ICLR 2026 Papers

One of the major challenges in causal inference with observational data is handling missing confounders. Latent variable modeling offers a valid framework to address this challenge, but existing approaches within this framework often suffer from consistency issues in causal effect estimation and are difficult to extend to more complex application scenarios. To bridge this gap, we propose a new latent variable modeling approach, Confounder Imputation with Stochastic Neural Networks (CI-StoNet). The CI-StoNet utilizes a stochastic neural network to jointly model the outcome function and the missing confounders, and employs an adaptive stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm to impute the missing confounders and train the neural networks simultaneously. Under mild conditions, we show that the causal effect remains identifiable through CI-StoNet, even though the missing confounders are non-identifiable -- these confounders can only be identified up to an unknown loss-invariant transformation due to the non-identifiability inherent in neural network models. The CI-StoNet provides state-of-the-art performance on benchmarks for causal effect estimation and showcases its adaptability to proxy variable and multiple-cause scenarios. This new approach also serves as a versatile tool for modeling various causal relationships, leveraging the flexibility of stochastic neural networks in natural process modeling.

Small influential data subsets can dramatically impact model conclusions, with a few data points overturning key findings. While recent work identifies these most influential sets, there is no formal way to tell when maximum influence is excessive rather than expected under natural random sampling variation. We address this gap by developing a principled framework for most influential sets. Focusing on linear least-squares, we derive a convenient exact influence formula and identify the extreme value distributions of maximal influence – the heavy-tailed Fréchet for constant‑size sets and heavy tailed data, and the well-behaved Gumbel for growing sets or light tails. This allows us to conduct rigorous hypothesis tests for excessive influence. We demonstrate through applications across economics, biology, and machine learning benchmarks, resolving contested findings and replacing ad‑hoc heuristics with rigorous inference.