Multi-Domain Causal Discovery in Bijective Causal Models

TL;DR

This work addresses causal structure learning across multiple domains by assuming invariant causal mechanisms and domain-varying noise under Bijective Generation Mechanisms (BGM). It introduces a practical test based on pairwise similarity of conditional distributions, encoded through density-vectorization and a special random variable $\Psi_V$, and derives a domain-wise independence criterion $\Gamma_{X\to Y}\perp X$ to identify the true causal direction. The framework extends to discrete, multivariate, and mixed-type data, and is implemented in a two-step algorithm combining sampling from $\Gamma_{X\to Y}$ with domain-specific hypothesis tests to infer parent sets. Empirical results on synthetic and real-world datasets show improved causal-direction identification and more accurate parent recovery compared with existing multi-domain methods. The approach contributes a theoretically grounded, practically scalable path for invariant-causal-discovery across heterogeneous environments.

Abstract

We consider the problem of causal discovery (a.k.a., causal structure learning) in a multi-domain setting. We assume that the causal functions are invariant across the domains, while the distribution of the exogenous noise may vary. Under causal sufficiency (i.e., no confounders exist), we show that the causal diagram can be discovered under less restrictive functional assumptions compared to previous work. What enables causal discovery in this setting is bijective generation mechanisms (BGM), which ensures that the functional relation between the exogenous noise $E$ and the endogenous variable $Y$ is bijective and differentiable in both directions at every level of the cause variable $X = x$. BGM generalizes a variety of models including additive noise model, LiNGAM, post-nonlinear model, and location-scale noise model. Further, we derive a statistical test to find the parents set of the target variable. Experiments on various synthetic and real-world datasets validate our theoretical findings.

Figures (3)

• Figure 1: Left: Density functions $p^1_V,p^2_V$ for random variable $V$. Values of density functions at point $v$ are also shown in this plot. Right: Points on the gray curve are vectorization of values of density functions $p^1_V,p^2_V$ for each point $v$ in $\mathbb{R}$. Projecting the density vector $(p^1_V(v),p^2_V(v))^T$ on the simplex $S_2$ yields $\Phi_V(v)$ which is shown on the plot.
• Figure 2: 1000 datapoints in two domains, the marginals of $X$ and $Y$, the regression lines in each domain, and the true invariant component $f$.
• Figure 3: Multivariate case: Performance of our methods (H1 and H2) as well as previous work.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5: Identical r.v.s
• Definition 6: Similar r.v.s
• Proposition 1
• Definition 7
• Definition 8
• Proposition 2