Causal Relations Among N Variables

By combining experimental interventions with search procedures for graphical causal models we show that under familiar assumptions, with perfect data, N − 1 experiments suffice to determine the causal relations among N > 2 variables when each experiment randomizes at most one variable. We show the same bound holds for adaptive learners, but does not hold for N > 4 when each experiment can simultaneously randomize more than one variable. This bound provides a type of ideal for the measure of success of heuristic approaches in active learning methods of causal discovery, which currently use less informative measures. 1 Three Methods and Their Limitations Consider situations in which the aim of inquiry is to determine the causal structure of a kind of system with many variables, for example the gene regulation network of a species in a particular environment. The aim in other words is to determine for each pair X, Y of variables in a set of variables, S, whether X directly causes Y (or vice-versa), with respect to the remaining variables in S, i.e., for some assignment of values V to all the remaining variables in S, if we were to intervene to hold those variables fixed at values V while randomizing X, Y would covary with X , or vice versa. Such a system of causal relations can be represented by a directed graph, in which the variables are nodes or vertices of the graph, and X → Y indicates that X is a direct cause of Y . If there are no feedback relations among the variables, the graph is acyclic. We are concerned with the most efficient way to determine the complete structure of such a directed acyclic graph, under some simplifying assumptions. Suppose that, Second affiliation: Florida Institute for Human and Machine Cognition