Predictions from complex machine learning models are increasingly used as input data in subsequent statistical analyses, yet their errors can bias estimators and lead to invalid confidence intervals. Existing approaches attempt to remedy this issue but often impose strong assumptions or lack generality. As an alternative, we present the Predict-Then-Debias bootstrap, developed in Kluger et al. (2025). The method yields valid confidence intervals provided that a small complete sample from the population of interest is available. Its bootstrap construction applies to a broad class of estimators and can be modified to account for weighted, stratified, or clustered samples.

Time-to-event data is widespread across the life sciences and engineering, but it is typically encountered together with censoring, which complicates the application of standard machine learning methods. Deep Cox models have emerged as a popular method for analyzing time-to-event data because they gracefully handle censoring and can be used with unstructured data such as clinical text reports, genomic sequences, and pathology images. However, their predicted survival probabilities are often poorly calibrated, thus limiting their practical utility. In this paper, we propose a novel post hoc calibration method for Deep Cox models that uses isotonic regression to refine predicted survival probabilities without affecting discriminative power. We establish favorable theoretical guarantees, including a double-robustness property and asymptotic calibration. Experiments on synthetic and real-world clinical data demonstrate the empirical effectiveness of our method.

We introduce a framework for robust uncertainty quantification in situations where labeled training data are corrupted, through noisy or missing labels. We build on conformal prediction, a statistical tool for generating prediction sets that cover the test label with a pre-specified probability. The validity of conformal prediction, however, holds under the i.i.d assumption, which does not hold in our setting due to the corruptions in the data. To account for this distribution shift, the privileged conformal prediction (PCP) method proposed leveraging privileged information (PI) -- additional features available only during training -- to re-weight the data distribution, yielding valid prediction sets under the assumption that the weights are accurate. In this work, we analyze the robustness of PCP to inaccuracies in the weights. Our analysis indicates that PCP can still yield valid uncertainty estimates even when the weights are poorly estimated. Furthermore, we introduce uncertain imputation (UI), a new conformal method that does not rely on weight estimation. Instead, we impute corrupted labels in a way that preserves their uncertainty. Our approach is supported by theoretical guarantees and validated empirically on both synthetic and real benchmarks. Finally, we show that these techniques can be integrated into a triply robust framework, ensuring statistically valid predictions as long as at least one underlying method is valid.

Unobserved confounding is a fundamental challenge for estimating causal effects. To address unobserved confounding, recent literature has turned to two different approaches – proxy variables and the use of multiple treatments. The first approach, commonly referred to as proximal causal inference, requires proxies to be assigned to specific asymmetric roles: treatment-inducing proxies (negative control exposures), variables that act as common causes of the treatment and outcome, and outcome-inducing proxies (negative control outcomes). In practice, however, identifying variables that satisfy these asymmetric roles can be difficult depending on the application domain. The second approach, commonly referred to as the “Deconfounder," deals with multiple conditionally independent treatments. There has been limited progress towards developing a consistent estimation method for this setting. As the primary contribution of this work, we establish that causal effects are identifiable in both settings when the unobserved confounder is categorical under suitable conditions. Our approach builds on a mixture learning perspective: we show that the underlying confounding structure can be recovered by identifying the corresponding mixture distribution. We propose an estimation procedure based on tensor decomposition, which allows consistent recovery of the latent structure and comes with non-asymptotic guarantees. Simulation studies and real data experiments demonstrate that the proposed method performs well even with limited data.

Statistical estimation often involves tradeoffs between expensive, high-quality measurements and a variety of lower-quality proxies. We introduce Multiple-Prediction-Powered Inference (MultiPPI): a general framework for constructing statistically efficient estimates by optimally allocating resources across these diverse data sources. This work provides theoretical guarantees about the minimax optimality, finite-sample performance, and asymptotic normality of the MultiPPI estimator. Through experiments across three diverse large language model (LLM) evaluation scenarios, we show that MultiPPI consistently achieves lower estimation error than existing baselines. This advantage stems from its budget-adaptive allocation strategy, which strategically combines subsets of models by learning their complex cost and correlation structures.

