We study the mediation analysis under the distributed framework, where data are stored and processed across different worker machines due to storage limitations or privacy concerns. Building upon the classic Sobel's test and MaxP test, we introduce the distributed Sobel's test and distributed MaxP test, respectively. These tests are both communication-efficient and easy to implement. Theoretical analysis and numerical experiments show that, compared to the global test obtained by pooling all data together, the proposed tests achieve nearly identical power, independent of the number of machines. Furthermore, based on these two distributed test statistics, many enhanced mediation tests derived from the Sobel's or MaxP tests can be easily adapted to the distributed system. We apply our method to an educational study, testing whether the effect of high school mathematics on college-level Probability and Mathematical Statistics courses is mediated by Calculus. Our method successfully detects the mediation effect, which would not be identifiable using data from only the first or second class, highlighting the advantage of our approach.

Causal mediation analysis is crucial for deconstructing complex mechanisms of action. However, in current mediation analysis, complex structures derived from causal discovery lack direct interpretation of mediation pathways, while traditional mediation analysis and effect estimation are limited by the reliance on pre-specified pathways, leading to a disconnection between structure discovery and causal mechanism understanding. Therefore, a unified framework integrating structure discovery, pathway identification, and effect estimation systematically quantifies mediation pathways under structural uncertainty, enabling automated identification and inference of mediation pathways. To this end, we propose Structure-Informed Guided Mediation Analysis (SIGMA), which guides automated mediation pathway identification through probabilistic causal structure discovery and uncertainty quantification, enabling end-to-end propagation of structural uncertainty from structure learning to effect estimation. Specifically, SIGMA employs differentiable Flow-Structural Equation Models to learn structural posteriors, generating diverse Directed Acyclic Graphs (DAGs) to quantify structural uncertainty. Based on these DAGs, we introduce the Path Stability Score to evaluate the marginal probability of pathways, identifying high-confidence mediation paths. For identified mediation pathways, we integrate Efficient Influence Functions with Bayesian model averaging to fuse within-structure estimation uncertainty and between-structure effect variation, propagating uncertainty to the final effect estimates. In synthetic data experiments, SIGMA achieves state-of-the-art performance in pathway identification accuracy and effect quantification precision under structural uncertainty, concurrent multiple pathways, and nonlinear scenarios. In real-world applications using Human Phenotype Project data, SIGMA identifies mediation effects of sleep quality on cardiovascular health through inflammatory and metabolic pathways, uncovering previously unspecified multiple mediation paths.

In this work, we prove that agents capable of adapting to distribution shifts must have learned the causal model of their environment even in the presence of mediation. This term describes situations where an agent's actions affect its environment, a dynamic common to most real-world settings. For example, a robot in an industrial plant might interact with tools, move through space, and transform products to complete its task. We introduce an algorithm for eliciting causal knowledge from robust agents using optimal policy oracles, with the flexibility to incorporate prior causal knowledge. We further demonstrate its effectiveness in mediated singleagent scenarios and multi-agent environments. We identify conditions under which the presence of a single robust agent is sufficient to recover the full causal model and derive optimal policies for other agents in the same environment. Finally, we show how to apply these results to sequential decision-making tasks modeled as Partially Observable Markov Decision Processes (POMDPs).

Mediation analysis is essential for decomposing the causal effect of a treatment into direct and indirect pathways. However, many practical settings rely on the stringent assumption that recanting witnesses, defined as treatment-induced mediator-outcome confounders, are either absent or fully known a priori. Such a requirement is often untenable, especially when these variables remain unobservable due to measurement difficulties or privacy constraints. In this paper, we leverage proximal causal inference to develop three novel identification strategies to address the challenge of identifying path-specific effects in the presence of unknown recanting witnesses. Building on this, we develop a semiparametric inference framework that derives the efficient influence function and proposes a proximal multiply robust estimator, which remains consistent if at least one set of nuisance models is correctly specified. When all nuisance models are correctly specified and converge at appropriate rates, the estimator is asymptotically normal and achieves the semiparametric efficiency bound. We provide a minimax optimization-based debiased machine learning procedure for point estimation and constructing valid confidence intervals. The performance of the proposed methods is demonstrated by simulation studies and a real data application.

Mediation analysis has traditionally focused on outcome-level summary contrasts, such as mean effects, which may obscure substantial distributional changes induced by complex and nonlinear causal mechanisms. We propose Distributional Causal Mediation Analysis (DCMA), a generative learning framework for identifying and estimating treatment effects on entire outcome distributions transmitted through multiple mediators. DCMA learns conditional generative models for the mediators and the outcome, recovering the relevant conditional distributions from observational data. Leveraging the identification formulas, it reconstructs interventional outcome distributions via Monte Carlo forward simulation by noise resampling, enabling the capture of both classical summary effects and rich distributional contrasts such as energy distance and the Wasserstein distance. Analytical error bounds are derived to decompose how estimation errors in the learned conditional models propagate to the reconstructed interventional outcome distributions. The empirical effectiveness of DCMA is demonstrated through numerical experiments and real-world data applications.

