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1. Bayesian semiparametric causal inference: Targeted doubly robust estimation of treatment effects

   Sert, Gözde; Chakrabortty, Abhishek; Bhattacharya, Anirban (November 2025, arXiv.org)

   We propose a semiparametric Bayesian methodology for estimating the average treatment effect (ATE) within the potential outcomes framework using observational data with high-dimensional nuisance parameters. Our method introduces a Bayesian debiasing procedure that corrects for bias arising from nuisance estimation and employs a targeted modeling strategy based on summary statistics rather than the full data. These summary statistics are identified in a debiased manner, enabling the estimation of nuisance bias via weighted observables and facilitating hierarchical learning of the ATE. By combining debiasing with sample splitting, our approach separates nuisance estimation from inference on the target parameter, reducing sensitivity to nuisance model specification. We establish that, under mild conditions, the marginal posterior for the ATE satisfies a Bernstein-von Mises theorem when both nuisance models are correctly specified and remains consistent and robust when only one is correct, achieving Bayesian double robustness. This ensures asymptotic efficiency and frequentist validity. Extensive simulations confirm the theoretical results, demonstrating accurate point estimation and credible intervals with nominal coverage, even in high-dimensional settings. The proposed framework can also be extended to other causal estimands, and its key principles offer a general foundation for advancing Bayesian semiparametric inference more broadly. 

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2. Bayesian semi-supervised inference via a debiased modeling approach

   https://doi.org/10.1016/j.ecosta.2025.05.001  

   Sert, Gözde; Chakrabortty, Abhishek; Bhattacharya, Anirban (May 2025, Econometrics and Statistics)

   Inference in semi-supervised (SS) settings has gained substantial attention in recent years due to increased relevance in modern big-data problems. In a typical SS setting, there is a much larger-sized unlabeled data, containing only observations of predictors, and a moderately sized labeled data containing observations for both an outcome and the set of predictors. Such data naturally arises when the outcome, unlike the predictors, is costly or difficult to obtain. One of the primary statistical objectives in SS settings is to explore whether parameter estimation can be improved by exploiting the unlabeled data. We propose a novel Bayesian method for estimating the population mean in SS settings. The approach yields estimators that are both efficient and optimal for estimation and inference. The method itself has several interesting artifacts. The central idea behind the method is to model certain summary statistics of the data in a targeted manner, rather than the entire raw data itself, along with a novel Bayesian notion of debiasing. Specifying appropriate summary statistics crucially relies on a debiased representation of the population mean that incorporates unlabeled data through a flexible nuisance function while also learning its estimation bias. Combined with careful usage of sample splitting, this debiasing approach mitigates the effect of bias due to slow rates or misspecification of the nuisance parameter from the posterior of the final parameter of interest, ensuring its robustness and efficiency. Concrete theoretical results, via Bernstein--von Mises theorems, are established, validating all claims, and are further supported through extensive numerical studies. To our knowledge, this is possibly the first work on Bayesian inference in SS settings, and its central ideas also apply more broadly to other Bayesian semi-parametric inference problems. 

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3. On Differentially Private U Statistics

   Chaudhuri, Kamalika; Loh, Po-Ling; Pandey, Shourya; Sarkar, Purnamrita (July 2024, https://doi.org/10.48550/arXiv.2407.04945)

   We consider the problem of privately estimating a parameter 𝔼[h(X1,…,Xk)], where X1, X2, …, Xk are i.i.d. data from some distribution and h is a permutation-invariant function. Without privacy constraints, standard estimators are U-statistics, which commonly arise in a wide range of problems, including nonparametric signed rank tests, symmetry testing, uniformity testing, and subgraph counts in random networks, and can be shown to be minimum variance unbiased estimators under mild conditions. Despite the recent outpouring of interest in private mean estimation, privatizing U-statistics has received little attention. While existing private mean estimation algorithms can be applied to obtain confidence intervals, we show that they can lead to suboptimal private error, e.g., constant-factor inflation in the leading term, or even Θ(1/n) rather than O(1/n²) in degenerate settings. To remedy this, we propose a new thresholding-based approach using local Hájek projections to reweight different subsets of the data. This leads to nearly optimal private error for non-degenerate U-statistics and a strong indication of near-optimality for degenerate U-statistics. 

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4. Minimax Kernel Machine Learning for a Class of Doubly Robust Functionals with Application to Proximal Causal Inference

   Amiremad Ghassami, Andrew Ying (April 2022, Proceedings of The 25th International Conference on Artificial Intelligence and Statistics)

   Gustau Camps-Valls; Francisco J. R. Ruiz; Isabel Valera (Ed.)

   Robins et al. (2008) introduced a class of influence functions (IFs) which could be used to obtain doubly robust moment functions for the corresponding parameters. However, that class does not include the IF of parameters for which the nuisance functions are solutions to integral equations. Such parameters are particularly important in the field of causal inference, specifically in the recently proposed proximal causal inference framework of Tchetgen Tchetgen et al. (2020), which allows for estimating the causal effect in the presence of latent confounders. In this paper, we first extend the class of Robins et al. to include doubly robust IFs in which the nuisance functions are solutions to integral equations. Then we demonstrate that the double robustness property of these IFs can be leveraged to construct estimating equations for the nuisance functions, which enables us to solve the integral equations without resorting to parametric models. We frame the estimation of the nuisance functions as a minimax optimization problem. We provide convergence rates for the nuisance functions and conditions required for asymptotic linearity of the estimator of the parameter of interest. The experiment results demonstrate that our proposed methodology leads to robust and high-performance estimators for average causal effect in the proximal causal inference framework. 

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5. Differentially private estimation of U statistics

   Kamalika Chaudhuri, Po-Ling Loh (January 2024, Proceedings of Machine Learning Research)

   Under Review for COLT 2024 (Ed.)

   In this paper, we consider the problem of private estimation of U statistics. U statistics are widely used estimators that naturally arise in a broad class of problems, from nonparametric signed rank tests to subgraph counts in random networks. They are simply averages of an appropriate kernel applied to all subsets of a given size (also known as the degree) of sample size n. However, despite the recent outpouring of interest in private mean estimation, private algorithms for more general U statistics have received little attention. We propose a framework where, for a broad class of U statistics, one can use existing tools in private mean estimation to obtain confidence intervals where the private error does not overwhelm the irreducible error resulting from the variance of the U statistics. However, in specific cases that arise when the U statistics degenerate or have vanishing moments, the private error may be of a larger order than the non-private error. To remedy this, we propose a new thresholding-based approach that uses Hajek projections to re-weight different subsets. As we show, this leads to more accurate inference in certain settings. 

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