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On Learning Causal Structures from Non-Experimental Data without Any Faithfulness Assumption

Hanti Lin, Jiji Zhang

TL;DR

This paper tackles learning the causal structure of a fixed set of variables from non-experimental data without assuming faithfulness. It introduces a framework of nine joint modes of convergence, focusing on almost-everywhere convergence, maximal domain, and adherent locally uniform convergence, and proves that achieving the best combination of these modes forces convergence on all faithful CBNs, thereby justifying the standard faithfulness-based design. A strengthened result further ties the required convergence to the class of $\text{u}$-minimal causal states, clarifying where convergence must or may be sacrificed. The work combines topological and statistical methods to establish existence results for learning procedures and to explain why sacrificing convergence on unfaithful models is unavoidable under strong convergence criteria. Overall, the paper provides a principled rationale for the common practice of relying on faithfulness in causal discovery from non-experimental data and charts precise conditions under which this practice becomes mandatory.

Abstract

Consider the problem of learning, from non-experimental data, the causal (Markov equivalence) structure of the true, unknown causal Bayesian network (CBN) on a given, fixed set of (categorical) variables. This learning problem is known to be so hard that there is no learning algorithm that converges to the truth for all possible CBNs (on the given set of variables). So the convergence property has to be sacrificed for some CBNs---but for which? In response, the standard practice has been to design and employ learning algorithms that secure the convergence property for at least all the CBNs that satisfy the famous faithfulness condition, which implies sacrificing the convergence property for some CBNs that violate the faithfulness condition (Spirtes et al. 2000). This standard design practice can be justified by assuming---that is, accepting on faith---that the true, unknown CBN satisfies the faithfulness condition. But the real question is this: Is it possible to explain, without assuming the faithfulness condition or any of its weaker variants, why it is mandatory rather than optional to follow the standard design practice? This paper aims to answer the above question in the affirmative. We first define an array of modes of convergence to the truth as desiderata that might or might not be achieved by a causal learning algorithm. Those modes of convergence concern (i) how pervasive the domain of convergence is on the space of all possible CBNs and (ii) how uniformly the convergence happens. Then we prove a result to the following effect: for any learning algorithm that tackles the causal learning problem in question, if it achieves the best achievable mode of convergence (considered in this paper), then it must follow the standard design practice of converging to the truth for at least all CBNs that satisfy the faithfulness condition---it is a requirement, not an option.

On Learning Causal Structures from Non-Experimental Data without Any Faithfulness Assumption

TL;DR

This paper tackles learning the causal structure of a fixed set of variables from non-experimental data without assuming faithfulness. It introduces a framework of nine joint modes of convergence, focusing on almost-everywhere convergence, maximal domain, and adherent locally uniform convergence, and proves that achieving the best combination of these modes forces convergence on all faithful CBNs, thereby justifying the standard faithfulness-based design. A strengthened result further ties the required convergence to the class of -minimal causal states, clarifying where convergence must or may be sacrificed. The work combines topological and statistical methods to establish existence results for learning procedures and to explain why sacrificing convergence on unfaithful models is unavoidable under strong convergence criteria. Overall, the paper provides a principled rationale for the common practice of relying on faithfulness in causal discovery from non-experimental data and charts precise conditions under which this practice becomes mandatory.

Abstract

Consider the problem of learning, from non-experimental data, the causal (Markov equivalence) structure of the true, unknown causal Bayesian network (CBN) on a given, fixed set of (categorical) variables. This learning problem is known to be so hard that there is no learning algorithm that converges to the truth for all possible CBNs (on the given set of variables). So the convergence property has to be sacrificed for some CBNs---but for which? In response, the standard practice has been to design and employ learning algorithms that secure the convergence property for at least all the CBNs that satisfy the famous faithfulness condition, which implies sacrificing the convergence property for some CBNs that violate the faithfulness condition (Spirtes et al. 2000). This standard design practice can be justified by assuming---that is, accepting on faith---that the true, unknown CBN satisfies the faithfulness condition. But the real question is this: Is it possible to explain, without assuming the faithfulness condition or any of its weaker variants, why it is mandatory rather than optional to follow the standard design practice? This paper aims to answer the above question in the affirmative. We first define an array of modes of convergence to the truth as desiderata that might or might not be achieved by a causal learning algorithm. Those modes of convergence concern (i) how pervasive the domain of convergence is on the space of all possible CBNs and (ii) how uniformly the convergence happens. Then we prove a result to the following effect: for any learning algorithm that tackles the causal learning problem in question, if it achieves the best achievable mode of convergence (considered in this paper), then it must follow the standard design practice of converging to the truth for at least all CBNs that satisfy the faithfulness condition---it is a requirement, not an option.

Paper Structure

This paper contains 14 sections, 19 theorems, 23 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Review of Standard Definitions
  3. Key Definitions: Modes of Convergence
  4. Main Result & Discussion
  5. Main Result Strengthened
  6. Proof of an Important Lemma
  7. Closing: Some Possibilities for Future Research
  8. Proof of the Existence Result (Clause 2) of Theorem \ref{['the:general1']}
  9. Topological Preliminaries
  10. Statistical Preliminaries
  11. Construction of Learning Methods
  12. Proof of Theorem \ref{['the:general2']}
  13. An Illustrated Explanation of Why Clause 3 of Theorem \ref{['the:general1']} Holds
  14. Some More Details of the Example in Section \ref{['sec:setup1']}

Key Result

Theorem 1

Let $\hbox{$\left( \mathcal{V}, \mathcal{S}, \mathcal{H} \right)$}$ be any causal learning problem such that $\mathcal{V}$ is a finite set of categorical variables, $\mathcal{S}$ is the state space consisting of all causal states on $\mathcal{V}$ (i.e., all causal Bayesian networks on $\mathcal{V}$)

Figures (8)

  • Figure 1: Modes of convergence to the truth
  • Figure 2: Two spaces of causal Bayesian networks
  • Figure 3: Summary of theorem \ref{['the:general2']}
  • Figure 4: Constructions in the proof of lemma \ref{['lem-crucial']}
  • Figure 5: Conditional probability tables
  • ...and 3 more figures

Theorems & Definitions (19)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • Lemma 10
  • ...and 9 more