Linear-Time Algorithms for Front-Door Adjustment in Causal Graphs

TL;DR

Problem: Identify and estimate the total causal effect $P({\bf y} | do({\bf x}))$ when unobserved confounding precludes covariate adjustment, by using front-door sets in a DAG. Approach: develop linear-time algorithms that (i) find a front-door adjustment set ${\bf Z}$ in $O(n+m)$, (ii) enumerate all front-door sets with $O(n(n+m))$ delay, and (iii) compute an inclusion-minimal front-door set in $O(n+m)$ time, exploiting Bayes-Ball for $d$-separation and a linear-time forbidden-vertex propagation. Contributions: first linear-time FD set finder, $O(n(n+m))$-delay enumeration, and a linear-time minimal-FD-set finder, with multi-language implementations and large-scale empirical validation. Significance: speeds up front-door identifications to be comparable with back-door methods, enabling practical causal effect estimation in large DAGs; future work includes finding true minimum-size FD sets efficiently and handling causal discovery over Markov-equivalence classes.

Abstract

Causal effect estimation from observational data is a fundamental task in empirical sciences. It becomes particularly challenging when unobserved confounders are involved in a system. This paper focuses on front-door adjustment -- a classic technique which, using observed mediators allows to identify causal effects even in the presence of unobserved confounding. While the statistical properties of the front-door estimation are quite well understood, its algorithmic aspects remained unexplored for a long time. In 2022, Jeong, Tian, and Bareinboim presented the first polynomial-time algorithm for finding sets satisfying the front-door criterion in a given directed acyclic graph (DAG), with an $O(n^3(n+m))$ run time, where $n$ denotes the number of variables and $m$ the number of edges of the causal graph. In our work, we give the first linear-time, i.e., $O(n+m)$, algorithm for this task, which thus reaches the asymptotically optimal time complexity. This result implies an $O(n(n+m))$ delay enumeration algorithm of all front-door adjustment sets, again improving previous work by a factor of $n^3$. Moreover, we provide the first linear-time algorithm for finding a minimal front-door adjustment set. We offer implementations of our algorithms in multiple programming languages to facilitate practical usage and empirically validate their feasibility, even for large graphs.

Figures (8)

• Figure 1: Causal graphs, where $X$ is the treatment, $Y$ the outcome, and $U$ represents an unobserved confounder. Graph (i) is a canonical example, with the (unique) set $\{Z\}$ satisfying the front-door (FD) criterion relative to $(X,Y)$. For graph (ii), there exist 13 FD sets; Both the algorithm of jeong2022finding as well as our basic Algorithm \ref{['alg:finding']}, output ${\bf Z}=\{A,B,C,D\}$ of maximum size. In contrast, Algorithm \ref{['alg:finding-minimal']} computes minimal FD set $\{D\}$ of size 1. Graph (iii) illustrates the non-monotonicity of the FD criterion: while both $\{A,B,C\}$ and $\{A\}$ are FD sets, neither $\{A,B\}$ nor $\{A,C\}$ nor $\{B,C\}$ satisfy the FD criterion.
• Figure 2: Running example for the algorithms for finding FD sets in $O(n+m)$ given in this section. Nodes in ${\bf Z}_{(\mathrm{i})}$ are marked green and nodes in ${\bf Z}_{(\mathrm{ii})}$ are marked blue.
• Figure 3: Example graph for finding a minimal FD set.
• Figure 4: Log-Log plot of the average run time in seconds for jeong2022finding (jtb), Algorithm \ref{['alg:finding']} (find) and Algorithm \ref{['alg:finding-minimal']} (min) on Erdős-Rényi graphs with $1.5n$ (left) and $5n$ (right) edges, corresponding to expected vertex degree $3$ and $10$. $|{\bf X}|, |{\bf Y}|$ are random integers between $1$ and $3$; $|{\bf I}| = 0$ and $|{\bf R}| = 0.5n$. For each choice of $n$, we average over 50 graphs.
• Figure 5: Run time comparison between find (Algorithm \ref{['alg:finding']}), min (Algorithm \ref{['alg:finding-minimal']}) and jtb (jeong2022finding) for additional parameter choices. In particular, we choose $|{\bf E}| = 2.5n$ (corresponding to expected degree $5$) and vary the choices of $|{\bf R}|$ and $|{\bf X}|$, $|{\bf Y}|$. As can be seen the run time differences are minor.

Theorems & Definitions (24)

• Lemma 1
• proof
• Definition 1
• Lemma 2
• proof
• Theorem 1
• Theorem 2
• proof
• Corollary 1
• Definition 2