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1. Combining Non-Probability and Probability Survey Samples Through Mass Imputation

   https://doi.org/10.1111/rssa.12696  

   Kim, Jae Kwang; Park, Seho; Chen, Yilin; Wu, Changbao (July 2021, Journal of the Royal Statistical Society Series A: Statistics in Society)

   Abstract Analysis of non-probability survey samples requires auxiliary information at the population level. Such information may also be obtained from an existing probability survey sample from the same finite population. Mass imputation has been used in practice for combining non-probability and probability survey samples and making inferences on the parameters of interest using the information collected only in the non-probability sample for the study variables. Under the assumption that the conditional mean function from the non-probability sample can be transported to the probability sample, we establish the consistency of the mass imputation estimator and derive its asymptotic variance formula. Variance estimators are developed using either linearization or bootstrap. Finite sample performances of the mass imputation estimator are investigated through simulation studies. We also address important practical issues of the method through the analysis of a real-world non-probability survey sample collected by the Pew Research Centre. 

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2. Trace abstraction modulo probability

   https://doi.org/10.1145/3290352  

   Smith, Calvin; Hsu, Justin; Albarghouthi, Aws (January 2019, Proceedings of the ACM on Programming Languages)

   We propose trace abstraction modulo probability, a proof technique for verifying high-probability accuracy guarantees of probabilistic programs. Our proofs overapproximate the set of program traces using failure automata, finite-state automata that upper bound the probability of failing to satisfy a target specification. We automate proof construction by reducing probabilistic reasoning to logical reasoning: we use program synthesis methods to select axioms for sampling instructions, and then apply Craig interpolation to prove that traces fail the target specification with only a small probability. Our method handles programs with unknown inputs, parameterized distributions, infinite state spaces, and parameterized specifications. We evaluate our technique on a range of randomized algorithms drawn from the differential privacy literature and beyond. To our knowledge, our approach is the first to automatically establish accuracy properties of these algorithms. 

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3. Kinetic equations for two-photon light in random media

   https://doi.org/10.1063/5.0106535  

   Kraisler, Joseph; Schotland, John C. (November 2023, Journal of Mathematical Physics)

   We consider the propagation of light in a random medium of two-level atoms. We investigate the dynamics of the field and atomic probability amplitudes for a two-photon state and show that at long times and large distances, the corresponding average probability densities can be determined from the solutions to a system of kinetic equations. 

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4. Identifying and bounding the probability of necessity for causes of effects with ordinal outcomes

   https://doi.org/10.1093/biomet/asaf049  

   Zhang, Chao; Geng, Zhi; Li, Wei; Ding, Peng (July 2025, Biometrika)

   Abstract Although the existing causal inference literature focuses on the forward-looking perspective by estimating effects of causes, the backward-looking perspective can provide insights into causes of effects. In backward-looking causal inference, the probability of necessity measures the probability that a certain event is caused by the treatment given the observed treatment and outcome. Most existing results focus on binary outcomes. Motivated by applications with ordinal outcomes, we propose a general definition of the probability of necessity. However, identifying the probability of necessity is challenging because it involves the joint distribution of the potential outcomes. We propose a novel assumption of monotonic incremental treatment effect to identify the probability of necessity with ordinal outcomes. We also discuss the testable implications of this key identification assumption. When it fails, we derive explicit formulas of the sharp large-sample bounds on the probability of necessity. 

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5. Gradient descent algorithms for Bures-Wasserstein barycenters

   Chewi, Sinh; Maunu, Tyler; Rigollet, Philipp; Stromme, Austin J. (January 2020, Proceedings of Machine Learning Research)

   Abernethy, Jacob; Agarwal, Shivani (Ed.)

   We study first order methods to compute the barycenter of a probability distribution $$P$$ over the space of probability measures with finite second moment. We develop a framework to derive global rates of convergence for both gradient descent and stochastic gradient descent despite the fact that the barycenter functional is not geodesically convex. Our analysis overcomes this technical hurdle by employing a Polyak-Ł{}ojasiewicz (PL) inequality and relies on tools from optimal transport and metric geometry. In turn, we establish a PL inequality when $$P$$ is supported on the Bures-Wasserstein manifold of Gaussian probability measures. It leads to the first global rates of convergence for first order methods in this context. 

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