Identification and Bounding of Central Moments of Causal Effects Using Marginal Moments Information

Evaluating the causal effect of a treatment on an outcome is a central objective in causal inference. While the average causal effect summarizes the mean impact of treatment, the central moments of the individual causal effect (ICE) characterize the shape of the ICE distribution, thereby revealing the extent and structure of treatment effect heterogeneity across individuals. This paper investigates the identification and bounding of the central moments of the ICE using only the marginal central moments of each potential outcome (PO). Compared with existing approaches that require knowledge of the full marginal distributions of the POs, marginal moment information is often substantially easier to obtain in empirical applications. Finally, we illustrate the practical relevance of our results through two empirical case studies.

Fixed-Confidence Best-Arm Identification for Causal Mediation Analysis

This paper studies the problem of identifying the treatment that maximizes the expected natural direct potential outcome (NDPO), which captures the potential outcome of an intervention while excluding the pathway transmitted through a mediator that researchers may wish to remove from evaluation. We first establish population-level identification of the expected NDPO in a causal bandit setting using observable interventional distributions. We then develop a fixed-confidence best-arm identification (BAI) algorithm based on the Track-and-Stop (TaS) framework, employing a cutting-set method to solve the resulting semi-infinite optimization problem. The proposed algorithm achieves sample-efficient identification with a high-probability correctness guarantee. We prove that it satisfies $δ$-correctness and asymptotic optimality. Finally, we validate the approach through empirical evaluations on a large-scale real-world advertising dataset (IPinYou).

Nonlinear Data Integration via Kernel Methods for Data Collaboration Analysis

Collaborative analysis of decentralized confidential datasets is important, but direct sharing of original datasets is often restricted by privacy and institutional constraints. Data collaboration (DC) analysis transforms each dataset into privacy-preserving intermediate representations via party-specific obfuscation functions and integrates them into common collaboration representations using an anchor dataset. However, many existing DC analysis methods rely on linear transformations for data obfuscation and integration, which may increase reconstruction risk. Although nonlinear dimensionality reduction can mitigate this risk, conventional linear integration methods cannot accurately align intermediate representations produced by nonlinear transformations. Moreover, existing integration methods mainly minimize discrepancies among parties and do not explicitly incorporate geometric or target-variable information useful for downstream analysis. To overcome these limitations, we first formulate linear kernel integration (LKI) as a linear integration method and then kernelize it to obtain nonlinear kernel integration (NKI). NKI admits a globally optimal solution via kernel ridge regression and an eigenvalue problem. We also introduce graph regularization and a centering constraint so that the target representation can capture geometric and target-variable information useful for downstream analysis. Experiments on image classification tasks demonstrate that NKI improves classification accuracy over existing linear integration methods under nonlinear dimensionality reduction, with further gains from target-variable-aware graph regularization and centering. The results also show that dimensionality reduction choices substantially affect both classification accuracy and reconstruction risk.

Potential Outcome Rankings for Counterfactual Decision Making

Counterfactual decision-making in the face of uncertainty involves selecting the optimal action from several alternatives using causal reasoning. Decision-makers often rank expected potential outcomes (or their corresponding utility and desirability) to compare the preferences of candidate actions. In this paper, we study new counterfactual decision-making rules by introducing two new metrics: the probabilities of potential outcome ranking (PoR) and the probability of achieving the best potential outcome (PoB). PoR reveals the most probable ranking of potential outcomes for an individual, and PoB indicates the action most likely to yield the top-ranked outcome for an individual. We then establish identification theorems and derive bounds for these metrics, and present estimation methods. Finally, we perform numerical experiments to illustrate the finite-sample properties of the estimators and demonstrate their application to a real-world dataset.

