On the Number of Experiments Sufficient and in the Worst Case Necessary to Identify All Causal Relations Among N Variables

We show that if any number of variables are allowed to be simultaneously and independently randomized in any one experiment, log2(N) + 1 experiments are sufficient and in the worst case necessary to determine the causal relations among N ≥ 2 variables when no latent variables, no sample selection bias and no feedback cycles are present. For all K, 0 < K < 1 2N we provide an upper bound on the number experiments required to determine causal structure when each experiment simultaneously randomizes K variables. For large N , these bounds are significantly lower than the N − 1 bound required when each experiment randomizes at most one variable. For kmax < N 2 , we show that ( N kmax −1)+ N 2kmax log2(kmax) experiments are sufficient and in the worst case necessary. We offer a conjecture as to the minimal number of experiments that are in the worst case sufficient to identify all causal relations among N observed variables that are a subset of the vertices of a DAG.