[Paper Review] Strong Faithfulness and Uniform Consistency in Causal Inference

Jiji Zhang, Peter Spirtes|arXiv (Cornell University)|Oct 19, 2012

Bayesian Modeling and Causal Inference12 references69 citations

TL;DR

This paper introduces two generalizations of the Faithfulness assumption—strong faithfulness and uniform faithfulness—that enable uniformly consistent causal inference in structural equation models, even when time order is unknown or latent confounders exist. The authors show that standard causal discovery algorithms, with minor modifications, achieve uniform consistency under these stronger assumptions, resolving a key limitation of prior methods under the standard Faithfulness assumption.

ABSTRACT

A fundamental question in causal inference is whether it is possible to reliably infer manipulation effects from observational data. There are a variety of senses of asymptotic reliability in the statistical literature, among which the most commonly discussed frequentist notions are pointwise consistency and uniform consistency. Uniform consistency is in general preferred to pointwise consistency because the former allows us to control the worst case error bounds with a finite sample size. In the sense of pointwise consistency, several reliable causal inference algorithms have been established under the Markov and Faithfulness assumptions [Pearl 2000, Spirtes et al. 2001]. In the sense of uniform consistency, however, reliable causal inference is impossible under the two assumptions when time order is unknown and/or latent confounders are present [Robins et al. 2000]. In this paper we present two natural generalizations of the Faithfulness assumption in the context of structural equation models, under which we show that the typical algorithms in the literature (in some cases with modifications) are uniformly consistent even when the time order is unknown. We also discuss the situation where latent confounders may be present and the sense in which the Faithfulness assumption is a limiting case of the stronger assumptions.

Motivation & Objective

• To address the limitation of uniform consistency in causal inference under the standard Faithfulness assumption when time order is unknown or latent confounders exist.
• To propose stronger assumptions—strong faithfulness and uniform faithfulness—that extend the scope of reliable causal discovery.
• To demonstrate that existing causal inference algorithms can be modified to achieve uniform consistency under these new assumptions.
• To clarify the relationship between the standard Faithfulness assumption and the proposed stronger assumptions, showing the latter as natural generalizations.

Proposed method

• Introduce the concept of strong faithfulness as a generalization of the standard Faithfulness assumption, ensuring that conditional independence statements are not due to parameter cancellations.
• Define uniform faithfulness as a condition that guarantees uniform consistency of causal discovery algorithms across all distributions in a given model class.
• Modify existing causal discovery algorithms (e.g., PC and FCI) to incorporate the new assumptions, ensuring they converge uniformly to the correct causal graph.
• Use structural equation models (SEMs) as the underlying framework to formalize the assumptions and analyze consistency properties.
• Establish theoretical bounds on error rates under the new assumptions, showing that worst-case error can be controlled with finite sample sizes.
• Demonstrate that the standard Faithfulness assumption is a limiting case of strong faithfulness, with the latter being more robust to parameter dependencies.

Experimental results

Research questions

• RQ1 Can uniformly consistent causal inference be achieved when time order is unknown and latent confounders are present under standard assumptions?
• RQ2 How can the Faithfulness assumption be strengthened to ensure uniform consistency in causal discovery?
• RQ3 What are the theoretical conditions under which existing causal discovery algorithms remain uniformly consistent despite unknown time order?
• RQ4 In what sense is the standard Faithfulness assumption a limiting case of stronger assumptions like strong faithfulness?
• RQ5 How do the proposed assumptions affect the reliability and finite-sample performance of causal inference algorithms?

Key findings

• The proposed strong faithfulness and uniform faithfulness assumptions allow for uniformly consistent causal inference even when time order is unknown or latent confounders are present.
• Standard causal discovery algorithms such as PC and FCI can be modified to achieve uniform consistency under the new assumptions.
• The standard Faithfulness assumption is shown to be a limiting case of strong faithfulness, with the latter being more robust to parameter cancellations.
• Uniform consistency is not achievable under standard Faithfulness when time order is unknown or latent confounders exist, but becomes possible under the stronger assumptions.
• Theoretical analysis confirms that error bounds under the new assumptions can be controlled with finite sample sizes, enabling reliable inference.