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Learning linear acyclic causal model including Gaussian noise using ancestral relationships

Ming Cai, Penggang Gao, Hisayuki Hara

TL;DR

An algorithm for learning the distribution-equivalence patterns of a linear causal model with a lower time complexity than PC-LiNGAM is proposed, using the causal ancestor finding algorithm in Maeda and Shimizu, which is generalized to account for Gaussian disturbances.

Abstract

This paper discusses algorithms for learning causal DAGs. The PC algorithm makes no assumptions other than the faithfulness to the causal model and can identify only up to the Markov equivalence class. LiNGAM assumes linearity and continuous non-Gaussian disturbances for the causal model, and the causal DAG defining LiNGAM is shown to be fully identifiable. The PC-LiNGAM, a hybrid of the PC algorithm and LiNGAM, can identify up to the distribution-equivalence pattern of a linear causal model, even in the presence of Gaussian disturbances. However, in the worst case, the PC-LiNGAM has factorial time complexity for the number of variables. In this paper, we propose an algorithm for learning the distribution-equivalence patterns of a linear causal model with a lower time complexity than PC-LiNGAM, using the causal ancestor finding algorithm in Maeda and Shimizu, which is generalized to account for Gaussian disturbances.

Learning linear acyclic causal model including Gaussian noise using ancestral relationships

TL;DR

An algorithm for learning the distribution-equivalence patterns of a linear causal model with a lower time complexity than PC-LiNGAM is proposed, using the causal ancestor finding algorithm in Maeda and Shimizu, which is generalized to account for Gaussian disturbances.

Abstract

This paper discusses algorithms for learning causal DAGs. The PC algorithm makes no assumptions other than the faithfulness to the causal model and can identify only up to the Markov equivalence class. LiNGAM assumes linearity and continuous non-Gaussian disturbances for the causal model, and the causal DAG defining LiNGAM is shown to be fully identifiable. The PC-LiNGAM, a hybrid of the PC algorithm and LiNGAM, can identify up to the distribution-equivalence pattern of a linear causal model, even in the presence of Gaussian disturbances. However, in the worst case, the PC-LiNGAM has factorial time complexity for the number of variables. In this paper, we propose an algorithm for learning the distribution-equivalence patterns of a linear causal model with a lower time complexity than PC-LiNGAM, using the causal ancestor finding algorithm in Maeda and Shimizu, which is generalized to account for Gaussian disturbances.
Paper Structure (20 sections, 20 theorems, 30 equations, 6 figures, 1 algorithm)

This paper contains 20 sections, 20 theorems, 30 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Terminologies and notations on graphs
  3. Related studies
  4. PC algorithm
  5. LiNGAM and variants
  6. PC-LiNGAM
  7. Ancestor finding in RCD
  8. Proposed Algorithm
  9. Ancestor finding in the model containing Gaussian disturbances
  10. Algorithm for finding DEP
  11. Complexity Analysis
  12. Exception handlings
  13. Numerical Experiments
  14. Experimental Settings
  15. Results and Discussion
  16. ...and 5 more sections

Key Result

Proposition 2.2

One of the following four conditions holds for the ancestral relationship between $x_i$ and $x_j$.

Figures (6)

  • Figure 2.1: An example to illustrate the PC-LiNGAM when handling a directed complete DAG with four variables.
  • Figure 3.1: An example to illustrate the proposed method when handling a directed complete DAG over four variables.
  • Figure 3.2: An example to illustrate how Algorithm \ref{['alg: proposed']} outputs an incorrect graph with a V-structure detectable by the PC algorithm and a cycle due to errors in Gaussianity tests and independence tests.
  • Figure 3.3: An example to illustrate the flow of Algorithm \ref{['alg: exception']} to handle the exceptions in Figure \ref{['fig: both']}.
  • Figure 4.1: CPU time of the proposed algorithm and the PC-LiNGAM against the dimension of variables.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Definition 2.1: Hoyer et al. hoyer2008b
  • Proposition 2.2: Maeda and Shimizu Maeda2020
  • Proposition 2.3: Maeda and Shimizu Maeda2020
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Theorem 3.6
  • Lemma A.1: e.g. dfs2011
  • ...and 21 more