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Parameter identification in linear non-Gaussian causal models under general confounding

Daniele Tramontano, Mathias Drton, Jalal Etesami

Abstract

Linear non-Gaussian causal models postulate that each random variable is a linear function of parent variables and non-Gaussian exogenous error terms. We study identification of the linear coefficients when such models contain latent variables. Our focus is on the commonly studied acyclic setting, where each model corresponds to a directed acyclic graph (DAG). For this case, prior literature has demonstrated that connections to overcomplete independent component analysis yield effective criteria to decide parameter identifiability in latent variable models. However, this connection is based on the assumption that the observed variables linearly depend on the latent variables. Departing from this assumption, we treat models that allow for arbitrary non-linear latent confounding. Our main result is a graphical criterion that is necessary and sufficient for deciding the generic identifiability of direct causal effects. Moreover, we provide an algorithmic implementation of the criterion with a run time that is polynomial in the number of observed variables. Finally, we report on estimation heuristics based on the identification result and explore a generalization to models with feedback loops.

Parameter identification in linear non-Gaussian causal models under general confounding

Abstract

Linear non-Gaussian causal models postulate that each random variable is a linear function of parent variables and non-Gaussian exogenous error terms. We study identification of the linear coefficients when such models contain latent variables. Our focus is on the commonly studied acyclic setting, where each model corresponds to a directed acyclic graph (DAG). For this case, prior literature has demonstrated that connections to overcomplete independent component analysis yield effective criteria to decide parameter identifiability in latent variable models. However, this connection is based on the assumption that the observed variables linearly depend on the latent variables. Departing from this assumption, we treat models that allow for arbitrary non-linear latent confounding. Our main result is a graphical criterion that is necessary and sufficient for deciding the generic identifiability of direct causal effects. Moreover, we provide an algorithmic implementation of the criterion with a run time that is polynomial in the number of observed variables. Finally, we report on estimation heuristics based on the identification result and explore a generalization to models with feedback loops.
Paper Structure (29 sections, 25 theorems, 100 equations, 16 figures)

This paper contains 29 sections, 25 theorems, 100 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Related Work
  3. Organization of the Paper
  4. Notation
  5. Linear Mixed Graph Models and Identifiability
  6. Necessary and Sufficient Conditions for Generic Identifiability of Direct Causal Effects
  7. Certifying Identifiability
  8. The Maximum Flow Problem
  9. Deciding Generic Identifiability
  10. The Genericity Condition for the Error Distribution
  11. Linear Factor Models
  12. Random Variables with Finite Moments
  13. Cyclic Graphs
  14. Computational Experiments
  15. Certifying Identifiability
  16. ...and 14 more sections

Key Result

Lemma 3.1

The entries of matrix $A$ defined in eq:A:matrix can be written as In particular, we have $a_{vu}=0$ if $v\notin\mathop{\rm de}\nolimits(u)$, and $a_{uu}=1$ for every $u\in V$.

Figures (16)

  • Figure 1: Instrumental variable graph based on evans:1999.
  • Figure 2: An acyclic directed mixed graphs (ADMG) with 4 nodes.
  • Figure 3: (a) A non-intersecting systems of directed paths from $\{u_1,u_2\}$ to $\{w_1,w_2\}$. (b) Two intersecting directed paths with node $c$ in their intersection.
  • Figure 4: An ADMG for which only one of the causal effects is identifiable.
  • Figure 5: Double confounder graph.
  • ...and 11 more figures

Theorems & Definitions (76)

  • Example 1.1
  • Definition 2.1
  • Example 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Example 3.1
  • Lemma 3.3
  • proof
  • ...and 66 more