A general framework for synthetic-augmented inference across a population of related tasks is developed, which characterizes synthetic augmentation by the number of synthetic observations and their weight and specifies a size-weight frontier that specifies, for each weight, the largest synthetic sample size for which all smaller sizes attain the target task-marginal coverage.
Abstract
Synthetic data can improve statistical inference when real data are scarce, but naively treating synthetic samples as real data can introduce bias and lead to unreliable inference. We develop a general framework for synthetic-augmented inference across a population of related tasks. It characterizes synthetic augmentation by the number of synthetic observations and their weight. Central to our framework is a size-weight frontier that specifies, for each weight, the largest synthetic sample size for which all smaller sizes attain the target task-marginal coverage. We estimate this frontier from historical tasks, and establish a finite-sample coverage guarantee simultaneously for all size-weight configurations on or below the estimated frontier. In experiments using large language model responses to augment opinion survey data, our procedure achieves target coverage and substantially narrows confidence intervals.
In transfer-learning settings, a model derived from abundant surrogate labels may be deployed in a target population where gold-standard outcomes are unobserved. Evaluating its target performance is essential for determining whether decisions based on the model remain reliable, yet it is difficult when gold labels are scarce, and covariate distributions differ across data sources. We study a three-sample setting with a small gold-labeled source, a larger surrogate-labeled source, and an unlabeled target. Under conditional transportability, we evaluate the surrogate-derived model against the latent gold-standard outcome in the target population. We propose cross-fitted estimators that transport information from the two labeled sources through source-specific density ratios. We also combine outcome-regression augmentation with a kernel correction for estimating the model near a threshold, accounting for uncertainty from all three samples. We establish asymptotically linear inference for TPR and FPR, consistency and pointwise inference for the ROC curve, and asymptotically normal inference for AUC. Simulations assess bias, coverage, and sensitivity to bandwidth and relative sample sizes. A retrospective temporal validation on Chatbot Arena and a semi-synthetic ACS-Income study provide validation in real-world AI applications.
A novel perturbation test based on a nonsmooth max-difference revenue statistic comparing the best null assortment with the best alternative assortment and asymptotic validity of the proposed p-value under adaptive assortment selection is proposed.
Reweighting source samples to match a target covariate distribution is a standard response to distribution shift when generalizing evidence from one population to another. This strategy is well suited to deterministic, learnable covariate discrepancies, but can be insufficient when source--target population differences also contain changes beyond covariate shift or when estimation of the density-ratio weights is unstable. To address this challenge, we introduce a new model that allows non-systematic changes between two population laws after systematic shifts are accounted for. Such residual shift is modeled as random perturbations to the probability space that cannot be represented in a learnable way. In this way, we separate systematic shifts, treated as bias and corrected by reweighting, from residual random perturbations, treated as distributional uncertainty and handled through dataset pooling. Under pure random perturbations, this principle yields Augmented Inverse Distance Weighting (AIDW), which uses regression augmentation and variance-optimal dataset-level pooling. For mixed shifts, we develop Augmented Inverse Hybrid Weighting (AIHW), which interpolates between AIDW and standard augmented importance weighting. Both methods trade off sampling uncertainty and distributional uncertainty via a \emph{distributional distance} that describes the strength of random perturbations. We establish asymptotic properties of the methods, together with plug-in guidance for choosing tuning parameters and model diagnostic tools. Experiments on three real-world multi-site datasets demonstrate consistent reductions in mean-squared error compared with standard weighting baselines, along with substantially improved empirical coverage in settings where covariate-shift adjustment alone undercovers, showing the robustness of the proposed methods across diverse distribution shift scenarios.
Kernel density estimation converts finite samples into probability densities, but its performance depends critically on bandwidth selection. Classical selectors prescribe the sample-to-bandwidth rule analytically or asymptotically, or solve a new optimization for each sample. An amortized framework is proposed that instead learns this mapping across a distribution of density-estimation tasks by optimizing the logarithmic score. A truncated-and-renormalized bounded-support formulation enables stable learning across heterogeneous tasks, while affine standardization allows a selector trained on a single reference interval to transfer across bounded intervals. Experiments under Gaussian sampling, a multi-family benchmark, and randomized Gaussian-mixture training show that the amortized selector consistently and substantially outperforms Silverman's rule, the Sheather--Jones selector, and least-squares cross-validation, with especially large gains in small and heterogeneous samples. Finite Gaussian mixtures provide a generic training mechanism supported by their $L^1$ approximation property. Selectors trained in this way generalize strongly across different density structures, allowing the same trained selector to be applied directly to finite samples from unknown densities without specifying or fitting a distributional family. This combination of broad applicability and strong empirical performance makes the framework attractive for a wide range of applications in which finite samples or ensembles must be converted into continuous probability densities.
Conditional generative models have emerged as powerful tools for sampling from target conditional distributions, driving substantial advances across a wide range of scientific and applied domains. As these models proliferate, practitioners often face multiple plausible generators whose performance can vary with the task, data, or input condition. We propose an optimal model averaging framework for conditional generative models, allowing candidate generators to be combined even when they are accessible only through conditional samples without tractable densities. Specifically, we use a sample-based maximum mean discrepancy between conditional distributions, which first leads to a static model averaging method, StaticMA, assigning fixed weights to different candidates. In addition, we develop MoEMA (mixture-of-experts model averaging), an input-adaptive method that parameterizes covariate-dependent weights through a softmax neural-network gate. We establish in-sample and out-of-sample asymptotic optimality for the proposed methods, together with consistency of the estimated adaptive weight function under regularity conditions. The framework applies directly to Euclidean responses and extends to unstructured data by combining our formulation with fixed representation maps. Across a broad set of simulations and real-data studies spanning tabular, image, and text modalities, MoEMA generally improves over competing baselines, demonstrating the effectiveness of our proposed methods.
In this paper, we study the problem of augmenting a tiny target sample with a massive auxiliary sample. Utilizing Tukey's factorization, there are two popular approaches: the inverse probability weight (IPW) and the full-likelihood (FL) methods. We show that the IPW approach suffers from the limited target sample problem while the FL method may estimate some model parameters at the rate of the massive auxiliary sample size, a phenomenon we call full efficiency gain. We study the theory behind the full efficiency gain for exponential families and mixtures of exponential families. We also study the efficiency gain for the IPW method under a nonparametric procedure and show how it can achieve a parametric rate of the target sample size. As a side note, we also discuss how one may use FL to train neural network models simultaneously for both the target distribution and the odds model.
A new method for surgically removing training examples from a model reveals that as datasets grow, the link between what a model learns and what it produces dissolves.
