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Identification for Tree-shaped Structural Causal Models in Polynomial Time

Aaryan Gupta, Markus Bläser

TL;DR

This work presents a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs whose directed component forms a tree and is not only polynomial time, but also complete.

Abstract

Linear structural causal models (SCMs) are used to express and analyse the relationships between random variables. Direct causal effects are represented as directed edges and confounding factors as bidirected edges. Identifying the causal parameters from correlations between the nodes is an open problem in artificial intelligence. In this paper, we study SCMs whose directed component forms a tree. Van der Zander et al. (AISTATS'22, PLMR 151, pp. 6770--6792, 2022) give a PSPACE-algorithm for the identification problem in this case, which is a significant improvement over the general Gröbner basis approach, which has doubly-exponential time complexity in the number of structural parameters. In this work, we present a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs. For every structural parameter, our algorithms decides whether it is generically identifiable, generically 2-identifiable, or generically unidentifiable. (No other cases can occur.) In the first two cases, it provides one or two fractional affine square root terms of polynomials (FASTPs) for the corresponding parameter, respectively.

Identification for Tree-shaped Structural Causal Models in Polynomial Time

TL;DR

This work presents a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs whose directed component forms a tree and is not only polynomial time, but also complete.

Abstract

Linear structural causal models (SCMs) are used to express and analyse the relationships between random variables. Direct causal effects are represented as directed edges and confounding factors as bidirected edges. Identifying the causal parameters from correlations between the nodes is an open problem in artificial intelligence. In this paper, we study SCMs whose directed component forms a tree. Van der Zander et al. (AISTATS'22, PLMR 151, pp. 6770--6792, 2022) give a PSPACE-algorithm for the identification problem in this case, which is a significant improvement over the general Gröbner basis approach, which has doubly-exponential time complexity in the number of structural parameters. In this work, we present a randomized polynomial-time algorithm, which solves the identification problem for tree-shaped SCMs. For every structural parameter, our algorithms decides whether it is generically identifiable, generically 2-identifiable, or generically unidentifiable. (No other cases can occur.) In the first two cases, it provides one or two fractional affine square root terms of polynomials (FASTPs) for the corresponding parameter, respectively.
Paper Structure (17 sections, 16 theorems, 20 equations, 6 figures, 2 algorithms)

This paper contains 17 sections, 16 theorems, 20 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Results by Van der Zander et al.
  5. Polynomial Identity Testing
  6. The Rank of Edges
  7. Rank-1 Edges
  8. Rank-2 Edges
  9. Computing the Rank of an Edge
  10. Composition and Elimination
  11. Identification Using Cycles
  12. Finding Identifying Cycles
  13. Polynomial Identity Testing
  14. Selfreducibility
  15. An Algorithm for Finding Identifying Cycles
  16. ...and 2 more sections

Key Result

Lemma 1

Let $p(x_1,...,x_n)$ be a non-zero polynomial of total degree $\leq d$ over a field $K$. Let $S\subseteq K$ be a finite set and let $a_1,...,a_n \in S$ be selected uniformly at random. Then $\Pr(p(a_1,a_2,...,a_n)\not= 0))\geq 1- \frac{d}{|S|}$.

Figures (6)

  • Figure 1: SCM $M_1$, a classical IV example
  • Figure 2: Tree-shaped SCM $M_2$
  • Figure 3: An example of the graph $\vec{G}$. For simplicity, we show only one weight for each pair of edges between two nodes. The other edge has the corresponding adjoint matrix as weight.
  • Figure 4: The layered graph $\vec{G}^{(3)}$ corresponding to $\vec{G}$ from Figure \ref{['fig:SCM:ex']}. Weights are not drawn for simplicity. The bold blue path in this graph corresponds to the identifying cycle $x \to u \to z \to x$.
  • Figure 5: If the missing edge $i \leftrightarrow j$ has rank 1, then all the other blue edges have to be missing by trek-separability. The best choice for $T$ is $T = \{q\}$ (drawn yellow).
  • ...and 1 more figures

Theorems & Definitions (33)

  • Lemma 1: pipSchwartz1980ZippelpipZippel1979Schwartz
  • Definition 2
  • Lemma 3
  • proof
  • Lemma 4
  • Remark 5
  • proof
  • Lemma 6
  • Lemma 7
  • proof
  • ...and 23 more