arXiv subjects

Edoardo M. Airoldi

Publications and source records attributed to Edoardo M. Airoldi.

At least 19 recordsLinked to original sources

arXiv2026

Causal inference with bipartite designs: A generalized propensity score approach

Bipartite experiments, in which one set of units receives a treatment while outcomes are measured on another set, have recently garnered attention for their ability to capture interference across two distinct populations, such as buyers and sellers in online marketplaces. However, analyzing these experiments can be challenging, given that exposure is neither purely binary nor independent across units. In this paper, we propose a unified framework for causal inference in bipartite designs that leverages generalized propensity scores (GPS) to estimate exposure-response functions. Under standard unconfoundedness assumptions, we show that our GPS-based estimators are unbiased and derive theoretical bounds on their variance. We further introduce practical modeling and weighting strategies, along with double deconfounding methods, that integrate the GPS into both the outcome model and the assignment mechanism. Through extensive simulations, we demonstrate that these approaches achieve substantial bias reduction compared to naive methods. We also illustrate their effectiveness in a real-world application using an Amazon Pet Supplies dataset, where controlling for network structure proves critical to drawing valid causal conclusions. Our results underscore the importance of bipartite designs in contexts with significant interference and highlight how GPS-based methods can bolster the reliability of causal effect estimates in such settings.

stat.ME↗

arXiv2026

Inward and Outward Spillover Effects of One Unit's Treatment on Network Neighbors under Partial Interference

In settings where interference is present, direct effects are commonly defined as the average effect of a unit's treatment on their own outcome while fixing the treatment status or probability among interfering units. Spillover effects measure the average effect of a change in the latter while the individual's treatment status is kept fixed. Here, we define the average causal effect of a unit's treatment status on the outcome of their network neighbors, while fixing the treatment probability in the remaining interference set. We propose two different weighting schemes defining two causal effects: i) the outward spillover effect, which represents the average effect of a unit's treatment on their neighbors' potential outcomes, and ii) the inward spillover effect, which represents the impact of each neighbor's treatment on an individual's own potential outcome. We provide a necessary and sufficient condition for when outward and inward spillover effects differ--even in undirected networks--and show that they are equivalent under specific conditions. We provide numerous examples illustrating the conditions for equivalence or discrepancy of the two spillover effects. We then compare their Horvitz-Thompson estimators, examining their relative variance under various graph structures and structural assumptions on potential outcomes.

stat.ME↗

arXiv2026

Dynamic causal inference with time series data

We generalize the potential outcome framework to time series with an intervention by defining causal effects on stochastic processes. Interventions in dynamic systems alter not only outcome levels but also evolutionary dynamics -- changing persistence and transition laws. Our framework treats potential outcomes as entire trajectories, enabling causal estimands, identification conditions, and estimators to be formulated directly on path space. The resulting Dynamic Average Treatment Effect (DATE) characterizes how causal effects evolve through time and reduces to the classical average treatment effect under one period of time. For observational data, we derive a dynamic inverse-probability weighting estimator that is unbiased under dynamic ignorability and positivity. When treated units are scarce, we show that conditional mean trajectories underlying the DATE admit a linear state-space representation, yielding a dynamic linear model implementation. Simulations demonstrate that modeling time as intrinsic to the causal mechanism exposes dynamic effects that static methods systematically misestimate. An empirical study of COVID-19 lockdowns illustrates the framework's practical value for estimating and decomposing treatment effects.

stat.ME↗

arXiv2025

On the admissibility of Horvitz-Thompson estimator for estimating causal effects under network interference

