Predicting counterfactual outcomes in longitudinal data, where sequential treatment decisions heavily depend on evolving patient states, is critical yet notoriously challenging due to complex time-dependent confounding and inadequate uncertainty quantification in existing methods. We introduce the Causal Diffusion Model (CDM), the first denoising diffusion probabilistic approach explicitly designed to generate full probabilistic distributions of counterfactual outcomes under sequential interventions. CDM employs a novel residual denoising architecture with relational self-attention, capturing intricate temporal dependencies and multimodal outcome trajectories without requiring explicit adjustments (e.g., inverse-probability weighting or adversarial balancing) for confounding. In rigorous evaluation on a pharmacokinetic-pharmacodynamic tumor-growth simulator widely adopted in prior work, CDM consistently outperforms state-of-the-art longitudinal causal inference methods, achieving a 15-30% relative improvement in distributional accuracy (1-Wasserstein distance) while maintaining competitive or superior point-estimate accuracy (RMSE) under high-confounding regimes. By unifying uncertainty quantification and robust counterfactual prediction in complex, sequentially confounded settings, without tailored deconfounding, CDM offers a flexible, high-impact tool for decision support in medicine, policy evaluation, and other longitudinal domains.

Retrospective causal questions ask what would have happened to an observed individual had they received a different treatment. We study the problem of estimating $μ(x,y)=\mathbb{E}[Y(1)\mid X=x,Y(0)=y]$, the expected counterfactual outcome for an individual with covariates $x$ and observed outcome $y$, and constructing valid prediction intervals under the Neyman-Rubin superpopulation model. This quantity is generally not identified without additional assumptions. To link the observed and unobserved potential outcomes, we work with a cross-world correlation $ρ(x)=cor(Y(1),Y(0)\mid X=x)$; plausible bounds on $ρ(x)$ enable a principled approach to this otherwise unidentified problem. We introduce retrospective counterfactual estimators $\hatμ_ρ(x,y)$ and prediction intervals $C_ρ(x,y)$ that asymptotically satisfy $P[Y(1)\in C_ρ(x,y)\mid X=x, Y(0)=y]\ge1-α$ under standard causal assumptions. Many common baselines implicitly correspond to endpoint choices $ρ=0$ or $ρ=1$ (ignoring the factual outcome or treating the counterfactual as a shifted factual outcome). Interpolating between these cases through cross-world dependence yields substantial gains in both theory and practice.

Kernel methods are widely used in causal inference for tasks such as treatment effect estimation, policy evaluation, and policy learning. The bootstrap is a standard tool for uncertainty quantification because of its broad applicability. As increasingly large datasets become available, such as the 2023 U.S. Natality data from the National Vital Statistics System (NVSS), which includes 3,596,017 registered births, the computational demands of these methods increase substantially. Kernel methods are known to scale poorly with sample size, and this limitation is further exacerbated by the repeated re-fitting required by the bootstrap. As a result, bootstrap-based inference for kernel-based estimators can become computationally infeasible in large-scale settings. In this paper, we address these challenges by extending the causal Bag of Little Bootstraps (cBLB) algorithm to kernel methods. Our approach achieves computational scalability by combining subsampling and resampling while preserving first-order uncertainty quantification and asymptotically correct coverage. We evaluate the method across three representative implementations: kernelized augmented outcome-weighted learning, kernel-based minimax weighting, and double machine learning with kernel support vector machines. We show in simulations that our method yields confidence intervals with nominal coverage at a fraction of the computational cost. We further demonstrate its utility in a real-world application by estimating the effect of any amount of smoking on birth weight, as well as the optimal treatment regime, using the NVSS dataset, where the standard bootstrap is prohibitively expensive computationally and effectively infeasible at this scale.

The assumption that data samples are independent and identically distributed (iid) is standard in many areas of statistics and machine learning. Nevertheless, in some settings, such as social networks, infectious disease modeling, and reasoning with spatial and temporal data, this assumption is false. An extensive literature exists on making causal inferences under the iid assumption [12, 8, 21, 16], but, as pointed out in [14], causal inference in non-iid contexts is challenging due to the combination of unobserved confounding bias and data dependence. In this paper we develop a general theory describing when causal inferences are possible in such scenarios. We use segregated graphs [15], a generalization of latent projection mixed graphs [23], to represent causal models of this type and provide a complete algorithm for non-parametric identification in these models. We then demonstrate how statistical inferences may be performed on causal parameters identified by this algorithm, even in cases where parts of the model exhibit full interference, meaning only a single sample is available for parts of the model [19]. We apply these techniques to a synthetic data set which considers the adoption of fake news articles given the social network structure, articles read by each person, and baseline demographics and socioeconomic covariates.

Manyimportant questions posed in the natural and social sciences are causal in nature:What are the long-term effects of mild Covid-19 infection on lung and brain functions?

Second, they lack a specific focus on selecting a robust CA TE estimator.

Accurately predicting conditional average treatment effects (CA TEs) is crucial in personalized medicine and digital platform analytics.

RCM. Finally, we illustrate the power of this conciliatory perspective by pinpointing