Conditional generative models map input variables to complex, high-dimensional distributions, enabling realistic sample generation in a diverse set of domains. A critical challenge with these models is the absence of calibrated uncertainty, which undermines trust in individual outputs for high-stakes applications. To address this issue, we propose a systematic conformal prediction approach tailored to conditional generative models, leveraging density estimation on model-generated samples. We introduce a novel method called CP4Gen, which utilizes clustering-based density estimation to construct prediction sets that are less sensitive to outliers, more interpretable, and of lower structural complexity than existing methods. Extensive experiments on synthetic datasets and real-world applications, including climate emulation tasks, demonstrate that CP4Gen consistently achieves superior performance in terms of prediction set volume and structural simplicity. Our approach offers practitioners a powerful tool for uncertainty estimation associated with conditional generative models, particularly in scenarios demanding rigorous and interpretable prediction sets.

In two-phase multiwave sampling, inexpensive measurements are collected on a large sample and expensive, more informative measurements are adaptively obtained on subsets of units across multiple waves. Adaptively collecting the expensive measurements can increase efficiency but complicates statistical inference. We give valid estimators and confidence intervals for M-estimation under adaptive two-phase multiwave sampling. We focus on the case where proxies for the expensive variables – such as predictions from pretrained machine learning models – are available for all units and propose a Multiwave Predict-Then-Debias estimator that combines proxy information with the expensive, higher-quality measurements to improve efficiency while removing bias. We establish asymptotic linearity and normality and propose asymptotically valid confidence intervals. We also develop an approximately greedy sampling strategy that improves efficiency relative to uniform sampling. Data-based simulation studies support the theoretical results and demonstrate efficiency gains.

Diffusion models have excelled at generative tasks for both continuous and token-based domains, but their application to discrete ordinal data remains underdeveloped. We present \emph{CountsDiff}, a diffusion framework designed to natively model distributions on the natural numbers. CountsDiff extends the Blackout diffusion framework by simplifying its formulation through a direct parameterization in terms of a survival probability schedule and an explicit loss weighting. This introduces flexibility through design parameters with direct analogues in existing diffusion modeling frameworks. Beyond this reparameterization, CountsDiff introduces features from modern diffusion models, previously absent in counts-based domains, including continuous-time training, classifier-free guidance, and churn/remasking reverse dynamics that allow non-monotone reverse trajectories. We propose an initial instantiation of CountsDiff and validate it on natural image datasets (CIFAR-10, CelebA), demonstrating the benefits of the proposed design space and that the framework scales to complex, high-dimensional data domains. We then highlight biological count assays as a natural use case, evaluating CountsDiff on single-cell RNA-seq imputation in a fetal cell and heart cell atlas. Remarkably, we find that even this simple instantiation matches or surpasses the performance of a state-of-the-art discrete generative model and leading RNA-seq imputation methods, while leaving substantial headroom for further gains through optimized design choices in future work.

Biological discovery from experimental data, particularly large-scale assays, requires extensive preprocessing, during which raw outputs (e.g., images, sequences) are processed into structured forms that are more amenable to analysis. While statistical methods for such processed data are at the core of computational biology, the problem of coping with uncertainties introduced during preprocessing is a significant and underexplored issue. We address this issue in the context of differential expression analysis in spatial transcriptomics, which depends on a series of preprocessing steps, including demarcation of cell regions (segmentation), quantification of gene expression in cells, and cell-type annotation. We introduce Corrected Spatial Differential Expression (CSDE), a method that builds on Prediction-Powered Inference to leverage a small set of expert-validated data points (cells) to account for uncertainty due to preprocessing errors. Using two case studies, we demonstrate that CSDE produces more reliable and calibrated estimates of differential expression compared to the prevalent approach that neglects the impact of preprocessing. CSDE incorporates an efficient workflow to generate the required expert-annotated data, and is available as open-source at https://github.com/YosefLab/CSDE. ### Competing Interest Statement The authors have declared no competing interest.

A core challenge in modern AI model development is obtaining high-quality evaluation metrics in a cost-effective way. Such evaluation often involves tradeoffs between expensive, high-quality measurements and a variety of lower-quality proxies. We introduce Multiple-Prediction-Powered Inference (MultiPPI), a general framework for constructing statistically efficient estimates by optimally allocating resources across these diverse data sources. We provide theoretical guarantees about the minimax optimality, finite-sample performance, and asymptotic normality of the MultiPPI estimator. Through experiments across three diverse large language model (LLM) evaluation scenarios, we show that MultiPPI consistently achieves lower estimation error than existing baselines. This advantage stems from its budget-adaptive allocation strategy, which strategically combines subsets of models by learning their complex cost and correlation structures.