The Horvitz-Thompson (H-T) estimator is widely used for estimating network causal effects. We study its optimality properties by embedding it in the class of all linear estimators. We show that, under any form of interference, the H-T estimator is inadmissible with respect to mean squared error for designs that generate a random number of units for a given treatment-exposure combination, which includes completely randomized and Bernoulli designs. In contrast, the Hajek and Ratio estimators are admissible under such designs as long as their weights exceed one. We show that the H-T estimator becomes admissible under restricted randomization schemes that fix treatment-exposure counts. These findings motivate a new estimator, the conditional horvitz-Thompson (CHT) estimator, which uses propensity scores conditional on the realized number of units in each treatment-exposure combination. We prove that the CHT estimator is unbiased and admissible for network causal effects, for any design, within the class of linear estimators. Because computing conditional propensity scores can be challenging in practice, we develop three practical approximations: an MCMC approximation and two closed-form approximations based on the difference-in-means and Ratio estimators, the first one avoids computing any propensity scores. Numerical experiments on the Add Health network illustrate the theoretical results.

math.ST↗

arXiv2025

Educational Intervention Re-Wires Social Interactions in Isolated Village Networks

Social networks shape behavior, disseminate information, and undergird collective action within communities. Consequently, they can be very valuable in the design of effective interventions to improve community well-being. But any exogenous intervention in networked groups, including ones that just involve the provision of information, can also possibly modify the underlying network structure itself, and some interventions are indeed designed to do so. While social networks obey certain fundamental principles (captured by network-level statistics, such as the degree distribution or transitivity level), they can nevertheless undergo change across time, as people form and break ties with each other within an overall population. Here, using a randomized controlled trial in 110 remote Honduran villages involving 8,331 people, we evaluated the effects of a 22-month public health intervention on pre-existing social network structures. We leverage a two-stage randomized design, where a varying fraction of households received the intervention in each village. In low-dosage villages (5%, 10%, 20%, and 30%) compared to untreated villages (0%), over a two-year period, individuals who received the intervention tended to sever both inbound and outbound ties with untreated individuals whom they previously trusted for health advice. Conversely, in high-dosage villages (50%, 75%, and 100%), treated individuals increased both their inbound and outbound ties. Furthermore, although the intervention was health-focused, it also reshaped broader friendship and financial ties. In aggregate, the imposition of a novel health information regime in rural villages (as a kind of social institution) led to a significant rewiring in individuals particular connections, but it still had a limited effect on the overall global structure of the village-wide social networks.

stat.AP↗

arXiv2024

Bayesian Causal Synthesis for Meta-Inference on Heterogeneous Treatment Effects

The estimation of heterogeneous treatment effects in the potential outcome setting is biased when there exists model misspecification or unobserved confounding. As these biases are unobservable, what model to use when remains a critical open question. In this paper, we propose a novel Bayesian methodology to mitigate misspecification and improve estimation via a synthesis of multiple causal estimates, which we call Bayesian causal synthesis. Our development is built upon identifying a synthesis function that correctly specifies the heterogeneous treatment effect under no unobserved confounding, and achieves the irreducible bias under unobserved confounding. We show that our proposed method results in consistent estimates of the heterogeneous treatment effect; either with no bias or with irreducible bias. We provide a computational algorithm for fast posterior sampling. Several benchmark simulations and an empirical study highlight the efficacy of the proposed approach compared to existing methodologies, providing improved point and density estimation of the heterogeneous treatment effect, even under unobserved confounding.

stat.ME↗

arXiv2023

Independent-Set Design of Experiments for Estimating Treatment and Spillover Effects under Network Interference

Interference is ubiquitous when conducting causal experiments over networks. Except for certain network structures, causal inference on the network in the presence of interference is difficult due to the entanglement between the treatment assignments and the interference levels. In this article, we conduct causal inference under interference on an observed, sparse but connected network, and we propose a novel design of experiments based on an independent set. Compared to conventional designs, the independent-set design focuses on an independent subset of data and controls their interference exposures through the assignments to the rest (auxiliary set). We provide a lower bound on the size of the independent set from a greedy algorithm , and justify the theoretical performance of estimators under the proposed design. Our approach is capable of estimating both spillover effects and treatment effects. We justify its superiority over conventional methods and illustrate the empirical performance through simulations.

