[Paper Review] A Generalization of the Chow-Liu Algorithm and its Application to Statistical Learning

Joe Suzuki|arXiv (Cornell University)|Feb 10, 2010

Bayesian Modeling and Causal Inference5 references3 citations

TL;DR

This paper generalizes the Chow-Liu algorithm to handle both Gaussian and finite-valued random variables by extending mutual information estimation and incorporating a penalty term based on model complexity. The key contribution is a unified algorithm that balances data fitness and structural simplicity using the Minimum Description Length principle, enabling accurate dependency structure learning in mixed-variable settings without overfitting.

ABSTRACT

We extend the Chow-Liu algorithm for general random variables while the previous versions only considered finite cases. In particular, this paper applies the generalization to Suzuki's learning algorithm that generates from data forests rather than trees based on the minimum description length by balancing the fitness of the data to the forest and the simplicity of the forest. As a result, we successfully obtain an algorithm when both of the Gaussian and finite random variables are present.

Motivation & Objective

• To extend the Chow-Liu algorithm beyond finite random variables to include Gaussian and mixed-variable cases.
• To address overfitting in dependency structure learning by incorporating model complexity into the selection criterion.
• To develop a unified framework for learning Markov networks (forests) from data using Minimum Description Length (MDL) principles.
• To enable efficient and accurate learning of dependency structures when both discrete and continuous variables coexist in a dataset.

Proposed method

• Generalizes the Chow-Liu algorithm by defining mutual information for mixed Gaussian and finite variables using conditional density estimation.
• Applies the MDL principle by subtracting a complexity penalty from mutual information: $ J_n(i,j) = I_n(i,j) - \frac{\text{params}}{2} d_n $, where $ d_n $ is a normalization factor.
• For Gaussian given finite, estimates conditional mean $ \hat{g}(y) $ and variance $ \hat{\phi}_i $ via maximum likelihood, with $ \alpha^{(j)} - 1 $ parameters per value of the discrete variable.
• Uses a modified Kruskal-style algorithm that selects edges based on $ J_n(i,j) $, prioritizing high fitness and low complexity.
• Handles edge selection with loop detection to ensure forest structure, generalizing the original tree-based Chow-Liu approach.
• Derives asymptotic consistency of the estimated mutual information and structure under regularity conditions, proving convergence to the true dependency graph.

Experimental results

Research questions

• RQ1 How can the Chow-Liu algorithm be generalized to handle both Gaussian and finite-valued random variables?
• RQ2 What penalty term should be subtracted from mutual information to balance model fitness and complexity in mixed-variable settings?
• RQ3 How does the inclusion of complexity penalties affect the order of edge selection compared to standard Chow-Liu?
• RQ4 Can the MDL-based approach avoid overfitting when learning forests instead of trees from data?
• RQ5 What are the correct parameter counts and estimation procedures for conditional Gaussian models given discrete parents?

Key findings

• The generalized algorithm successfully learns dependency structures in mixed-variable datasets by combining mutual information with complexity penalties.
• For Gaussian given finite variables, the penalty term is $ \frac{(\alpha^{(j)} - 1)}{2} d_n $, reflecting the number of conditional mean parameters to estimate.
• For two finite variables, the penalty is $ \frac{(\alpha^{(i)} - 1)(\alpha^{(j)} - 1)}{2} d_n $, accounting for joint parameter estimation.
• For two Gaussian variables, the penalty is $ \frac{1}{2} d_n $, corresponding to estimating the correlation coefficient.
• The algorithm maintains computational efficiency by using a modified Kruskal-style greedy selection with loop detection.
• Theoretical analysis confirms consistency of the estimated structure under regularity conditions, ensuring convergence to the true dependency graph as sample size increases.