Sharp Results for Hypothesis Testing with Risk-Sensitive Agents

Statistical protocols are often used for decision-making involving multiple parties, each with their own incentives, private information, and ability to influence the distributional properties of the data. We study a game-theoretic version of hypothesis testing in which a statistician, also known as a principal, interacts with strategic agents that can generate data. The statistician seeks to design a testing protocol with controlled error, while the data-generating agents, guided by their utility and prior information, choose whether or not to opt in based on expected utility maximization. This strategic behavior affects the data observed by the statistician and, consequently, the associated testing error. We analyze this problem for general concave and monotonic utility functions and prove an upper bound on the Bayes false discovery rate (FDR). Underlying this bound is a form of prior elicitation: we show how an agent's choice to opt in implies a certain upper bound on their prior null probability. Our FDR bound is unimprovable in a strong sense, achieving equality at a single point for an individual agent and at any countable number of points for a population of agents. We also demonstrate that our testing protocols exhibit a desirable maximin property when the principal's utility is considered. To illustrate the qualitative predictions of our theory, we examine the effects of risk aversion, reward stochasticity, and signal-to-noise ratio, as well as the implications for the Food and Drug Administration's testing protocols.

Theoretical Foundations of Conformal Prediction

This book is about conformal prediction and related inferential techniques that build on permutation tests and exchangeability. These techniques are useful in a diverse array of tasks, including hypothesis testing and providing uncertainty quantification guarantees for machine learning systems. Much of the current interest in conformal prediction is due to its ability to integrate into complex machine learning workflows, solving the problem of forming prediction sets without any assumptions on the form of the data generating distribution. Since contemporary machine learning algorithms have generally proven difficult to analyze directly, conformal prediction's main appeal is its ability to provide formal, finite-sample guarantees when paired with such methods. The goal of this book is to teach the reader about the fundamental technical arguments that arise when researching conformal prediction and related questions in distribution-free inference. Many of these proof strategies, especially the more recent ones, are scattered among research papers, making it difficult for researchers to understand where to look, which results are important, and how exactly the proofs work. We hope to bridge this gap by curating what we believe to be some of the most important results in the literature and presenting their proofs in a unified language, with illustrations, and with an eye towards pedagogy.

Quantifying uncertainty in area and regression coefficient estimation from remote sensing maps

Remote sensing map products are used to obtain estimates of environmental quantities, such as deforested area or the effect of conservation zones on deforestation. However, the quality of map products varies, and - because maps are outputs of complex machine learning algorithms that take in a variety of remotely sensed variables as inputs - errors are difficult to characterize. Without capturing the biases that may be present, naive calculations of population-level estimates from such maps are statistically invalid. In this paper, we compare several uncertainty quantification methods - stratification, Olofsson area estimation method, and prediction-powered inference - that combine a small amount of randomly sampled ground truth data with large-scale remote sensing map products to generate statistically valid estimates. Applying these methods across four remote sensing use cases in area and regression coefficient estimation, we find that they result in estimates that are more reliable than naively using the map product as if it were 100% accurate and have lower uncertainty than using only the ground truth and ignoring the map product. Prediction-powered inference uses ground truth data to correct for bias in the map product estimate and (unlike stratification) does not require us to choose a map product before sampling. This is the first work to (1) apply prediction-powered inference to remote sensing estimation tasks, and (2) perform uncertainty quantification on remote sensing regression coefficients without assumptions on the structure of map product errors. To improve the utility of machine learning-generated remote sensing maps for downstream applications, we recommend that map producers provide a holdout ground truth dataset to be used for calibration in uncertainty quantification alongside their maps.

Data-Adaptive Tradeoffs among Multiple Risks in Distribution-Free Prediction

Decision-making pipelines are generally characterized by tradeoffs among various risk functions. It is often desirable to manage such tradeoffs in a data-adaptive manner. As we demonstrate, if this is done naively, state-of-the art uncertainty quantification methods can lead to significant violations of putative risk guarantees. To address this issue, we develop methods that permit valid control of risk when threshold and tradeoff parameters are chosen adaptively. Our methodology supports monotone and nearly-monotone risks, but otherwise makes no distributional assumptions. To illustrate the benefits of our approach, we carry out numerical experiments on synthetic data and the large-scale vision dataset MS-COCO.

