[Paper Review] Identifying Causal Effects with Computer Algebra

Luis David García Puente, Sarah Spielvogel+1 more|arXiv (Cornell University)|Jul 22, 2010

Bayesian Modeling and Causal Inference6 references20 citations

TL;DR

This paper presents a computer algebra framework using Gröbner bases to systematically determine whether causal effects in linear structural equation models (SEMs) are generically identifiable. It applies this method to all 4096 directed and bidirected graphs on four variables, finding that 1,246 are generically identifiable, 6 require algebraic identification beyond standard criteria, and 2,844 are not generically identifiable, offering the first complete classification for such models on four nodes.

ABSTRACT

The long-standing identification problem for causal effects in graphical models has many partial results but lacks a systematic study. We show how computer algebra can be used to either prove that a causal effect can be identified, generically identified, or show that the effect is not generically identifiable. We report on the results of our computations for linear structural equation models, where we determine precisely which causal effects are generically identifiable for all graphs on three and four vertices.

Motivation & Objective

• To develop a systematic computational framework for determining generic identifiability of causal effects in linear SEMs.
• To classify all possible causal effects in linear SEMs on three and four variables for generic identifiability.
• To identify cases where standard graphical criteria (e.g., back-door, single-door) fail and algebraic methods are required.
• To analyze the computational complexity and structural properties of non-identifiable models.
• To provide a searchable online resource of identifiability results for small graphical models.

Proposed method

• The method uses polynomial mappings from parameters to moments (e.g., covariance matrices) to model SEMs.
• It formulates the identifiability problem as determining whether a parameter lies in the image of the moment map, using algebraic geometry techniques.
• Gröbner basis computations are used to compute the vanishing ideal of the image, which encodes all constraints on the moment parameters.
• Generic identifiability is determined by checking whether the parameter function is algebraically expressible in terms of the moments via the ideal.
• The framework leverages computational algebra systems to automate the identification process across all graphs on three and four vertices.
• The method detects when identifiability requires solving polynomial equations (e.g., quadratic), beyond standard graphical criteria.

Experimental results

Research questions

• RQ1 Which causal effects in linear SEMs on four variables are generically identifiable?
• RQ2 Are there models where standard graphical criteria (e.g., back-door, single-door) fail to identify causal effects, requiring algebraic computation?
• RQ3 What is the computational complexity of proving non-identifiability in SEMs, and what structural features cause long-running times?
• RQ4 How many of the 4,096 four-variable SEMs have non-determinantal vanishing ideals, indicating non-standard constraints?
• RQ5 Can a complete classification of identifiability be achieved for small graphical models using computer algebra?

Key findings

• Out of 4,096 graphs on four variables, exactly 1,246 are generically identifiable, 6 are algebraically 2-identified, and 2,844 are not generically identifiable.
• Among the 1,246 generically identifiable models, 1,093 are identifiable via standard criteria (e.g., single-door, instrumental variables), while 153 require algebraic methods.
• There are exactly 729 bow-free models on four variables, all of which are generically identifiable via the single-door criterion.
• The vanishing ideal of every three-variable SEM is determinantal, but 33 of the 4,096 four-variable SEMs have non-determinantal ideals.
• Some models, such as the one with directed edges 1→2, 1→4, 2→3, 3→4 and bidirected edges 1↔2, 1↔3, 1↔4, required over 75 days to prove non-identifiability of ω₄₄.
• The six algebraically 2-identified models all involve parameters that are solutions to quadratic equations derived from the covariance matrix, even when standard criteria fail.