stat.ME↗

arXiv2022

Optimizing Randomized and Deterministic Saturation Designs under Interference

Randomized saturation designs are a family of designs which assign a possibly different treatment proportion to each cluster of a population at random. As a result, they generalize the well-known (stratified) completely randomized designs and the cluster-based randomized designs, which are included as special cases. We show that, under the stable unit treatment value assumption, either the cluster-based or the stratified completely randomized design are in fact optimal for the bias and variance of the difference-in-means estimator among randomized saturation designs. However, this is no longer the case when interference is present. We provide the closed form of the bias and variance of the difference-in-means estimator under a linear model of interference and investigate the optimization of each of these objectives. In addition to the randomized saturation designs, we propose a deterministic saturation design, where the treatment proportion for clusters are fixed, rather than randomized, in order to further improve the estimator under correct model specification. Through simulations, we illustrate the merits of optimizing randomized saturation designs to the graph and potential outcome structure, as well as showcasing the additional improvements yielded by well-chosen deterministic saturation designs.

stat.ME↗

arXiv2020

The Proximal Robbins-Monro Method

The need for parameter estimation with massive datasets has reinvigorated interest in stochastic optimization and iterative estimation procedures. Stochastic approximations are at the forefront of this recent development as they yield procedures that are simple, general, and fast. However, standard stochastic approximations are often numerically unstable. Deterministic optimization, on the other hand, increasingly uses proximal updates to achieve numerical stability in a principled manner. A theoretical gap has thus emerged. While standard stochastic approximations are subsumed by the framework of Robbins and Monro (1951), there is no such framework for stochastic approximations with proximal updates. In this paper, we conceptualize a proximal version of the classical Robbins-Monro procedure. Our theoretical analysis demonstrates that the proposed procedure has important stability benefits over the classical Robbins-Monro procedure, while it retains the best known convergence rates. Exact implementations of the proximal Robbins-Monro procedure are challenging, but we show that approximate implementations lead to procedures that are easy to implement, and still dominate classical procedures by achieving numerical stability, practically without tradeoffs. Moreover, approximate proximal Robbins-Monro procedures can be applied even when the objective cannot be calculated analytically, and so they generalize stochastic proximal procedures currently in use.

math.ST↗

arXiv2019

Evaluating and Optimizing Network Sampling Designs: Decision Theory and Information Theory Perspectives

Some of the most used sampling mechanisms that implicitly leverage a social network depend on tuning parameters; for instance, Respondent-Driven Sampling (RDS) is specified by the number of seeds and maximum number of referrals. We are interested in the problem of optimizing these sampling mechanisms with respect to their tuning parameters in order to optimize the inference on a population quantity, where such quantity is a function of the network and measurements taken at the nodes. This is done by formulating the problem in terms of decision theory and information theory, in turn. We discuss how the approaches discussed in this paper relate, via theoretical results, to other formalisms aimed to compare sampling designs, namely sufficiency and the Goel-DeGroot Criterion. The optimization procedure for different network sampling mechanisms is illustrated via simulations in the fashion of the ones used for Bayesian clinical trials.

stat.ME↗

arXiv2019

Stacking Models for Nearly Optimal Link Prediction in Complex Networks

Most real-world networks are incompletely observed. Algorithms that can accurately predict which links are missing can dramatically speedup the collection of network data and improve the validity of network models. Many algorithms now exist for predicting missing links, given a partially observed network, but it has remained unknown whether a single best predictor exists, how link predictability varies across methods and networks from different domains, and how close to optimality current methods are. We answer these questions by systematically evaluating 203 individual link predictor algorithms, representing three popular families of methods, applied to a large corpus of 548 structurally diverse networks from six scientific domains. We first show that individual algorithms exhibit a broad diversity of prediction errors, such that no one predictor or family is best, or worst, across all realistic inputs. We then exploit this diversity via meta-learning to construct a series of "stacked" models that combine predictors into a single algorithm. Applied to a broad range of synthetic networks, for which we may analytically calculate optimal performance, these stacked models achieve optimal or nearly optimal levels of accuracy. Applied to real-world networks, stacked models are also superior, but their accuracy varies strongly by domain, suggesting that link prediction may be fundamentally easier in social networks than in biological or technological networks. These results indicate that the state-of-the-art for link prediction comes from combining individual algorithms, which achieves nearly optimal predictions. We close with a brief discussion of limitations and opportunities for further improvement of these results.