Decomposition of Probabilities of Causation with Two Mediators

Mediation analysis for probabilities of causation (PoC) provides a fundamental framework for evaluating the necessity and sufficiency of treatment in provoking an event through different causal pathways. One of the primary objectives of causal mediation analysis is to decompose the total effect into path-specific components. In this study, we investigate the path-specific probability of necessity and sufficiency (PNS) to decompose the total PNS into path-specific components along distinct causal pathways between treatment and outcome, incorporating two mediators. We define the path-specific PNS for decomposition and provide an identification theorem. Furthermore, we conduct numerical experiments to assess the properties of the proposed estimators from finite samples and demonstrate their practical application using a real-world educational dataset.

Moments of Causal Effects

The moments of random variables are fundamental statistical measures for characterizing the shape of a probability distribution, encompassing metrics such as mean, variance, skewness, and kurtosis. Additionally, the product moments, including covariance and correlation, reveal the relationships between multiple random variables. On the other hand, the primary focus of causal inference is the evaluation of causal effects, which are defined as the difference between two potential outcomes. While traditional causal effect assessment focuses on the average causal effect, this work provides definitions, identification theorems, and bounds for moments and product moments of causal effects to analyze their distribution and relationships. We conduct experiments to illustrate the estimation of the moments of causal effects from finite samples and demonstrate their practical application using a real-world medical dataset.

Mediation Analysis for Probabilities of Causation

Probabilities of causation (PoC) offer valuable insights for informed decision-making. This paper introduces novel variants of PoC-controlled direct, natural direct, and natural indirect probability of necessity and sufficiency (PNS). These metrics quantify the necessity and sufficiency of a treatment for producing an outcome, accounting for different causal pathways. We develop identification theorems for these new PoC measures, allowing for their estimation from observational data. We demonstrate the practical application of our results through an analysis of a real-world psychology dataset.

Probabilities of Causation for Continuous and Vector Variables

Probabilities of causation (PoC) are valuable concepts for explainable artificial intelligence and practical decision-making. PoC are originally defined for scalar binary variables. In this paper, we extend the concept of PoC to continuous treatment and outcome variables, and further generalize PoC to capture causal effects between multiple treatments and multiple outcomes. In addition, we consider PoC for a sub-population and PoC with multi-hypothetical terms to capture more sophisticated counterfactual information useful for decision-making. We provide a nonparametric identification theorem for each type of PoC we introduce. Finally, we illustrate the application of our results on a real-world dataset about education.

New Solutions Based on the Generalized Eigenvalue Problem for the Data Collaboration Analysis

In recent years, the accumulation of data across various institutions has garnered attention for the technology of confidential data analysis, which improves analytical accuracy by sharing data between multiple institutions while protecting sensitive information. Among these methods, Data Collaboration Analysis (DCA) is noted for its efficiency in terms of computational cost and communication load, facilitating data sharing and analysis across different institutions while safeguarding confidential information. However, existing optimization problems for determining the necessary collaborative functions have faced challenges, such as the optimal solution for the collaborative representation often being a zero matrix and the difficulty in understanding the process of deriving solutions. This research addresses these issues by formulating the optimization problem through the segmentation of matrices into column vectors and proposing a solution method based on the generalized eigenvalue problem. Additionally, we demonstrate methods for constructing collaborative functions more effectively through weighting and the selection of efficient algorithms suited to specific situations. Experiments using real-world datasets have shown that our proposed formulation and solution for the collaborative function optimization problem achieve superior predictive accuracy compared to existing methods.

Identification and Estimation of Conditional Average Partial Causal Effects via Instrumental Variable

There has been considerable recent interest in estimating heterogeneous causal effects. In this paper, we introduce conditional average partial causal effects (CAPCE) to reveal the heterogeneity of causal effects with continuous treatment. We provide conditions for identifying CAPCE in an instrumental variable setting. We develop three families of CAPCE estimators: sieve, parametric, and reproducing kernel Hilbert space (RKHS)-based, and analyze their statistical properties. We illustrate the proposed CAPCE estimators on synthetic and real-world data.