[Paper Review] Causal inference via algebraic geometry: necessary and sufficient conditions for the feasibility of discrete causal models

Ciarán M. Lee, Robert W. Spekkens|arXiv (Cornell University)|Jun 12, 2015

Bayesian Modeling and Causal Inference3 citations

TL;DR

This paper develops a causal inference framework using algebraic geometry to determine necessary and sufficient conditions for the feasibility of discrete causal models with two binary observed variables, allowing arbitrary functional dependencies and latent variables without additive noise. It provides an inductive classification of observational equivalence classes and derives algebraic constraints on joint distributions to test model feasibility.

ABSTRACT

We provide a scheme for inferring causal relations from uncontrolled statistical data which makes use of all of the information in the joint probability distribution over the observed variables rather than just the conditional independence relations. We focus on causal models containing just two observed variables, each of which is binary. We allow any number of latent variables and we do not impose any restriction on the manner in which the observed variables may depend functionally on the latent ones. In particular, the noise need not be additive. We provide an inductive scheme for classifying causal models into distinct observational equivalence classes. For each observational equivalence class, we provide a procedure for deriving, using techniques from algebraic geometry, necessary and sufficient conditions on the joint distribution for the feasibility of the class. Connections and applications of these results to the emerging field of quantum causal models are also discussed.

Motivation & Objective

• To develop a comprehensive method for causal inference from uncontrolled statistical data using the full joint distribution, not just conditional independence.
• To classify causal models with two binary observed variables into observational equivalence classes, regardless of functional form or latent structure.
• To derive necessary and sufficient algebraic conditions on the joint distribution for the feasibility of each equivalence class using algebraic geometry.
• To extend the applicability of causal modeling to non-additive noise and complex functional dependencies in latent variable models.
• To explore connections between classical discrete causal models and emerging quantum causal models through algebraic constraints.

Proposed method

• The method uses an inductive scheme to group causal models into observational equivalence classes based on their observable distributions.
• For each equivalence class, it applies techniques from algebraic geometry to derive polynomial constraints on the joint probability distribution that must be satisfied for the model to be feasible.
• It leverages the structure of the joint distribution over two binary observed variables to characterize all possible functional dependencies on latent variables, including non-additive noise.
• The approach identifies algebraic varieties corresponding to each equivalence class, enabling feasibility testing via polynomial ideal membership.
• It constructs a systematic procedure to determine whether a given joint distribution lies within the algebraic variety defined by a causal model.
• The method does not require parametric assumptions or conditional independence testing, relying instead on full distributional information.

Experimental results

Research questions

• RQ1 What are the necessary and sufficient algebraic conditions for a joint distribution to be compatible with a given discrete causal model involving two binary observed variables and arbitrary latent structures?
• RQ2 How can causal models be systematically grouped into observational equivalence classes when functional dependencies on latent variables are non-additive and unspecified?
• RQ3 What role does algebraic geometry play in characterizing the set of feasible joint distributions for a given causal structure with latent variables?
• RQ4 How do the derived algebraic constraints generalize or differ from those based on conditional independence in traditional causal discovery?
• RQ5 In what ways can these algebraic methods be extended to or inform the development of quantum causal models?

Key findings

• The paper establishes that the set of feasible joint distributions for a given causal model forms a semi-algebraic set, characterized by polynomial inequalities and equalities derived from the model's structure.
• It provides a complete classification of observational equivalence classes for two-binary-variable models, even when functional dependencies on latent variables are arbitrary and noise is non-additive.
• For each equivalence class, the method derives explicit polynomial constraints on the joint distribution that are both necessary and sufficient for model feasibility.
• The approach enables the detection of causal structure without relying on conditional independence tests, using the full joint distribution as input.
• The framework reveals that algebraic geometry offers a more comprehensive characterization of causal feasibility than traditional methods based on d-separation or conditional independence.
• Connections to quantum causal models are identified, suggesting that similar algebraic techniques may underlie the feasibility conditions in quantum contexts.