approfondimento - Causal Inference - # Leaky Instrumental Variables

Bounding Causal Effects with Leaky Instruments: A Novel Approach to Partial Identification

Q: How can the leaky IV approach be extended to nonlinear structural equation models

To extend the leaky IV approach to nonlinear structural equation models, we can reformulate the τ-exclusion criterion to accommodate nonlinear dependencies between variables. One way to do this is by placing an upper bound on the conditional mutual information between the candidate instruments Z and the outcome Y given the treatment X and unobserved confounders U. This generalized τ-exclusion criterion, denoted as (A3′′), would ensure that the leaky instruments have a limited impact on the outcome Y, considering the nonlinear relationships in the model. Additionally, we can explore measures such as the Wasserstein distance or the KL-divergence to quantify the discrepancy between the conditional distributions of Y given X, U and Y given X, U, Z. By incorporating these nonlinear considerations into the τ-exclusion criterion, we can adapt the leaky IV framework to nonlinear structural equation models effectively.

Q: What are the implications of relaxing the assumption of linear effects in the leaky IV model

Relaxing the assumption of linear effects in the leaky IV model has several implications. Firstly, allowing for nonlinear effects between the treatment X, candidate instruments Z, and the outcome Y can lead to more flexible modeling of complex relationships in the data. Nonlinear effects can capture intricate interactions and dependencies that may not be adequately represented by linear models. However, incorporating nonlinear effects can also introduce additional complexity in the estimation and interpretation of causal effects. Nonlinear relationships may require more sophisticated modeling techniques and potentially larger sample sizes to accurately estimate causal effects and derive valid bounds. Moreover, the presence of nonlinear effects may impact the identifiability of causal effects and the precision of the estimated bounds, requiring careful consideration and validation of the modeling assumptions.

Q: How can the leaky IV framework be applied to causal discovery problems, where the goal is to infer the underlying causal structure rather than just estimate a specific causal effect

The leaky IV framework can be applied to causal discovery problems by leveraging the concept of partial identification to infer the underlying causal structure from observational data. In causal discovery, the goal is not only to estimate specific causal effects but also to uncover the causal relationships among variables in the system. By relaxing the assumption of strict exclusion criteria and allowing for leaky instruments, the framework can provide insights into the causal pathways and dependencies between variables, even in the presence of unobserved confounding. By formulating the τ-exclusion criterion to account for potential information leakage from candidate instruments to the outcome, the leaky IV approach can help identify causal relationships and infer the causal structure of the system. This can be particularly valuable in settings where the true causal relationships are complex, nonlinear, or involve latent variables that are not directly observable.

Concetti Chiave

This paper proposes a novel method for bounding causal effects in linear structural equation models when the exclusion criterion for instrumental variables is violated to some limited degree. The authors derive a convex optimization objective that provides provably sharp bounds on the average treatment effect under common forms of information leakage from instruments to outcome.

Sintesi

The paper introduces the concept of "leaky instruments" - variables that satisfy the relevance and unconfoundedness assumptions for instrumental variables (IVs) but may violate the exclusion criterion to some extent. The authors derive formal results for partially identifying the average treatment effect (ATE) in linear structural equation models with leaky instruments.

Key highlights:

The authors relax the exclusion criterion by allowing instruments to have a bounded direct effect on the outcome, either through a scalar threshold (τ-exclusion) or separate thresholds for each instrument (vector τ-exclusion).
They show that under these relaxed assumptions, the ATE is partially identifiable and derive convex optimization objectives to compute provably sharp bounds on the ATE.
For the L2 norm case, they provide a closed-form solution for the ATE bounds.
The authors propose a Monte Carlo test for the exclusion criterion and a bootstrapping procedure to quantify uncertainty around the estimated ATE bounds.
Experiments on simulated data demonstrate that the proposed method outperforms existing approaches designed for invalid instruments, providing valid and informative bounds on the ATE in a wide range of settings.

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Approfondimenti chiave tratti da

Bounding Causal Effects with Leaky Instruments

by David S. Wat... alle arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.04446.pdf

Bounding Causal Effects with Leaky Instruments

Domande più approfondite

How can the leaky IV approach be extended to nonlinear structural equation models

To extend the leaky IV approach to nonlinear structural equation models, we can reformulate the τ-exclusion criterion to accommodate nonlinear dependencies between variables. One way to do this is by placing an upper bound on the conditional mutual information between the candidate instruments Z and the outcome Y given the treatment X and unobserved confounders U. This generalized τ-exclusion criterion, denoted as (A3′′), would ensure that the leaky instruments have a limited impact on the outcome Y, considering the nonlinear relationships in the model. Additionally, we can explore measures such as the Wasserstein distance or the KL-divergence to quantify the discrepancy between the conditional distributions of Y given X, U and Y given X, U, Z. By incorporating these nonlinear considerations into the τ-exclusion criterion, we can adapt the leaky IV framework to nonlinear structural equation models effectively.

What are the implications of relaxing the assumption of linear effects in the leaky IV model

Relaxing the assumption of linear effects in the leaky IV model has several implications. Firstly, allowing for nonlinear effects between the treatment X, candidate instruments Z, and the outcome Y can lead to more flexible modeling of complex relationships in the data. Nonlinear effects can capture intricate interactions and dependencies that may not be adequately represented by linear models. However, incorporating nonlinear effects can also introduce additional complexity in the estimation and interpretation of causal effects. Nonlinear relationships may require more sophisticated modeling techniques and potentially larger sample sizes to accurately estimate causal effects and derive valid bounds. Moreover, the presence of nonlinear effects may impact the identifiability of causal effects and the precision of the estimated bounds, requiring careful consideration and validation of the modeling assumptions.

How can the leaky IV framework be applied to causal discovery problems, where the goal is to infer the underlying causal structure rather than just estimate a specific causal effect

The leaky IV framework can be applied to causal discovery problems by leveraging the concept of partial identification to infer the underlying causal structure from observational data. In causal discovery, the goal is not only to estimate specific causal effects but also to uncover the causal relationships among variables in the system. By relaxing the assumption of strict exclusion criteria and allowing for leaky instruments, the framework can provide insights into the causal pathways and dependencies between variables, even in the presence of unobserved confounding. By formulating the τ-exclusion criterion to account for potential information leakage from candidate instruments to the outcome, the leaky IV approach can help identify causal relationships and infer the causal structure of the system. This can be particularly valuable in settings where the true causal relationships are complex, nonlinear, or involve latent variables that are not directly observable.