High-dimensional mediation analysis is often associated with a multiple testing problem for detecting significant mediators. Assessing the uncertainty of this detecting process via false discovery rate (FDR) has garnered great interest. To control the FDR in multiple testing, two essential steps are involved: ranking and selection. Existing approaches either construct p-values without calibration or disregard the joint information across tests, leading to conservation in FDR control or non-optimal ranking rules for multiple hypotheses. In this paper, we develop an adaptive mediation detection procedure (referred to as "AMDP") to identify relevant mediators while asymptotically controlling the FDR in high-dimensional mediation analysis. AMDP produces the optimal rule for ranking hypotheses and proposes a data-driven strategy to determine the threshold for mediator selection. This novel method captures information from the proportions of composite null hypotheses and the distribution of p-values, which turns the high dimensionality into an advantage instead of a limitation. The numerical studies on synthetic and real data sets illustrate the performances of AMDP compared with existing approaches.

We study high-dimensional mediation analysis in which exposures, mediators, and outcomes are all multivariate, and both exposures and mediators may be high-dimensional. We formalize this as a many (exposures)-to-many (mediators)-to-many (outcomes) (MMM) mediation analysis problem. Methodologically, MMM mediation analysis simultaneously performs variable selection for high-dimensional exposures and mediators, estimates the indirect effect matrix (i.e., the coefficient matrices linking exposure-to-mediator and mediator-to-outcome pathways), and enables prediction of multivariate outcomes. Theoretically, we show that the estimated indirect effect matrices are consistent and element-wise asymptotically normal, and we derive error bounds for the estimators. To evaluate the efficacy of the MMM mediation framework, we first investigate its finite-sample performance, including convergence properties, the behavior of the asymptotic approximations, and robustness to noise, via simulation studies. We then apply MMM mediation analysis to data from the Alzheimer's Disease Neuroimaging Initiative to study how cortical thickness of 202 brain regions may mediate the effects of 688 genome-wide significant single nucleotide polymorphisms (SNPs) (selected from approximately 1.5 million SNPs) on eleven cognitive-behavioral and diagnostic outcomes. The MMM mediation framework identifies biologically interpretable, many-to-many-to-many genetic-neural-cognitive pathways and improves downstream out-of-sample classification and prediction performance. Taken together, our results demonstrate the potential of MMM mediation analysis and highlight the value of statistical methodology for investigating complex, high-dimensional multi-layer pathways in science. The MMM package is available at https://github.com/THELabTop/MMM-Mediation.

High-dimensional mediation analysis is often associated with a multiple testing problem for detecting significant mediators. Assessing the uncertainty of this detecting process via false discovery rate (FDR) has garnered great interest. To control the FDR in multiple testing, two essential steps are involved: ranking and selection. Existing approaches either construct p-values without calibration or disregard the joint information across tests, leading to conservation in FDR control or non-optimal ranking rules for multiple hypotheses. In this paper, we develop an adaptive mediation detection procedure (referred to as AMDP) to identify relevant mediators while asymptotically controlling the FDR in high-dimensional mediation analysis. AMDP produces the optimal rule for ranking hypotheses and proposes a data-driven strategy to determine the threshold for mediator selection. This novel method captures information from the proportions of composite null hypotheses and the distribution of p-values, which turns the high dimensionality into an advantage instead of a limitation. The numerical studies on synthetic and real data sets illustrate the performances of AMDP compared with existing approaches.

Deciding on an appropriate intervention requires a causal model of a treatment, the outcome, and potential mediators. Causal mediation analysis lets us distinguish between direct and indirect effects of the intervention, but has mostly been studied in a static setting. In healthcare, data come in the form of complex, irregularly sampled time-series, with dynamic interdependencies between a treatment, outcomes, and mediators across time. Existing approaches to dynamic causal mediation analysis are limited to regular measurement intervals, simple parametric models, and disregard long-range mediator--outcome interactions. To address these limitations, we propose a non-parametric mediator--outcome model where the mediator is assumed to be a temporal point process that interacts with the outcome process. With this model, we estimate the direct and indirect effects of an external intervention on the outcome, showing how each of these affects the whole future trajectory. We demonstrate on semi-synthetic data that our method can accurately estimate direct and indirect effects.