Online conformal prediction with decaying step sizes

We introduce a method for online conformal prediction with decaying step sizes. Like previous methods, ours possesses a retrospective guarantee of coverage for arbitrary sequences. However, unlike previous methods, we can simultaneously estimate a population quantile when it exists. Our theory and experiments indicate substantially improved practical properties: in particular, when the distribution is stable, the coverage is close to the desired level for every time point, not just on average over the observed sequence.

Delegating Data Collection in Decentralized Machine Learning

Motivated by the emergence of decentralized machine learning ecosystems, we study the delegation of data collection. Taking the field of contract theory as our starting point, we design optimal and near-optimal contracts that deal with two fundamental machine learning challenges: lack of certainty in the assessment of model quality and lack of knowledge regarding the optimal performance of any model. We show that lack of certainty can be dealt with via simple linear contracts that achieve 1-1/e fraction of the first-best utility, even if the principal has a small test set. Furthermore, we give sufficient conditions on the size of the principal's test set that achieves a vanishing additive approximation to the optimal utility. To address the lack of a priori knowledge regarding the optimal performance, we give a convex program that can adaptively and efficiently compute the optimal contract.

Incentive-Theoretic Bayesian Inference for Collaborative Science

Contemporary scientific research is a distributed, collaborative endeavor, carried out by teams of researchers, regulatory institutions, funding agencies, commercial partners, and scientific bodies, all interacting with each other and facing different incentives. To maintain scientific rigor, statistical methods should acknowledge this state of affairs. To this end, we study hypothesis testing when there is an agent (e.g., a researcher or a pharmaceutical company) with a private prior about an unknown parameter and a principal (e.g., a policymaker or regulator) who wishes to make decisions based on the parameter value. The agent chooses whether to run a statistical trial based on their private prior and then the result of the trial is used by the principal to reach a decision. We show how the principal can conduct statistical inference that leverages the information that is revealed by an agent's strategic behavior -- their choice to run a trial or not. In particular, we show how the principal can design a policy to elucidate partial information about the agent's private prior beliefs and use this to control the posterior probability of the null. One implication is a simple guideline for the choice of significance threshold in clinical trials: the type-I error level should be set to be strictly less than the cost of the trial divided by the firm's profit if the trial is successful.

Class-Conditional Conformal Prediction With Many Classes

Standard conformal prediction methods provide a marginal coverage guarantee, which means that for a random test point, the conformal prediction set contains the true label with a user-chosen probability. In many classification problems, we would like to obtain a stronger guarantee -- that for test points of a specific class, the prediction set contains the true label with the same user-chosen probability. Existing conformal prediction methods do not work well when there is a limited amount of labeled data per class, as is often the case in real applications where the number of classes is large. We propose a method called clustered conformal prediction, which clusters together classes that have "similar" conformal scores and then performs conformal prediction at the cluster level. Based on empirical evaluation across four image data sets with many (up to 1000) classes, we find that clustered conformal typically outperforms existing methods in terms of class-conditional coverage and set size metrics.

Operationalizing Counterfactual Metrics: Incentives, Ranking, and Information Asymmetry

From the social sciences to machine learning, it has been well documented that metrics to be optimized are not always aligned with social welfare. In healthcare, Dranove et al. [12] showed that publishing surgery mortality metrics actually harmed the welfare of sicker patients by increasing provider selection behavior. Using a principal-agent model, we directly study the incentive misalignments that arise from such average treated outcome metrics, and show that the incentives driving treatment decisions would align with maximizing total patient welfare if the metrics (i) accounted for counterfactual untreated outcomes and (ii) considered total welfare instead of average welfare among treated patients. Operationalizing this, we show how counterfactual metrics can be modified to satisfy desirable properties when used for ranking. Extending to realistic settings when the providers observe more about patients than the regulatory agencies do, we bound the decay in performance by the degree of information asymmetry between the principal and the agent. In doing so, our model connects principal-agent information asymmetry with unobserved heterogeneity in causal inference.

Prediction-Powered Inference

We introduce prediction-powered inference $\unicode{x2013}$ a framework for performing valid statistical inference when an experimental data set is supplemented with predictions from a machine-learning system. Our framework yields provably valid conclusions without making any assumptions on the machine-learning algorithm that supplies the predictions. Higher accuracy of the predictions translates to smaller confidence intervals, permitting more powerful inference. Prediction-powered inference yields simple algorithms for computing valid confidence intervals for statistical objects such as means, quantiles, and linear and logistic regression coefficients. We demonstrate the benefits of prediction-powered inference with data sets from proteomics, genomics, electronic voting, remote sensing, census analysis, and ecology.