[Paper Review] Strong Completeness and Faithfulness in Bayesian Networks

Christopher Meek|arXiv (Cornell University)|Feb 20, 2013

Bayesian Modeling and Causal InferenceComputer Science13 references199 citations

TL;DR

This paper establishes strong completeness and faithfulness in discrete Bayesian networks by proving that, in a measure-theoretic sense, almost all parameterized distributions over a given network structure are faithful—meaning all conditional independence relations in the distribution are exactly those implied by the d-separation criterion. The result confirms that d-separation is both sound and complete for almost all distributions in the space of discrete Bayesian networks.

ABSTRACT

A completeness result for d-separation applied to discrete Bayesian networks is presented and it is shown that in a strong measure-theoretic sense almost all discrete distributions for a given network structure are faithful; i.e. the independence facts true of the distribution are all and only those entailed by the network structure.

Motivation & Objective

• To establish the completeness of the d-separation criterion in discrete Bayesian networks.
• To investigate whether all conditional independence relations in a distribution are fully captured by the network structure.
• To determine the measure-theoretic prevalence of faithful distributions within the space of all possible distributions for a given network structure.
• To provide theoretical justification for the faithfulness assumption commonly used in causal discovery and structure learning.
• To demonstrate that faithfulness is not an exception but the norm in generic discrete Bayesian networks.

Proposed method

• Uses measure-theoretic analysis to evaluate the set of all possible parameterized distributions for a given Bayesian network structure.
• Applies the d-separation criterion to identify all conditional independence relations entailed by the network structure.
• Proves that the set of distributions that violate faithfulness (i.e., have extra independence relations not implied by d-separation) has Lebesgue measure zero.
• Demonstrates that the space of faithful distributions is dense and full measure in the parameter space of the network.
• Employs algebraic geometry and polynomial constraints to characterize the set of non-faithful distributions as a lower-dimensional submanifold.
• Shows that faithfulness holds generically across all discrete conditional probability distributions compatible with the network structure.

Experimental results

Research questions

• RQ1 Is the d-separation criterion complete for capturing all conditional independence relations in discrete Bayesian networks?
• RQ2 What is the measure-theoretic prevalence of faithful distributions within the space of all distributions for a fixed network structure?
• RQ3 Are there significant classes of distributions that violate faithfulness, or is faithfulness the norm?
• RQ4 Can faithfulness be guaranteed for almost all parameter settings in a discrete Bayesian network?
• RQ5 To what extent does the structure of a Bayesian network determine the independence facts of its associated probability distribution?

Key findings

• For any discrete Bayesian network structure, the set of parameterized distributions that are not faithful has Lebesgue measure zero.
• Almost all distributions over a given Bayesian network structure are faithful to the d-separation criterion.
• The faithfulness assumption is not an arbitrary restriction but holds generically across the parameter space.
• The set of non-faithful distributions forms a lower-dimensional algebraic variety, implying they are rare in a measure-theoretic sense.
• The d-separation criterion is not only sound but also complete for almost all discrete distributions in the network's parameter space.
• This result supports the use of faithfulness as a foundational assumption in causal discovery and structure learning algorithms.