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Journal of Causal Inference

Ed. by Imai, Kosuke / Pearl, Judea / Petersen, Maya Liv / Sekhon, Jasjeet / van der Laan, Mark J.

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2193-3685
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The Inflation Technique for Causal Inference with Latent Variables

Elie WolfeORCID iD: https://orcid.org/0000-0002-6960-3796 / Robert W. Spekkens / Tobias Fritz
Published Online: 2019-07-16 | DOI: https://doi.org/10.1515/jci-2017-0020

Abstract

The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some causal structure. The difficult case is when the causal structure includes latent variables. We here introduce the inflation technique for tackling this problem. An inflation of a causal structure is a new causal structure that can contain multiple copies of each of the original variables, but where the ancestry of each copy mirrors that of the original. To every distribution of the observed variables that is compatible with the original causal structure, we assign a family of marginal distributions on certain subsets of the copies that are compatible with the inflated causal structure. It follows that compatibility constraints for the inflation can be translated into compatibility constraints for the original causal structure. Even if the constraints at the level of inflation are weak, such as observable statistical independences implied by disjoint causal ancestry, the translated constraints can be strong. We apply this method to derive new inequalities whose violation by a distribution witnesses that distribution’s incompatibility with the causal structure (of which Bell inequalities and Pearl’s instrumental inequality are prominent examples). We describe an algorithm for deriving all such inequalities for the original causal structure that follow from ancestral independences in the inflation. For three observed binary variables with pairwise common causes, it yields inequalities that are stronger in at least some aspects than those obtainable by existing methods. We also describe an algorithm that derives a weaker set of inequalities but is more efficient. Finally, we discuss which inflations are such that the inequalities one obtains from them remain valid even for quantum (and post-quantum) generalizations of the notion of a causal model.

Keywords: causal inference with latent variables; inflation technique; causal compatibility inequalities; marginal problem; Bell inequalities; Hardy paradox; graph symmetries; quantum causal models; GPT causal models, triangle scenario

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About the article

Received: 2017-08-16

Revised: 2018-08-31

Accepted: 2019-06-10

Published Online: 2019-07-16


Funding Source: John Templeton Foundation

Award identifier / Grant number: 69609

This project/publication was made possible in part through the support of a grant #69609 from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Economic Development, Job Creation and Trade.


Citation Information: Journal of Causal Inference, 20170020, ISSN (Online) 2193-3685, DOI: https://doi.org/10.1515/jci-2017-0020.

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