The power of intervention

Abstract

We further develop the mathematical theory of causal interventions, extending earlier results of Korb, Twardy, Handfield, & Oppy, (2005) and Spirtes, Glymour, Scheines (2000). Some of the skepticism surrounding causal discovery has concerned the fact that using only observational data can radically underdetermine the best explanatory causal model, with the true causal model appearing inferior to a simpler, faithful model (cf. Cartwright, (2001). Our results show that experimental data, together with some plausible assumptions, can reduce the space of viable explanatory causal models to one.

Appendix

Here we prove Theorem 3 for linear recursive path models with imperfect interventions. The proof assumes familiarity with path modeling and Bayesian networks.

Theorem 3

(Distinguishability under intervention) For any M ₁(θ₁), M ₂(θ₂) (which model the same space and are positive) ¹³ such that $M_1\neq M_2$, if $P_{M_1(\theta_1)} = P_{M_2(\theta_2)}$, then under interventions $P_{M^{\prime}_1(\theta_1)}\neq P_{M^{\prime}_2(\theta_2)}.$

Proof

Since M ₁ and M ₂ differ structurally, for some pair of nodes X and Y one contains $X\rightarrow Y$ while the other contains $X\leftarrow Y$ or else X:Y (i.e., they are not directly connected). In either case, we can find a probabilistic dependency induced by one model that must fail relative to the other. To fix ideas, let us suppose the former holds for M ₁ and the latter for M ₂.

Case 1

M ₂ contains X:Y. Let the set of variables Z be the set of all variables in M′₁ (equivalently, $M^{\prime}_2$) except for $\{X,I_Y,I_X\}$ (and so $Y\in {\bf Z}$). Then $r_{I_{Y} X\cdot {\bf Z}}\neq 0$ in $M^{\prime}_1(\theta_1)$ and $r_{I_{Y} X\cdot {\bf Z}} = 0$ in M′₂(θ₂). This follows from Wright’s theory of path models, given that $p_{YI_Y}p_{YX}\neq 0$ in M′₁(θ₁) and $p_{YI_Y}p_{YX} = 0$ in M′₂(θ₂). (Given Z, there is only one d-connecting path between X and I _Y in M ₁ and none in M ₂.)

Case 2

M ₂ contains X← Y. Let the set of variables Z be the set of all variables in M′₁ except for $\{X,I_{Y},I_{X}\}$ (and so $Y\in {\bf Z}$). Then $r_{I_Y X\cdot {\bf Z}}\neq 0$ in $M^{\prime}_1(\theta_1)$ and $r_{I_Y X\cdot {\bf Z}}= 0$ in $M^{\prime}_2(\theta_2)$. This follows from Wright’s theory of path models, given that $p_{YI_Y}p_{YX}\neq 0$ in $M^{\prime}_1(\theta_1)$. (As for Case 1, given Z, there is only one d-connecting path between X and I _Y in M ₁ and none in M ₂.) Similarly, letting Z be the set of all variables in $M^{\prime}_1$ except for $\{Y,I_Y,I_X\}$ (and so $X\in{\bf Z}$), $r_{I_X Y\cdot {\bf Z}}\neq 0$ in $M^{\prime}_2(\theta_2)$ and $r_{I_X Y\cdot {\bf Z}} = 0$ in $M^{\prime}_1(\theta_1)$. □

This implies, given perfect data, that distinguishability between any two distinct models of size n can be achieved with n independent interventions and O(n ²) conditional independence tests. Thus, the computational cost of finding such empirical distinctions is modest. ¹⁴