Mediation analysis (Robins & Greenland 1992, Pearl 2001, Imai, Keele & Tingley 2010, Tchetgen Tchetgen & Shpitser 2012) provides a principled framework for investigating causal mechanisms by decomposing the effect of a treatment A on an outcome Y into pathways operating through a mediator of interest M. Classical mediation analysis focuses on the natural indirect effect, corresponding to the pathway from Ato Y through M, and the natural direct effect, corresponding to pathways not through M. These estimands are well understood when a single mediator is present and strong identification assumptions hold. However, in many applications, there exist multiple intermediate variables between treatment and outcome. In such settings, conventional mediation analysis typically requires the absence of treatment-induced mediator-outcome confounders--often referred to as recanting witnesses--as well as the absence of unmeasured confounding. Under these circumstances, commonly used identification assumptions such as sequential ignorability (Imai, Keele & Yamamoto 2010) or nonparametric structural equation models with independent errors (NPSEM-IE) (Pearl 2009) no longer suffice to identify natural indirect effects (Avin et al. 2005, Tchetgen Tchetgen & VanderWeele 2014). Figure 1 illustrates this issue: the recanting witness D is directly affected by A and simultaneously confounds the relationship between M and Y. Such treatment-induced confounding is common in epidemiologic studies, particularly when the mediator of interest occurs long after the treatment initiation (Robins 1999). A motivating example arises in studies of preterm birth. Mediation analysis has been widely used to explore whether adequate prenatal care (A) reduces the risk of preterm birth (Y) through preeclampsia (M) (Vansteelandt & VanderWeele 2012, VanderWeele et al. 2014, Xia & Chan 2023).

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 introduce a new approach for computing optimal equilibria and mechanisms via learning in games. It applies to extensive-form settings with any number of players, including mechanism design, information design, and solution concepts such as correlated, communication, and certification equilibria. We observe that optimal equilibria are minimax equilibrium strategies of a player in an extensiveform zero-sum game. This reformulation allows us to apply techniques for learning in zero-sum games, yielding the first learning dynamics that converge to optimal equilibria, not only in empirical averages, but also in iterates. We demonstrate the practical scalability and flexibility of our approach by attaining state-of-the-art performance in benchmark tabular games, and by computing an optimal mechanism for a sequential auction design problem using deep reinforcement learning.

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.

Causal mediation analysis provides techniques for defining and estimating effects that may be endowed with mechanistic interpretations. With many scientific investigations seeking to address mechanistic questions, causal direct and indirect effects have garnered much attention. The natural direct and indirect effects, the most widely used among such causal mediation estimands, are limited in their practical utility due to stringent identification requirements. Accordingly, considerable effort has been invested in developing alternative direct and indirect effect decompositions with relaxed identification requirements. Such efforts often yield effect definitions with nuanced and challenging interpretations. By contrast, relatively limited attention has been paid to relaxing the identification assumptions of the natural direct and indirect effects. Motivated by a secondary aim of a recent non-randomized vaccine prospective cohort study (NCT05168813), we present a set of relaxed conditions under which the natural direct effect is identifiable in spite of unobserved baseline confounding of the exposure-mediator pathway; we use this result to investigate the effect mediated by putative immune correlates of protection. Relaxing the commonly used but restrictive cross-world counterfactual independence assumption, we discuss strategies for evaluating the natural direct effect in non-randomized settings that arise in the analysis of vaccine studies. We revisit prior studies of semi-parametric efficiency theory to demonstrate the construction of flexible, multiply robust estimators of the natural direct effect and discuss efficient estimation strategies that do not place restrictive modeling assumptions on nuisance functions.

In this paper, we develop efficient parameterized algorithms for regimes between these two extremes.