[Paper Review] Structure Learning for Cyclic Linear Causal Models

Carlos Améndola, Philipp Dettling+3 more|arXiv (Cornell University)|Jun 10, 2020

Bayesian Modeling and Causal Inference30 references4 citations

TL;DR

This paper proposes a score-based structure learning method for linear Gaussian causal models with feedback loops and latent confounders using simple mixed graphs (no multiple edges between nodes). Despite cyclic dependencies, the authors prove these models have expected dimension, enabling valid model selection scores; they further generalize a criterion for distributional equivalence from acyclic to cyclic graphs, validating greedy search on real protein expression data with consistent biological findings.

ABSTRACT

We consider the problem of structure learning for linear causal models based on observational data. We treat models given by possibly cyclic mixed graphs, which allow for feedback loops and effects of latent confounders. Generalizing related work on bow-free acyclic graphs, we assume that the underlying graph is simple. This entails that any two observed variables can be related through at most one direct causal effect and that (confounding-induced) correlation between error terms in structural equations occurs only in absence of direct causal effects. We show that, despite new subtleties in the cyclic case, the considered simple cyclic models are of expected dimension and that a previously considered criterion for distributional equivalence of bow-free acyclic graphs has an analogue in the cyclic case. Our result on model dimension justifies in particular score-based methods for structure learning of linear Gaussian mixed graph models, which we implement via greedy search.

Motivation & Objective

• To address the challenge of structure learning in linear Gaussian causal models with feedback loops and latent confounders.
• To establish conditions under which such models have the expected dimension, enabling valid score-based learning.
• To generalize a criterion for distributional equivalence from bow-free acyclic graphs to cyclic simple mixed graphs.
• To develop and evaluate a greedy search algorithm for structure learning in cyclic linear Gaussian models.
• To validate the method on real biological data, demonstrating consistency with known regulatory pathways.

Proposed method

• Models are represented using simple mixed graphs with directed edges (causal effects) and bidirected edges (correlated errors from latent confounders).
• The structural equation system is defined as $ X = \Lambda^T X + \varepsilon $, with $ \varepsilon \sim \mathcal{N}(0, \Omega) $, and $ \operatorname{Var}[X] = (I - \Lambda)^{-T} \Omega (I - \Lambda)^{-1} $.
• A greedy search algorithm is used to maximize a bias-corrected BIC-like score based on the estimated correlation matrix from data.
• The method uses Kendall’s tau correlation matrix as input, projected to a $ \frac{p(p-1)}{2} $-dimensional space to reduce dimensionality.
• The search is repeated from random initial graphs with increased penalty in BIC to avoid overfitting.
• Distributional equivalence is assessed using a generalized version of a criterion previously valid only for bow-free acyclic graphs.

Experimental results

Research questions

• RQ1 Can linear Gaussian mixed graph models with cycles still have the expected dimension despite non-identifiability issues?
• RQ2 Is there a tractable criterion for distributional equivalence in cyclic simple mixed graphs, analogous to the one in acyclic bow-free graphs?
• RQ3 Can score-based structure learning be effectively applied to cyclic linear Gaussian models with latent confounders?
• RQ4 How well do greedy search methods recover known biological regulatory relationships in real data?
• RQ5 What is the impact of cycle presence on the structure of learned models in real-world datasets?

Key findings

• Simple cyclic mixed graphs induce models of expected dimension, validating the use of score-based methods for structure learning.
• A previously known criterion for distributional equivalence in bow-free acyclic graphs has a natural generalization to the cyclic case.
• Greedy search based on bias-corrected Kendall’s tau correlation matrices successfully recovers known biological pathways in 14 protein expression datasets.
• The estimated graphs contain between 10 and 13 edges, with 4 out of 14 graphs containing cycles (3 of type 3-cycle, 1 of type 4-cycle).
• The method recovers 12 out of 14 known regulatory relationships (e.g., PLCG-PIP2-PIP3, PKC-P38-PJNK, P44/42-PKA-PAKTS473) consistently across all datasets.
• Three novel connections (PIP2→PKC, PLCG→PKC, PIP3→PAKTS473) were detected in multiple datasets, not previously reported in the literature.