Common methods of causal inference generate directed acyclic graphs (DAGs) that formalize causal relations between n variables. Given the joint distribution of all these variables, the DAG contains all information about how intervening on one variable would change the distribution of the other n-1 variables. It remains, however, a non-trivial question how to quantify the causal influence of one variable on another one.
Here we propose a measure for causal strength that refers to direct effects and measure the "strength of an arrow" or a set of arrows. It is based on a hypothetical intervention that modifies the joint distribution by cutting the corresponding edge. The causal strength is then the relative entropy distance between the old and the new distribution.
We discuss other measures of causal strength like the average causal effect, transfer entropy and information flow and describe their limitations. We argue that our measure is also more appropriate for time series than the known ones.
Finally, we discuss conceptual problems in defining the strength of indirect effects.
Quantifying causal influences
Dominik Janzing, David Balduzzi, Moritz Grosse-Wentrup, Bernhard Schoelkopf