The Annals of Statistics

Quantifying causal influences

Dominik Janzing, David Balduzzi, Moritz Grosse-Wentrup, and Bernhard Schölkopf

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Many methods for causal inference generate directed acyclic graphs (DAGs) that formalize causal relations between $n$ variables. Given the joint distribution on all these variables, the DAG contains all information about how intervening on one variable changes the distribution of the other $n-1$ variables. However, quantifying the causal influence of one variable on another one remains a nontrivial question.

Here we propose a set of natural, intuitive postulates that a measure of causal strength should satisfy. We then introduce a communication scenario, where edges in a DAG play the role of channels that can be locally corrupted by interventions. Causal strength is then the relative entropy distance between the old and the new distribution.

Many other measures of causal strength have been proposed, including average causal effect, transfer entropy, directed information, and information flow. We explain how they fail to satisfy the postulates on simple DAGs of $\leq3$ nodes. Finally, we investigate the behavior of our measure on time-series, supporting our claims with experiments on simulated data.

Article information

Ann. Statist., Volume 41, Number 5 (2013), 2324-2358.

First available in Project Euclid: 5 November 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62-09: Graphical methods 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Causality Bayesian networks information flow transfer entropy


Janzing, Dominik; Balduzzi, David; Grosse-Wentrup, Moritz; Schölkopf, Bernhard. Quantifying causal influences. Ann. Statist. 41 (2013), no. 5, 2324--2358. doi:10.1214/13-AOS1145.

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Supplemental materials

  • Supplementary material: Supplement to “Quantifying causal influences”. Three supplementary sections: (1) Generating an i.i.d. copy via random permutations; (2) Another option to define causal strength; and (3) The problem of defining total influence.