The following are equivalent: 1. X X as m, and X is a constant a.s., then X A, where A is a constant. The result follows by taking ϵ 0. 4. We adapt the proof of Fatou's lemma that holds for random variables that converge in E[f ( g (X))].

Counterfactual Explanations (CFEs) interpret machine learning models by identifying the smallest change to input features needed to change the model's prediction to a desired output. For classification tasks, CFEs determine how close a given sample is to the decision boundary of a trained classifier. Existing methods are often sample-inefficient, requiring numerous evaluations of a black-box model -- an approach that is both costly and impractical when access to the model is limited. We propose Adaptive sampling for Counterfactual Explanations (ACE), a sample-efficient algorithm combining Bayesian estimation and stochastic optimization to approximate the decision boundary with fewer queries. By prioritizing informative points, ACE minimizes evaluations while generating accurate and feasible CFEs. Extensive empirical results show that ACE achieves superior evaluation efficiency compared to state-of-the-art methods, while maintaining effectiveness in identifying minimal and actionable changes.

Predicting individualized potential outcomes in sequential decision-making is central for optimizing therapeutic decisions in personalized medicine (e.g., which dosing sequence to give to a cancer patient). However, predicting potential outcomes over long horizons is notoriously difficult. Existing methods that break the curse of the horizon typically lack strong theoretical guarantees such as orthogonality and quasi-oracle efficiency. In this paper, we revisit the problem of predicting individualized potential outcomes in sequential decision-making (i.e., estimating Q-functions in Markov decision processes with observational data) through a causal inference lens. In particular, we develop a comprehensive theoretical foundation for meta-learners in this setting with a focus on beneficial theoretical properties. As a result, we yield a novel meta-learner called DRQ-learner and establish that it is: (1) doubly robust (i.e., valid inference under the misspecification of one of the nuisances), (2) Neyman-orthogonal (i.e., insensitive to first-order estimation errors in the nuisance functions), and (3) achieves quasi-oracle efficiency (i.e., behaves asymptotically as if the ground-truth nuisance functions were known). Our DRQ-learner is applicable to settings with both discrete and continuous state spaces. Further, our DRQ-learner is flexible and can be used together with arbitrary machine learning models (e.g., neural networks). We validate our theoretical results through numerical experiments, thereby showing that our meta-learner outperforms state-of-the-art baselines.

Average and conditional treatment effects are fundamental causal quantities used to evaluate the effectiveness of treatments in various critical applications, including clinical settings and policy-making. Beyond the gold-standard estimators from randomized trials, numerous methods have been proposed to estimate treatment effects using observational data. In this paper, we provide a novel characterization of widely used causal inference techniques within the framework of staged event trees, demonstrating their capacity to enhance treatment effect estimation. These models offer a distinct advantage due to their interpretability, making them particularly valuable for practical applications. We implement classical estimators within the framework of staged event trees and illustrate their capabilities through both simulation studies and real-world applications. Furthermore, we showcase how staged event trees explicitly and visually describe when standard causal assumptions, such as positivity, hold, further enhancing their practical utility.

The Metropolis-within-Gibbs (MwG) algorithm is a widely used Markov Chain Monte Carlo method for sampling from high-dimensional distributions when exact conditional sampling is intractable. We study MwG with Random Walk Metropolis (RWM) updates, using proposal variances tuned to match the target's conditional variances. Assuming the target $π$ is a $d$-dimensional log-concave distribution with condition number $κ$, we establish a spectral gap lower bound of order $\mathcal{O}(1/κd)$ for the random-scan version of MwG, improving on the previously available $\mathcal{O}(1/κ^2 d)$ bound. This is obtained by developing sharp estimates of the conductance of one-dimensional RWM kernels, which can be of independent interest. The result shows that MwG can mix substantially faster with variance-adaptive proposals and that its mixing performance is just a constant factor worse than that of the exact Gibbs sampler, thus providing theoretical support to previously observed empirical behavior.

We survey the field of nonparametric inference under shape constraints, providing a historical overview and a perspective on its current state. An outlook and some open problems offer thoughts on future directions. 1 Introduction. Traditionally, we think of statistical methods as being divided into parametric approaches, which can be restrictive, but where estimation is typically straightforward (e.g. using maximum likelihood), and nonparametric methods, which are more flexible but often require careful choices of tuning parameters. Nonparametric inference under shape constraints sits somewhere in the middle, seeking in some ways the best of both worlds. The origins of the field are often traced to Grenander (1956), who proved that there exists a unique maximum likelihood estimator (MLE) of a decreasing density on the non-negative half-line (and was able to characterise it explicitly).

We introduce a formal active learning methodology for guiding the placement of Lagrangian observers to infer time-dependent vector fields -- a key task in oceanography, marine science, and ocean engineering -- using a physics-informed spatio-temporal Gaussian process surrogate model. The majority of existing placement campaigns either follow standard `space-filling' designs or relatively ad-hoc expert opinions. A key challenge to applying principled active learning in this setting is that Lagrangian observers are continuously advected through the vector field, so they make measurements at different locations and times. It is, therefore, important to consider the likely future trajectories of placed observers to account for the utility of candidate placement locations. To this end, we present BALLAST: Bayesian Active Learning with Look-ahead Amendment for Sea-drifter Trajectories. We observe noticeable benefits of BALLAST-aided sequential observer placement strategies on both synthetic and high-fidelity ocean current models.

Reinforcement learning in partially observable environments requires agents to act under uncertainty from noisy, incomplete observations. Asymmetric actor-critic methods leverage privileged information during training to improve learning under these conditions. However, existing approaches typically assume full-state access during training. In this work, we challenge this assumption by proposing a novel actor-critic framework, called informed asymmetric actor-critic, that enables conditioning the critic on arbitrary privileged signals without requiring access to the full state. We show that policy gradients remain unbiased under this formulation, extending the theoretical foundation of asymmetric methods to the more general case of privileged partial information. To quantify the impact of such signals, we propose informativeness measures based on kernel methods and return prediction error, providing practical tools for evaluating training-time signals. We validate our approach empirically on benchmark navigation tasks and synthetic partially observable environments, showing that our informed asymmetric method improves learning efficiency and value estimation when informative privileged inputs are available. Our findings challenge the necessity of full-state access and open new directions for designing asymmetric reinforcement learning methods that are both practical and theoretically sound.

Quantifying uncertainty of machine learning model predictions is essential for reliable decision-making, especially in safety-critical applications. Recently, uncertainty quantification (UQ) theory has advanced significantly, building on a firm basis of learning with proper scoring rules. However, these advances were focused on classification, while extending these ideas to regression remains challenging. In this work, we introduce a unified UQ framework for regression based on proper scoring rules, such as CRPS, logarithmic, squared error, and quadratic scores. We derive closed-form expressions for the resulting uncertainty measures under practical parametric assumptions and show how to estimate them using ensembles of models. In particular, the derived uncertainty measures naturally decompose into aleatoric and epistemic components. The framework recovers popular regression UQ measures based on predictive variance and differential entropy. Our broad evaluation on synthetic and real-world regression datasets provides guidance for selecting reliable UQ measures.