[Paper Review] Data Analysis with Bayesian Networks: A Bootstrap Approach

Nir Friedman, Moisés Goldszmidt+1 more|arXiv (Cornell University)|Jan 23, 2013

Bayesian Modeling and Causal Inference18 references266 citations

TL;DR

This paper introduces a bootstrap-based approach to assess confidence in Bayesian network structures derived from data, enabling reliable inference about edge existence, Markov blankets, and variable ordering—even with limited data. By resampling data and analyzing structural stability, the method enhances structure learning and detects latent variables, improving robustness and interpretability in complex data analysis.

ABSTRACT

In recent years there has been significant progress in algorithms and methods for inducing Bayesian networks from data. However, in complex data analysis problems, we need to go beyond being satisfied with inducing networks with high scores. We need to provide confidence measures on features of these networks: Is the existence of an edge between two nodes warranted? Is the Markov blanket of a given node robust? Can we say something about the ordering of the variables? We should be able to address these questions, even when the amount of data is not enough to induce a high scoring network. In this paper we propose Efron's Bootstrap as a computationally efficient approach for answering these questions. In addition, we propose to use these confidence measures to induce better structures from the data, and to detect the presence of latent variables.

Motivation & Objective

• Address the need for confidence measures in Bayesian network structures when data is scarce.
• Improve reliability of network features such as edges, Markov blankets, and variable orderings.
• Develop a computationally efficient method to quantify uncertainty in learned Bayesian networks.
• Enhance structure learning by incorporating confidence measures into the scoring process.
• Detect the presence of latent (unobserved) variables through structural instability analysis.

Proposed method

• Apply Efron's bootstrap resampling to generate multiple datasets from the original data.
• Learn a Bayesian network structure from each bootstrap sample to assess feature stability.
• Use the frequency of edge or structure feature occurrence across bootstrap samples as a confidence measure.
• Integrate confidence scores into the network scoring function to guide structure learning.
• Identify latent variables by detecting high structural variability in the Markov blankets of nodes.
• Apply the bootstrap framework to both score-based structure learning and model validation.

Experimental results

Research questions

• RQ1 How can we quantify confidence in the existence of edges between nodes in a Bayesian network?
• RQ2 To what extent is the Markov blanket of a node stable across different data samples?
• RQ3 Can we reliably infer the ordering of variables in a Bayesian network using bootstrap-based confidence measures?
• RQ4 How does incorporating confidence into structure learning improve the quality of induced networks?
• RQ5 Can structural instability detected via bootstrapping reveal the presence of latent variables?

Key findings

• Bootstrap resampling provides reliable confidence measures for network features even with small datasets.
• Edges that appear frequently across bootstrap samples are more likely to be statistically warranted.
• Markov blankets showing high variability across bootstrap samples suggest potential latent confounders.
• Incorporating confidence scores into structure learning leads to more robust and accurate network structures.
• The method successfully detects latent variables by identifying unstable or inconsistent network components.
• The approach is computationally efficient and scalable, making it practical for real-world data analysis tasks.