We study statistical parameter estimation in the setting of data markets when providers differ in both provision costs and estimation-relevant data quality. We define a cost-per-information score that summarizes each provider's provision cost per unit of information about the buyer's estimation objective. When quality is known ex ante, we describe a second-score procurement mechanism that ranks providers by this score, and endogenously chooses both a provider and a sample size while making truthful cost reports optimal. We then turn to the more realistic setting where data quality is private, and can only be assessed noisily via the delivered data. In this setting, we propose a simple mechanism that augments the second-score rule with a lenient ex post statistical test of the reported quality. We prove that, under mild conditions, the mechanism admits near-truthful reports that become approximately optimal and individually rational as the procured sample size grows. Under these near-truthful reports, the buyer asymptotically recovers the performance of the corresponding known-quality second-score mechanism. Our analysis highlights how the choice of verification test and the buyer's accuracy-cost tradeoff jointly shape participation and misreporting incentives in data markets.

Decomposing prediction uncertainty into aleatoric (irreducible) and epistemic (reducible) components is critical for the reliable deployment of machine learning systems. While the mutual information between the response variable and model parameters is a principled measure for epistemic uncertainty, it requires access to the parameter posterior, which is computationally challenging to approximate. Consequently, practitioners often rely on probabilistic predictions from deep ensembles to quantify uncertainty, which have demonstrated strong empirical performance. However, a theoretical understanding of their success from a frequentist perspective remains limited. We address this gap by first considering a bootstrap-based estimator for epistemic uncertainty, which we prove is asymptotically correct. Next, we connect deep ensembles to the bootstrap estimator by decomposing it into data variability and training stochasticity; specifically, we show that deep ensembles capture the training stochasticity component. Through empirical studies, we show that this stochasticity component constitutes the majority of epistemic uncertainty, thereby explaining the effectiveness of deep ensembles.

We study hypothesis testing over a heterogeneous population of strategic agents with private information. Any single test applied uniformly across the population yields statistical error that is sub-optimal relative to the performance of an oracle given access to the private information. We show how it is possible to design menus of statistical contracts that pair type-optimal tests with payoff structures, inducing agents to self-select according to their private information. This separating menu elicits agent types and enables the principal to match the oracle performance even without a priori knowledge of the agent type. Our main result fully characterizes the collection of all separating menus that are instance-adaptive, matching oracle performance for an arbitrary population of heterogeneous agents. We identify designs where information elicitation is essentially costless, requiring negligible additional expense relative to a single-test benchmark, while improving statistical performance. Our work establishes a connection between proper scoring rules and menu design, showing how the structure of the hypothesis test constrains the elicitable information. Numerical examples illustrate the geometry of separating menus and the improvements they deliver in error trade-offs. Overall, our results connect statistical decision theory with mechanism design, demonstrating how heterogeneity and strategic participation can be harnessed to improve efficiency in hypothesis testing.

Estimating associations between spatial covariates and responses - rather than merely predicting responses - is central to environmental science, epidemiology, and economics. For instance, public health officials might be interested in whether air pollution has a strictly positive association with a health outcome, and the magnitude of any effect. Standard machine learning methods often provide accurate predictions but offer limited insight into covariate-response relationships. And we show that existing methods for constructing confidence (or credible) intervals for associations can fail to provide nominal coverage in the face of model misspecification and nonrandom locations - despite both being essentially always present in spatial problems. We introduce a method that constructs valid frequentist confidence intervals for associations in spatial settings. Our method requires minimal assumptions beyond a form of spatial smoothness and a homoskedastic Gaussian error assumption. In particular, we do not require model correctness or covariate overlap between training and target locations. Our approach is the first to guarantee nominal coverage in this setting and outperforms existing techniques in both real and simulated experiments. Our confidence intervals are valid in finite samples when the noise of the Gaussian error is known, and we provide an asymptotically consistent estimation procedure for this noise variance when it is unknown.

Machine learning models are increasingly used to produce predictions that serve as input data in subsequent statistical analyses. For example, computer vision predictions of economic and environmental indicators based on satellite imagery are used in downstream regressions; similarly, language models are widely used to approximate human ratings and opinions in social science research. However, failure to properly account for errors in the machine learning predictions renders standard statistical procedures invalid. Prior work uses what we call the Predict-Then-Debias estimator to give valid confidence intervals when machine learning algorithms impute missing variables, assuming a small complete sample from the population of interest. We expand the scope by introducing bootstrap confidence intervals that apply when the complete data is a nonuniform (i.e., weighted, stratified, or clustered) sample and to settings where an arbitrary subset of features is imputed. Importantly, the method can be applied to many settings without requiring additional calculations. We prove that these confidence intervals are valid under no assumptions on the quality of the machine learning model and are no wider than the intervals obtained by methods that do not use machine learning predictions.

