Sequentially learning the topological ordering of causal directed acyclic graphs with likelihood ratio scores

TL;DR

This paper introduces a sequential likelihood ratio-based method for reliably learning the topological order of causal DAGs with general error distributions, scalable to high-dimensional data.

Contribution

It presents a novel, efficient algorithm for identifying the true DAG ordering using simple likelihood ratio scores, with proven consistency and practical scalability.

Findings

Algorithm accurately recovers DAG orderings in simulations.

Scales efficiently to thousands of nodes in high-dimensional settings.

Performs well on real single-cell gene expression data.

Abstract

Causal discovery, the learning of causality in a data mining scenario, has been of strong scientific and theoretical interest as a starting point to identify "what causes what?" Contingent on assumptions and a proper learning algorithm, it is sometimes possible to identify and accurately estimate a causal directed acyclic graph (DAG), as opposed to a Markov equivalence class of graphs that gives ambiguity of causal directions. The focus of this paper is in highlighting the identifiability and estimation of DAGs with general error distributions through a general sequential sorting procedure that orders variables one at a time, starting at root nodes, followed by children of the root nodes, and so on until completion. We demonstrate a novel application of this general approach to estimate the topological ordering of a DAG. At each step of the procedure, only simple likelihood ratio scores are calculated on regression residuals to decide the next node to append to the current partial ordering. The computational complexity of our algorithm on a p-node problem is O(pd), where d is the maximum neighborhood size. Under mild assumptions, the population version of our procedure provably identifies a true ordering of the underlying DAG. We provide extensive numerical evidence to demonstrate that this sequential procedure scales to possibly thousands of nodes and works well for high-dimensional data. We accompany these numerical experiments with an application to a single-cell gene expression dataset.