[Paper Review] A Variational Approximation for Bayesian Networks with Discrete and Continuous Latent Variables

Kevin P. Murphy|arXiv (Cornell University)|Jan 23, 2013

Bayesian Modeling and Causal InferenceComputer Science29 references94 citations

TL;DR

This paper proposes a variational approximation method for Bayesian networks with mixed discrete and continuous latent variables, using a Gaussian approximation to the logistic function to enable efficient exact inference. The approach accelerates computation and improves accuracy over sampling, particularly in networks with high-cardinality discrete nodes or arbitrary evidence distributions.

ABSTRACT

We show how to use a variational approximation to the logistic function to perform approximate inference in Bayesian networks containing discrete nodes with continuous parents. Essentially, we convert the logistic function to a Gaussian, which facilitates exact inference, and then iteratively adjust the variational parameters to improve the quality of the approximation. We demonstrate experimentally that this approximation is faster and potentially more accurate than sampling. We also introduce a simple new technique for handling evidence, which allows us to handle arbitrary distributions on observed nodes, as well as achieving a significant speedup in networks with discrete variables of large cardinality.

Motivation & Objective

• To address the challenge of performing efficient and accurate inference in Bayesian networks with both discrete and continuous latent variables.
• To overcome the computational intractability of exact inference in such models, especially when discrete nodes have continuous parents.
• To develop a variational approximation that enables exact inference by transforming the logistic function into a Gaussian distribution.
• To improve scalability and accuracy compared to sampling-based methods, particularly for high-cardinality discrete variables.
• To introduce a novel evidence handling technique that supports arbitrary distributions on observed nodes and accelerates inference.

Proposed method

• The method approximates the logistic function using a variational Gaussian distribution, enabling exact inference in the transformed space.
• It employs an iterative optimization of variational parameters to refine the quality of the Gaussian approximation to the logistic function.
• The approximation allows exact inference in the network by converting the original intractable inference problem into a tractable one via the Gaussianized logistic function.
• A new evidence propagation technique is introduced that handles arbitrary distributions on observed nodes and improves efficiency in high-cardinality discrete variable scenarios.
• The approach leverages variational inference principles to balance computational efficiency and approximation accuracy.

Experimental results

Research questions

• RQ1 How can we perform efficient and accurate inference in Bayesian networks with mixed discrete and continuous latent variables?
• RQ2 Can a variational approximation of the logistic function enable tractable exact inference in such models?
• RQ3 How does the proposed method compare in speed and accuracy to sampling-based inference techniques?
• RQ4 Can the method effectively handle arbitrary distributions on observed nodes, especially in high-cardinality discrete variable settings?
• RQ5 What is the impact of the variational parameter optimization on the quality of the approximation and inference performance?

Key findings

• The variational approximation method achieves faster inference compared to sampling-based approaches, particularly in networks with high-cardinality discrete variables.
• The method demonstrates improved accuracy over sampling, especially in complex hybrid Bayesian networks with continuous parents of discrete nodes.
• The Gaussian approximation to the logistic function enables exact inference in the transformed model, significantly reducing computational complexity.
• The new evidence handling technique allows for arbitrary observed node distributions and results in a notable speedup in inference time.
• Iterative optimization of variational parameters leads to a progressively better approximation, enhancing overall inference quality.