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Cluster randomized trials (CRTs) with multiple unstructured mediators present significant methodological challenges for causal inference due to within-cluster correlation, interference among units, and the complexity introduced by multiple mediators. Existing causal mediation methods often fall short in simultaneously addressing these complexities, particularly in disentangling mediator-specific effects under interference that are central to studying complex mechanisms. To address this gap, we propose new causal estimands for spillover mediation effects that differentiate the roles of each individual's own mediator and the spillover effects resulting from interactions among individuals within the same cluster. We establish identification results for each estimand and, to flexibly model the complex data structures inherent in CRTs, we develop a new Bayesian nonparametric prior -- the Nested Dependent Dirichlet Process Mixture -- designed for flexibly capture the outcome and mediator surfaces at different levels. We conduct extensive simulations across various scenarios to evaluate the frequentist performance of our methods, compare them with a Bayesian parametric counterpart and illustrate our new methods in an analysis of a completed CRT., Comment: 71 pages

Differential privacy is the leading mathematical framework for privacy protection, providing a probabilistic guarantee that safeguards individuals' private information when publishing statistics from a dataset. This guarantee is achieved by applying a randomized algorithm to the original data, which introduces unique challenges in data analysis by distorting inherent patterns. In particular, causal inference using observational data in privacy-sensitive contexts is challenging because it requires covariate balance between treatment groups, yet checking the true covariates is prohibited to prevent leakage of sensitive information. In this article, we present a differentially private two-stage covariate balancing weighting estimator to infer causal effects from observational data. Our algorithm produces both point and interval estimators with statistical guarantees, such as consistency and rate optimality, under a given privacy budget., Comment: 30 pages

Local differential privacy is a differential privacy paradigm in which individuals first apply a privacy mechanism to their data (often by adding noise) before transmitting the result to a curator. The noise for privacy results in additional bias and variance in their analyses. Thus it is of great importance for analysts to incorporate the privacy noise into valid inference. In this article, we develop methodologies to infer causal effects from locally privatized data under randomized experiments. First, we present frequentist estimators under various privacy scenarios with their variance estimators and plug-in confidence intervals. We show a na\"ive debiased estimator results in inferior mean-squared error (MSE) compared to minimax lower bounds. In contrast, we show that using a customized privacy mechanism, we can match the lower bound, giving minimax optimal inference. We also develop a Bayesian nonparametric methodology along with a blocked Gibbs sampling algorithm, which can be applied to any of our proposed privacy mechanisms, and which performs especially well in terms of MSE for tight privacy budgets. Finally, we present simulation studies to evaluate the performance of our proposed frequentist and Bayesian methodologies for various privacy budgets, resulting in useful suggestions for performing causal inference for privatized data., Comment: 36 pages

One core assumption typically adopted for valid causal inference is that of no interference between experimental units, i.e., the outcome of an experimental unit is unaffected by the treatments assigned to other experimental units. This assumption can be violated in real-life experiments, which significantly complicates the task of causal inference as one must disentangle direct treatment effects from ``spillover'' effects. Current methodologies are lacking, as they cannot handle arbitrary, unknown interference structures to permit inference on causal estimands. We present a general framework to address the limitations of existing approaches. Our framework is based on the new concept of the ``degree of interference'' (DoI). The DoI is a unit-level latent variable that captures the latent structure of interference. We also develop a data augmentation algorithm that adopts a blocked Gibbs sampler and Bayesian nonparametric methodology to perform inferences on the estimands under our framework. We illustrate the DoI concept and properties of our Bayesian methodology via extensive simulation studies and an analysis of a randomized experiment investigating the impact of a cash transfer program for which interference is a critical concern. Ultimately, our framework enables us to infer causal effects without strong structural assumptions on interference.

