice in the online setting since they characterize fast learning rates. By re-interpreting properness from the point of view of differential geometry, we provide a simple geometric characterization of mixability for the binary and multi-class cases: a proper loss function $\ell$ is $\eta$-mixable if and only if the superpredition set $\textrm{spr}(\eta \ell)$ of the scaled loss function $\eta \ell$ slides freely inside the superprediction set $\textrm{spr}(\ell_{\log})$ of the log loss $\ell_{\log}$, under fairly general assumptions on the differentiability of $\ell$. Our approach provides a way to treat some concepts concerning loss functions (like properness) in a ''coordinate-free'' manner and reconciles previous results obtained for mixable loss functions for the binary and the multi-class cases.