stat.ML↗

arXiv2018

A systematic investigation of classical causal inference strategies under mis-specification due to network interference

We systematically investigate issues due to mis-specification that arise in estimating causal effects when (treatment) interference is informed by a network available pre-intervention, i.e., in situations where the outcome of a unit may depend on the treatment assigned to other units. We develop theory for several forms of interference through the concept of exposure neighborhood, and develop the corresponding semi-parametric representation for potential outcomes as a function of the exposure neighborhood. Using this representation, we extend the definition of two popular classes of causal estimands, marginal and average causal effects, to the case of network interference. We characterize the bias and variance one incurs when combining classical randomization strategies (namely, Bernoulli, Completely Randomized, and Cluster Randomized designs) and estimators (namely, difference-in-means and Horvitz-Thompson) used to estimate average treatment effect and on the total treatment effect, under misspecification due to interference. We illustrate how difference-in-means estimators can have arbitrarily large bias when estimating average causal effects, depending on the form and strength of interference, which is unknown at design stage. Horvitz-Thompson (HT) estimators are unbiased when the correct weights are specified. Here, we derive the HT weights for unbiased estimation of different estimands, and illustrate how they depend on the design, the form of interference, which is unknown at design stage, and the estimand. More importantly, we show that HT estimators are in-admissible for a large class of randomization strategies, in the presence of interference. We develop new model-assisted and model-dependent strategies to improve HT estimators, and we develop new randomization strategies for estimating the average treatment effect and total treatment effect.

stat.ME↗

arXiv2018

Identification and estimation of treatment and interference effects in observational studies on networks

Causal inference on a population of units connected through a network often presents technical challenges, including how to account for interference. In the presence of local interference, for instance, potential outcomes of a unit depend on its treatment as well as on the treatments of other local units, such as its neighbors according to the network. In observational studies, a further complication is that the typical unconfoundedness assumption must be extended - say, to include the treatment of neighbors, and indi- vidual and neighborhood covariates - to guarantee identification and valid inference. Here, we propose new estimands that define treatment and interference effects. We then derive analytical expressions for the bias of a naive estimator that wrongly assumes away interference. The bias depends on the level of interference but also on the degree of association between individual and neighborhood treatments. We propose an extended unconfoundedness assumption that accounts for interference, and we develop new covariate-adjustment methods that lead to valid estimates of treatment and interference effects in observational studies on networks. Estimation is based on a generalized propensity score that balances individual and neighborhood covariates across units under different levels of individual treatment and of exposure to neighbors' treatment. We carry out simulations, calibrated using friendship networks and covariates in a nationally representative longitudinal study of adolescents in grades 7-12, in the United States, to explore finite-sample performance in different realistic settings.

stat.ME↗

arXiv2018

Optimizing cluster-based randomized experiments under a monotonicity assumption

Cluster-based randomized experiments are popular designs for mitigating the bias of standard estimators when interference is present and classical causal inference and experimental design assumptions (such as SUTVA or ITR) do not hold. Without an exact knowledge of the interference structure, it can be challenging to understand which partitioning of the experimental units is optimal to minimize the estimation bias. In the paper, we introduce a monotonicity condition under which a novel two-stage experimental design allows us to determine which of two cluster-based designs yields the least biased estimator. We then consider the setting of online advertising auctions and show that reserve price experiments verify the monotonicity condition and the proposed framework and methodology applies. We validate our findings on an advertising auction dataset.

stat.ME↗

arXiv2018

Propensity score methodology in the presence of network entanglement between treatments