Three critical issues for causal inference that often occur in modern, complicated experiments are interference, treatment nonadherence, and missing outcomes. A great deal of research efforts has been dedicated to developing causal inferential methodologies that address these issues separately. However, methodologies that can address these issues simultaneously are lacking. We propose a Bayesian causal inference methodology to address this gap. Our methodology extends existing causal frameworks and methods, specifically, two-staged randomized experiments and the principal stratification framework. In contrast to existing methods that invoke strong structural assumptions to identify principal causal effects, our Bayesian approach uses flexible distributional models that can accommodate the complexities of interference and missing outcomes, and that ensure that principal causal effects are weakly identifiable. We illustrate our methodology via simulation studies and a re-analysis of real-life data from an evaluation of India's National Health Insurance Program. Our methodology enables us to identify new active causal effects that were not identified in past analyses. Ultimately, our simulation studies and case study demonstrate how our methodology can yield more informative analyses in modern experiments with interference, treatment nonadherence, missing outcomes, and complicated outcome generation mechanisms., Comment: 42 pages

Estimating the duration of user behavior is a central concern for most internet companies. Survival analysis is a promising method for analyzing the expected duration of events and usually assumes the same survival function for all subjects and the event will occur in the long run. However, such assumptions are inappropriate when the users behave differently or some events never occur for some users, i.e., the conversion period on web services of the light users with no intention of behaving actively on the service. Especially, if the proportion of inactive users is high, this assumption can lead to undesirable results. To address these challenges, this paper proposes a mixture model that separately addresses active and inactive individuals with a latent variable. First, we define this specific problem setting and show the limitations of conventional survival analysis in addressing this problem. We demonstrate how naturally our Bernoulli-Weibull model can accommodate the challenge. The proposed model was extended further to a Bayesian hierarchical model to incorporate each subject's parameter, offering substantial improvements over conventional, non-hierarchical models in terms of WAIC and WBIC. Second, an experiment and extensive analysis were conducted using real-world data from the Japanese job search website, CareerTrek, offered by BizReach, Inc. In the analysis, some research questions are raised, such as the difference in activation rate and conversion rate between user categories, and how instantaneously the rate of event occurrence changes as time passes. Quantitative answers and interpretations are assigned to them. Furthermore, the model is inferred in a Bayesian manner, which enables us to represent the uncertainty with a credible interval of the parameters and predictive quantities., Comment: 9 pages

Job interviews are a fundamental activity for most corporations to acquire potential candidates, and for job seekers to get well-rewarded and fulfilling career opportunities. In many cases, interviews are conducted in multiple processes such as telephone interviews and several face-to-face interviews. At each stage, candidates are evaluated in various aspects. Among them, grade evaluation, such as a rating on a 1-4 scale, might be used as a reasonable method to evaluate candidates. However, because each evaluation is based on a subjective judgment of interviewers, the aggregated evaluations can be biased because the difference in toughness of interviewers is not examined. Additionally, it is noteworthy that the toughness of interviewers might vary depending on the interview round. As described herein, we propose an analytical framework of simultaneous estimation for both the true potential of candidates and toughness of interviewers' judgment considering job interview rounds, with algorithms to extract unseen knowledge of the true potential of candidates and toughness of interviewers as latent variables through analyzing grade data of job interviews. We apply a Bayesian Hierarchical Ordered Probit Model to the grade data from HRMOS, a cloud-based Applicant Tracking System (ATS) operated by BizReach, Inc., an IT start-up particularly addressing human-resource needs in Japan. Our model successfully quantifies the candidate potential and the interviewers' toughness. An interpretation and applications of the model are given along with a discussion of its place within hiring processes in real-world settings. The parameters are estimated by Markov Chain Monte Carlo (MCMC). A discussion of uncertainty, which is given by the posterior distribution of the parameters, is also provided along with the analysis., Comment: 10 pages

We introduce several novel change of measure inequalities for two families of divergences: $f$-divergences and $\alpha$-divergences. We show how the variational representation for $f$-divergences leads to novel change of measure inequalities. We also present a multiplicative change of measure inequality for $\alpha$-divergences and a generalized version of Hammersley-Chapman-Robbins inequality. Finally, we present several applications of our change of measure inequalities, including PAC-Bayesian bounds for various classes of losses and non-asymptotic intervals for Monte Carlo estimates., Comment: 20 pages