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

TL;DR

This paper determines the minimal number of experiments needed to identify all causal relations among N variables, showing that allowing multiple variables to be randomized simultaneously reduces the required experiments.

Contribution

It introduces bounds on the number of experiments needed for causal discovery when multiple variables are randomized simultaneously, improving over previous bounds.

Findings

Log2(N)+1 experiments are sufficient and necessary in the worst case for N variables.

Bounds are provided for experiments when multiple variables are randomized simultaneously.

For large N, fewer experiments are needed compared to the single-variable randomization case.

Abstract

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 aresufficient and in the worst case necessary. We over 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.