Generalization of LiNGAM that allows confounding

TL;DR

LiNGAM-MMI addresses recovering causal order under confounding in additive noise models by quantifying confounding with $K(e_1,\ldots,e_p)=E\left[\log \frac{P(e_1,\ldots,e_p)}{P(e_1)\cdots P(e_p)}\right]$ and selecting the order that minimizes it. It casts the search as a shortest-path problem and uses mutual-information-based distances estimated via copula entropy to achieve a globally optimal order with efficiency comparable to LiNGAM when confounding is absent. The key contributions are the KL-based confounding measure, the shortest-path formulation for global optimization, and empirical evidence showing improved order recovery in both simulated and real data. This approach removes the need to explicitly identify confounded variables and scales better than prior methods under realistic confounding, increasing robustness of causal discovery in practical settings.

Abstract

LiNGAM determines the variable order from cause to effect using additive noise models, but it faces challenges with confounding. Previous methods maintained LiNGAM's fundamental structure while trying to identify and address variables affected by confounding. As a result, these methods required significant computational resources regardless of the presence of confounding, and they did not ensure the detection of all confounding types. In contrast, this paper enhances LiNGAM by introducing LiNGAM-MMI, a method that quantifies the magnitude of confounding using KL divergence and arranges the variables to minimize its impact. This method efficiently achieves a globally optimal variable order through the shortest path problem formulation. LiNGAM-MMI processes data as efficiently as traditional LiNGAM in scenarios without confounding while effectively addressing confounding situations. Our experimental results suggest that LiNGAM-MMI more accurately determines the correct variable order, both in the presence and absence of confounding. The code is in the supplementary file in this link: https://github.com/SkyJoyTianle/ISIT2024.

Figures (12)

• Figure 1: Confounders in red of the right figure affect more than one variable. The previous methods can deal with only non-consecutive variables such as $X_1$ and $X_3$.
• Figure 2: Ancestral and bow-free graphs. (a) The vertices 1 and 3 are siblings, but none of them are ancestors of the other. (b) The vertices 1 and 3 are siblings, but the former is an ancestor of the latter. (c) The vertices 1 and 2 consist of a bow.
• Figure 3: Additive Noise Model: Either $e_1\perp\!\!\!\perp e_2$ ($X\rightarrow Y$) or $e_1'\perp\!\!\!\perp e_2'$ ($Y\rightarrow X$) must hold, but not both.
• Figure 4: In order to obtain the residue $z_{xy}^n$($=z_{yx}^n$), compute $\{y_x^n,z_x^n\}$ first and then obtain $\{z_{xy}^n\}$ via (\ref{['eq116']}), as shown in red, or compute $\{z_y^n, x_y^n\}$ first and then obtain $\{z_{yx}^n\}$ via (\ref{['eq112']}), as shown in blue.
• Figure 5: Suppose we have five variables $X,Y,Z,W,T$ are related by $Y=aX+f$, $Z=bY+g$, $W=cZ+f+g$, and $T=cX+dW$ for some $a,b,c,d\in {\mathbb R}$, and that the confounder $f$ and $g$ affect $Y,W$ and $Z,W$, respectively. The table on the right shows what pair of variables the order can be recovered for LvLiNGAM and ParceLiNGAM. For example, in LvLiNGAM, the order between $Y, Z$ can be recovered (denoted as "$\bigcirc$") even if $Y$ is affected by $f$.

Theorems & Definitions (4)

• Proposition 1: darmoisskito
• Proposition 2: shimizu06
• Theorem 1
• Theorem 2