Necessary and sufficient graphical conditions for optimal adjustment sets in causal graphical models with hidden variables (2102.10324v4)

Abstract: The problem of selecting optimal backdoor adjustment sets to estimate causal effects in graphical models with hidden and conditioned variables is addressed. Previous work has defined optimality as achieving the smallest asymptotic estimation variance and derived an optimal set for the case without hidden variables. For the case with hidden variables there can be settings where no optimal set exists and currently only a sufficient graphical optimality criterion of limited applicability has been derived. In the present work optimality is characterized as maximizing a certain adjustment information which allows to derive a necessary and sufficient graphical criterion for the existence of an optimal adjustment set and a definition and algorithm to construct it. Further, the optimal set is valid if and only if a valid adjustment set exists and has higher (or equal) adjustment information than the Adjust-set proposed in Perkovi{\'c} et al. [Journal of Machine Learning Research, 18: 1--62, 2018] for any graph. The results translate to minimal asymptotic estimation variance for a class of estimators whose asymptotic variance follows a certain information-theoretic relation. Numerical experiments indicate that the asymptotic results also hold for relatively small sample sizes and that the optimal adjustment set or minimized variants thereof often yield better variance also beyond that estimator class. Surprisingly, among the randomly created setups more than 90\% fulfill the optimality conditions indicating that also in many real-world scenarios graphical optimality may hold. Code is available as part of the python package \url{https://github.com/jakobrunge/tigramite}.

• The paper introduces an information-theoretic measure called 'adjustment information' to achieve optimal variance reduction in causal effect estimation.
• It establishes a necessary and sufficient graphical criterion to systematically identify optimal backdoor adjustment sets in the presence of hidden variables.
• Algorithm development and numerical experiments confirm that the proposed method outperforms traditional techniques in efficiency and practical variance reduction.

Overview of Optimal Adjustment Sets in Causal Graphical Models with Hidden Variables

The paper addresses the critical problem of selecting optimal backdoor adjustment sets to estimate causal effects in complex graphical models that include hidden variables. The main focus is the characterization and identification of adjustment sets that minimize asymptotic estimation variance, which is a significant challenge in causal inference.

Background

Causal graphical models are a staple in understanding causal relationships between variables, particularly when some variables may not be directly observable. Traditional methods rely heavily on criteria like the backdoor criterion to identify valid adjustment sets that lead to unbiased causal effect estimates. These methods, however, often struggle with variability in estimation variance across different adjustment sets, particularly in the presence of hidden variables.

Contributions

1. Optimality Criterion: The paper introduces an information-theoretic measure called "adjustment information," which quantifies the effectiveness of an adjustment set in maximizing constraint on the effect variable while minimizing constraint on the cause variable.
2. Graphical Criteria: It establishes a necessary and sufficient graphical criterion for the existence of an optimal adjustment set. This criterion, based on the adjustment information, offers a way to systematically identify the optimal set.
3. Algorithmic Development: An algorithm is developed to construct the optimal adjustment set, which is shown to be equivalent or superior to the existing Adjust-set for any graph, in terms of information content and estimation variance.
4. Practical Implications: Numerical experiments validate that the theoretically optimal sets also perform well in practice, even with smaller sample sizes, and provide better variance properties than previously known methods.

Numerical Results and Practical Outcomes

The numerical experiments conducted demonstrate that more than 90% of randomly generated configurations fulfill the optimality conditions proposed, indicating that in many real-world applications, these conditions may naturally hold. This suggests the widespread applicability of the proposed method in practical scenarios, potentially leading to more efficient and accurate causal inference.

Implications and Future Directions

The theoretical framework provided opens several avenues for future research, particularly in expanding the class of estimators that can benefit from the proposed optimization. While the results are robust for linear estimators, investigating applicability across a broader range of nonlinear or more complex estimators could further enhance the utility of the proposed methods. The relationship between adjustment information and estimators’ variance remains a rich area for exploration.

Moreover, extending the approach to handle scenarios such as models with dynamic treatments or graphs that are only partially known presents compelling future directions.

Conclusion

This paper makes a substantial contribution to the field of causal inference by providing a robust framework for identifying optimal adjustment sets in the presence of hidden variables. The methodologies developed are not only theoretically sound but also show promising practical performance, which could transform approaches to causal inference in complex graphs. The availability of code as part of a Python package will facilitate adoption and further experimentation by researchers in the field.