Trek-Based Parameter Identification for Linear Causal Models With Arbitrarily Structured Latent Variables

TL;DR

This paper introduces a new graphical criterion and algorithm for identifying causal effects in linear structural equation models with complex latent variables, extending beyond traditional assumptions of independent latent factors.

Contribution

The authors develop a novel latent-subgraph criterion and an integer linear programming algorithm for causal effect identifiability in models with arbitrarily structured latent variables.

Findings

The criterion is sufficient for causal effect identifiability via covariance matrix formulas.

The algorithm is sound and complete for checking the criterion.

The approach can simplify models to apply existing identification tools.

Abstract

We develop a criterion to certify whether causal effects are identifiable in linear structural equation models with latent variables. Linear structural equation models correspond to directed graphs whose nodes represent the random variables of interest and whose edges are weighted with linear coefficients that correspond to direct causal effects. In contrast to previous identification methods, we do not restrict ourselves to settings where the latent variables constitute independent latent factors (i.e., to source nodes in the graphical representation of the model). Our novel latent-subgraph criterion is a purely graphical condition that is sufficient for identifiability of causal effects by rational formulas in the covariance matrix. To check the latent-subgraph criterion, we provide a sound and complete algorithm that operates by solving an integer linear program. While it targets effects involving observed variables, our new criterion is also useful for identifying effects between latent variables, as it allows one to transform the given model into a simpler measurement model for which other existing tools become applicable.