# General Transportability of Soft Interventions: Completeness Results

NIPS 2020, 2020.

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Abstract:

The challenge of generalizing causal knowledge across different environments is pervasive in scientific explorations, including in AI, ML, and Data Science. Experiments are usually performed in one environment (e.g., in a lab, on Earth) with the intent, almost invariably, of being used elsewhere (e.g., outside the lab, on Mars), where the...More

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Introduction

- Generalizing causal knowledge across disparate domains is at the heart of many inferences across the empirical sciences as well as AI [26, 33, 29].
- The authors design an efficient algorithm to determine the existence of an estimand for the effect of a non-atomic intervention as a function of the available distributions.
- The authors prove that the σ-calculus is necessary and sufficient for the task of transportability when both the input and the output distributions involve soft interventions.

Highlights

- Generalizing causal knowledge across disparate domains is at the heart of many inferences across the empirical sciences as well as AI [26, 33, 29]
- We studied the problem of transporting effects of soft interventions from knowledge encoded in the form of a selection diagram and a combination of observational and experimental data from multiple, different domains
- We showed how the problem can be solved by transporting the effect of an atomic intervention from the same input (Thm. 1)
- We conclude that σ-calculus together with basic probability axioms are complete for the soft transportability task (Cor. 2)
- We described a complete graphical condition to determine the transportability of any transportability instance (Cor. 3)
- We hope that this series of results related to soft interventions, and knowledge of its relationship with atomic interventions, can help data scientists to apply causal inference in broader and more realistic scenarios

Results

- Conditional and stochastic interventions allow the intervened variable to change as a deterministic function or a conditional probability distribution of a set of observable parents.
- Given a causal diagram Gi = V, E and domain discrepancies ∆i, let S = {Sv | ∃ni=1V ∈ ∆i} be called selection variables.
- The effect intervention σX on a set of outcome variables Y, conditional on W, P ∗(y|w; σX), in a target environment π∗, is said to be transportable from G∆, Z , if it is uniquely computable from the set of distributions Z for every assignment (y, w) and every set of models {Mi}πi∈Π inducing G∆ and Z.
- Given the tightness of the reduction provided by Thm. 1, one may surmise that it is possible to blindly use existing transportability algorithms (e.g., GTR [22]) to solve for soft interventions.
- Both input and output of the transportability task refer to probability distributions within different domains and for different interventions.
- Let Y, X ⊆ V be any two sets of variables, and let σX=σX∗ be an atomic, conditional or stochastic intervention.
- Input: G∆ selection diagrams over variables V for domains Π; Y, W ⊆ V disjoint subsets of variables; an intervention σX∗ defined over a set X ⊆ V; and available distribution specification Z.
- Building on the observations and results the authors have so far, the authors design the algorithm σ-TR (Alg. 1) that takes as input the variables defining a query, the specification of σX, a set of available distributions (Z), and the selection diagrams.
- Σ-TR uses the subroutine IDENTIFY from [36] that applies Lemma 1 systematically to obtain a C-factor Q[A] from Q[B], where A ⊆ B, and the subroutine ‘REPLACE’ to determine the factors of intervened variables according to the particular type of intervention.

Conclusion

- Given query P ∗(y; σX=σX∗ ), selection diagram G∆, and the distribution specified by Z, let A be defined as in Thm. 2.
- The authors studied the problem of transporting effects of soft interventions from knowledge encoded in the form of a selection diagram and a combination of observational and experimental data from multiple, different domains.
- The authors hope that this series of results related to soft interventions, and knowledge of its relationship with atomic interventions, can help data scientists to apply causal inference in broader and more realistic scenarios

Summary

- Generalizing causal knowledge across disparate domains is at the heart of many inferences across the empirical sciences as well as AI [26, 33, 29].
- The authors design an efficient algorithm to determine the existence of an estimand for the effect of a non-atomic intervention as a function of the available distributions.
- The authors prove that the σ-calculus is necessary and sufficient for the task of transportability when both the input and the output distributions involve soft interventions.
- Conditional and stochastic interventions allow the intervened variable to change as a deterministic function or a conditional probability distribution of a set of observable parents.
- Given a causal diagram Gi = V, E and domain discrepancies ∆i, let S = {Sv | ∃ni=1V ∈ ∆i} be called selection variables.
- The effect intervention σX on a set of outcome variables Y, conditional on W, P ∗(y|w; σX), in a target environment π∗, is said to be transportable from G∆, Z , if it is uniquely computable from the set of distributions Z for every assignment (y, w) and every set of models {Mi}πi∈Π inducing G∆ and Z.
- Given the tightness of the reduction provided by Thm. 1, one may surmise that it is possible to blindly use existing transportability algorithms (e.g., GTR [22]) to solve for soft interventions.
- Both input and output of the transportability task refer to probability distributions within different domains and for different interventions.
- Let Y, X ⊆ V be any two sets of variables, and let σX=σX∗ be an atomic, conditional or stochastic intervention.
- Input: G∆ selection diagrams over variables V for domains Π; Y, W ⊆ V disjoint subsets of variables; an intervention σX∗ defined over a set X ⊆ V; and available distribution specification Z.
- Building on the observations and results the authors have so far, the authors design the algorithm σ-TR (Alg. 1) that takes as input the variables defining a query, the specification of σX, a set of available distributions (Z), and the selection diagrams.
- Σ-TR uses the subroutine IDENTIFY from [36] that applies Lemma 1 systematically to obtain a C-factor Q[A] from Q[B], where A ⊆ B, and the subroutine ‘REPLACE’ to determine the factors of intervened variables according to the particular type of intervention.
- Given query P ∗(y; σX=σX∗ ), selection diagram G∆, and the distribution specified by Z, let A be defined as in Thm. 2.
- The authors studied the problem of transporting effects of soft interventions from knowledge encoded in the form of a selection diagram and a combination of observational and experimental data from multiple, different domains.
- The authors hope that this series of results related to soft interventions, and knowledge of its relationship with atomic interventions, can help data scientists to apply causal inference in broader and more realistic scenarios

- Table1: Summary of the types of interventions considered. Each row contains the type of intervention, its representation using the regime indicator and the way the corresponding replacement function

Funding

- Acknowledgments and Disclosure of Funding This research was supported by grants from IBM Research, Adobe Research, NSF IIS-1704352, and IIS-1750807 (CAREER)

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