In experimental design and causal inference, it may happen that the treatment is not defined on individual experimental units, but rather on pairs or, more generally, on groups of units. For example, teachers may choose pairs of students who do not know each other to teach a new curriculum; regulators might allow or disallow merging of firms, and biologists may introduce or inhibit interactions between genes or proteins. In this paper, we formalize this experimental setting, and we refer to the individual treatments in such setting as entangled treatments. We then consider the special case where individual treatments depend on a common population quantity, and develop theory and methodology to deal with this case. In our target applications, the common population quantity is a network, and the individual treatments are defined as functions of the change in the network between two specific time points. Our focus is on estimating the causal effect of entangled treatments in observational studies where entangled treatments are endogenous and cannot be directly manipulated. When treatment cannot be manipulated, be it entangled or not, it is necessary to account for the treatment assignment mechanism to avoid selection bias, commonly through a propensity score methodology. In this paper, we quantify the extent to which classical propensity score methodology ignores treatment entanglement, and characterize the bias in the estimated causal effects. To characterize such bias we introduce a novel similarity function between propensity score models, and a practical approximation of it, which we use to quantify model misspecification of propensity scores due to entanglement. One solution to avoid the bias in the presence of entangled treatments is to model the change in the network, directly, and calculate an individual unit's propensity score by averaging treatment assignments over this change.

math.ST↗

arXiv2017

Model-assisted design of experiments in the presence of network correlated outcomes

We consider the problem of how to assign treatment in a randomized experiment, in which the correlation among the outcomes is informed by a network available pre-intervention. Working within the potential outcome causal framework, we develop a class of models that posit such a correlation structure among the outcomes. Then we leverage these models to develop restricted randomization strategies for allocating treatment optimally, by minimizing the mean square error of the estimated average treatment effect. Analytical decompositions of the mean square error, due both to the model and to the randomization distribution, provide insights into aspects of the optimal designs. In particular, the analysis suggests new notions of balance based on specific network quantities, in addition to classical covariate balance. The resulting balanced, optimal restricted randomization strategies are still design unbiased, in situations where the model used to derive them does not hold. We illustrate how the proposed treatment allocation strategies improve on allocations that ignore the network structure, with extensive simulations.

stat.ME↗

arXiv2017

Elements of estimation theory for causal effects in the presence of network interference

Randomized experiments in which the treatment of a unit can affect the outcomes of other units are becoming increasingly common in healthcare, economics, and in the social and information sciences. From a causal inference perspective, the typical assumption of no interference becomes untenable in such experiments. In many problems, however, the patterns of interference may be informed by the observation of network connections among the units of analysis. Here, we develop elements of optimal estimation theory for causal effects leveraging an observed network, by assuming that the potential outcomes of an individual depend only on the individual's treatment and on the treatment of the neighbors. We propose a collection of exclusion restrictions on the potential outcomes, and show how subsets of these restrictions lead to various parameterizations. Considering the class of linear unbiased estimators of the average direct treatment effect, we derive conditions on the design that lead to the existence of unbiased estimators, and offer analytical insights on the weights that lead to minimum integrated variance estimators. We illustrate the improved performance of these estimators when compared to more standard biased and unbiased estimators, using simulations.

stat.ME↗

arXiv2016

Asymptotic and finite-sample properties of estimators based on stochastic gradients

Stochastic gradient descent procedures have gained popularity for parameter estimation from large data sets. However, their statistical properties are not well understood, in theory. And in practice, avoiding numerical instability requires careful tuning of key parameters. Here, we introduce implicit stochastic gradient descent procedures, which involve parameter updates that are implicitly defined. Intuitively, implicit updates shrink standard stochastic gradient descent updates. The amount of shrinkage depends on the observed Fisher information matrix, which does not need to be explicitly computed; thus, implicit procedures increase stability without increasing the computational burden. Our theoretical analysis provides the first full characterization of the asymptotic behavior of both standard and implicit stochastic gradient descent-based estimators, including finite-sample error bounds. Importantly, analytical expressions for the variances of these stochastic gradient-based estimators reveal their exact loss of efficiency. We also develop new algorithms to compute implicit stochastic gradient descent-based estimators for generalized linear models, Cox proportional hazards, M-estimators, in practice, and perform extensive experiments. Our results suggest that implicit stochastic gradient descent procedures are poised to become a workhorse for approximate inference from large data sets

stat.ME↗