Generative and Nonparametric Approaches for Conditional Distribution Estimation: Methods, Perspectives, and Comparative Evaluations
Institute of Statistical Science, Academia Sinica（学术院统计科学研究所）；Department of Mathematics, National Taiwan University（台湾大学数学系）；Data Science Degree Program, National Taiwan University and Academia Sinica（台湾大学数据科学学士学位计划）；Department of Statistics, University of Michigan（密歇根大学统计学系）
The inference of conditional distributions is a fundamental problem in statistics, essential for prediction, uncertainty quantification, and probabilistic modeling. A wide range of methodologies have been developed for this task. This article reviews and compares several representative approaches spanning classical nonparametric methods and modern generative models. We begin with the single-index method of Hall and Yao (2005), which estimates the conditional distribution through a dimension-reducing index and nonparametric smoothing of the resulting one-dimensional cumulative conditional distribution function. We then examine the basis-expansion approaches, including FlexCode (Izbicki and Lee, 2017) and DeepCDE (Dalmasso et al., 2020), which convert conditional density estimation into a set of nonparametric regression problems. In addition, we discuss two recent generative simulation-based methods that leverage modern deep generative architectures: the generative conditional distribution sampler (Zhou et al., 2023) and the conditional denoising diffusion probabilistic model (Fu et al., 2024; Yang et al., 2025). A systematic numerical comparison of these approaches is provided using a unified evaluation framework that ensures fairness and reproducibility. The performance metrics used for the estimated conditional distribution include the mean-squared errors of conditional mean and standard deviation, as well as the Wasserstein distance. We also discuss their flexibility and computational costs, highlighting the distinct advantages and limitations of each approach.
Conditional Diffusion for Nonparametric Instrumental Variable Quantile Regression
Institute of Data Science and Statistics, Shanghai University of Finance and Economics（上海财经大学数据科学与统计研究院）；School of Mathematics and Statistics, Wuhan University（武汉大学数学与统计学院）；School of Artificial Intelligence, Wuhan University（武汉大学人工智能学院）
Conditional density estimation generalizes regression by modeling a full density f(yjx) rather than only the expected value E(yjx). This is important for many tasks, including handling multi-modality and generating prediction intervals. Though fundamental and widely applicable, nonparametric conditional density estimators have received relatively little attention from statisticians and little or none from the machine learning community. None of that work has been applied to greater than bivariate data, presumably due to the computational difficulty of data-driven bandwidth selection. We descri…
Southwestern University of Finance and Economics（西南财经大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Wuhan University（武汉大学）
We consider inference procedures, conditional on an observed ancillary statistic, for regression coefficients under a linear regression setup where the unknown error distribution is specified nonparametrically. We establish conditional asymptotic normality of the regression coefficient estimators under regularity conditions, and formally justify the approach of plugging in kernel-type density estimators in conditional inference procedures. Simulation results show that the approach yields accurate conditional coverage probabilities when used for constructing confidence intervals. The plug-in ap…
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North Carolina State University（北卡罗来纳州立大学）；University of Melbourne（墨尔本大学）
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In this paper, we consider the problem of estimating a conditional density in moderately large dimensions. Much more informative than regression functions, conditional densities are of main interest in recent methods, particularly in the Bayesian framework (studying the posterior distribution, finding its modes...). Considering a recently studied family of kernel estimators, we select a pointwise multivariate bandwidth by revisiting the greedy algorithm Rodeo (Regularisation Of Derivative Expectation Operator). The method addresses several issues: being greedy and computationally efficient by …
Deep conditional distribution learning via conditional Föllmer flow
Joint Laboratory of Data Science and Business Intelligence（数据科学与商务智能联合实验室）；Southwestern University of Finance and Economics（西南财经大学）；State Key Laboratory of Mathematical Sciences（数学科学国家重点实验室）；Academy of Mathematics and Systems Science（数学与系统科学学院）；Chinese Academy of Sciences（中国科学院）；School of Mathematics and Statistics（数学与统计学学院）；Wuhan University（武汉大学）；Hubei Key Laboratory of Computational Science（湖北省计算科学重点实验室）
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