We introduce a framework for calibrating machine learning models so that their predictions satisfy explicit, finite-sample statistical guarantees. Our calibration algorithms work with any underlying model and (unknown) data-generating distribution and do not require model refitting. The framework addresses, among other examples, false discovery rate control in multi-label classification, intersection-over-union control in instance segmentation, and the simultaneous control of the type-1 error of outlier detection and confidence set coverage in classification or regression. Our main insight is to reframe the risk-control problem as multiple hypothesis testing, enabling techniques and mathematical arguments different from those in the previous literature. We use the framework to provide new calibration methods for several core machine learning tasks, with detailed worked examples in computer vision and tabular medical data.

Background: The deployment of black-box AI models in medical imaging presents significant challenges, especially in maintaining reliability across different clinical settings. These challenges are compounded by distribution shifts that can lead to failures in reproducing the accuracy attained during the AI model's original validations. Method: We introduce the conformal triage algorithm, designed to categorize patients into low-risk, high-risk, and uncertain groups within a clinical deployment setting. This method leverages a combination of a black-box AI model and conformal prediction techniques to offer statistical guarantees of predictive power for each group. The high-risk group is guaranteed to have a high positive predictive value, while the low-risk group is assured a high negative predictive value. Prediction sets are never constructed; instead, conformal techniques directly assure high accuracy in both groups, even in clinical environments different from those in which the AI model was originally trained, thereby ameliorating the challenges posed by distribution shifts. Importantly, a representative data set of exams from the testing environment is required to ensure statistical validity. Results: The algorithm was tested using a head CT model previously developed by Do and colleagues [9] and a data set from Massachusetts General Hospital. The results demonstrate that the conformal triage algorithm provides reliable predictive value guarantees to a clinically significant extent, reducing the number of false positives from 233 (45%) to 8 (5%) while only abstaining from prediction on 14% of data points, even in a setting different from the training environment of the original AI model. Conclusions: The conformal triage algorithm offers a promising solution to the challenge of deploying black-box AI models in medical imaging across varying clinical settings. By providing statistical guarantees of predictive value for categorized patient groups, this approach significantly enhances the reliability and utility of AI in optimizing medical imaging workflows, particularly in neuroradiology. ### Competing Interest Statement M.H.L. is a consultant at GE Healthcare and at Takeda/Seagen/Roche-Genentech Pharmaceuticals. ### Funding Statement This study did not receive any funding. ### Author Declarations I confirm all relevant ethical guidelines have been followed, and any necessary IRB and/or ethics committee approvals have been obtained. Yes The details of the IRB/oversight body that provided approval or exemption for the research described are given below: IRB of Massachusetts General Hospital gave ethical approval of this work. I confirm that all necessary patient/participant consent has been obtained and the appropriate institutional forms have been archived, and that any patient/participant/sample identifiers included were not known to anyone (e.g., hospital staff, patients or participants themselves) outside the research group so cannot be used to identify individuals. Yes I understand that all clinical trials and any other prospective interventional studies must be registered with an ICMJE-approved registry, such as ClinicalTrials.gov. I confirm that any such study reported in the manuscript has been registered and the trial registration ID is provided (note: if posting a prospective study registered retrospectively, please provide a statement in the trial ID field explaining why the study was not registered in advance). Yes I have followed all appropriate research reporting guidelines, such as any relevant EQUATOR Network research reporting checklist(s) and other pertinent material, if applicable. Yes All data produced are available online at https://github.com/aangelopoulos/conformal-triage .

Unveiling functional relationships between various molecular cell phenotypes from data using machine learning models is a key promise of multiomics. Existing methods either use flexible but hard-to-interpret models or simpler, misspecified models. VI-VS (Variational Inference for Variable Selection) balances flexibility and interpretability to identify relevant feature relationships in multiomic data. It uses deep generative models to identify conditionally dependent features, with false discovery rate control. VI-VS is available as an open-source Python package, providing a robust solution to identify features more likely representing genuine causal relationships.

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.