Recursive Causal Discovery

TL;DR

A unified framework for the proposed algorithms is presented, refined with additional details and enhancements for a coherent presentation, and a comprehensive literature review is included, comparing the computational complexity of the methods with existing approaches, showcasing their state-of-the-art efficiency.

Abstract

Causal discovery, i.e., learning the causal graph from data, is often the first step toward the identification and estimation of causal effects, a key requirement in numerous scientific domains. Causal discovery is hampered by two main challenges: limited data results in errors in statistical testing and the computational complexity of the learning task is daunting. This paper builds upon and extends four of our prior publications (Mokhtarian et al., 2021; Akbari et al., 2021; Mokhtarian et al., 2022, 2023a). These works introduced the concept of removable variables, which are the only variables that can be removed recursively for the purpose of causal discovery. Presence and identification of removable variables allow recursive approaches for causal discovery, a promising solution that helps to address the aforementioned challenges by reducing the problem size successively. This reduction not only minimizes conditioning sets in each conditional independence (CI) test, leading to fewer errors but also significantly decreases the number of required CI tests. The worst-case performances of these methods nearly match the lower bound. In this paper, we present a unified framework for the proposed algorithms, refined with additional details and enhancements for a coherent presentation. A comprehensive literature review is also included, comparing the computational complexity of our methods with existing approaches, showcasing their state-of-the-art efficiency. Another contribution of this paper is the release of RCD, a Python package that efficiently implements these algorithms. This package is designed for practitioners and researchers interested in applying these methods in practical scenarios. The package is available at github.com/ban-epfl/rcd, with comprehensive documentation provided at rcdpackage.com.

Figures (6)

• Figure 1: Graphs in Example \ref{['example: must remove removable']}.
• Figure 2: Graphical criterion of removability. Asterisk (*) is used as a wildcard, which indicates that the edge endpoint can be either an arrowhead or a tail. In the case of MAGs, the path $(X,V_1,...,Y)$ is a collider path and $X,...,V_m\in\textit{Pa}_{\mathcal{G}}(Z)$. $X$ is removable if and only if for all such paths, $Y$ and $Z$ are adjacent. In the case of a DAG, it suffices to check this condition for vertices $Y$, where $Y$ is a parent, child, or co-parent of $X$.
• Figure 3: In this figure, $\{\mathcal{G}_1,\dots,\mathcal{G}_k\}$ denotes a set of Markov equivalent DAGs. $\Pi(\mathbf{V})$ denotes the set of orders over $\mathbf{V}$, which is the search space of ordering-based methods. $\Pi^c(\mathcal{G}_i)$ denotes the set of c-orders of $\mathcal{G}_i$, the target space of existing ordering-based methods in the literature. $\Pi^r(\mathcal{G}_i)$ denotes the set of r-orders of $\mathcal{G}_i$, which is the target space of ROL.
• Figure 4: Two Markov equivalent DAGs $\mathcal{G}_1$ and $\mathcal{G}_2$ that form a MEC together and their disjoint sets of c-orders. In this example, any order over $\mathbf{V} = \{X_1,X_2,X_3,X_4\}$ is an r-order, i.e., $\Pi^r(\mathcal{G}_1) = \Pi^r(\mathcal{G}_2) = \Pi(\mathbf{V})$. Note that $|\Pi^r(\mathcal{G}_1)| =|\Pi^r(\mathcal{G}_2)|=24 > 2 = |\Pi^c(\mathcal{G}_1)| =|\Pi^c(\mathcal{G}_2)|$.
• Figure 5: Diamond graphs.

Theorems & Definitions (57)

• Definition 1: $\mathcal{G}[\mathbf{W}]$
• Definition 2: $\Pi(\mathbf{W})$
• Definition 3: $\textit{Pa}_{\mathcal{G}}^+(X)$
• Definition 4: m-separation
• Definition 5: $\text{VS}_{\mathcal{G}}(X)$
• Definition 6: Co-parent
• Definition 7: Discriminating path
• Definition 8: Inducing path
• Definition 9: Latent projection
• Remark 10