[Paper Review] Variational Approximations between Mean Field Theory and the Junction Tree Algorithm

Wim Wiegerinck|arXiv (Cornell University)|Jan 16, 2013

Bayesian Modeling and Causal InferenceComputer Science24 references80 citations

TL;DR

This paper introduces a generalized mean field variational inference method that uses cluster-based factorizations of the approximating distribution, bridging the gap between standard mean field theory and the exact junction tree algorithm. By optimizing cluster potentials via generalized mean field equations, the approach enables flexible graphical structures and simplifies model design without sacrificing approximation quality.

ABSTRACT

Recently, variational approximations such as the mean field approximation have received much interest. We extend the standard mean field method by using an approximating distribution that factorises into cluster potentials. This includes undirected graphs, directed acyclic graphs and junction trees. We derive generalized mean field equations to optimize the cluster potentials. We show that the method bridges the gap between the standard mean field approximation and the exact junction tree algorithm. In addition, we address the problem of how to choose the graphical structure of the approximating distribution. From the generalised mean field equations we derive rules to simplify the structure of the approximating distribution in advance without affecting the quality of the approximation. We also show how the method fits into some other variational approximations that are currently popular.

Motivation & Objective

• To develop a variational inference framework that generalizes mean field theory by allowing cluster-based factorizations of the approximating distribution.
• To bridge the gap between the inexact mean field approximation and the exact junction tree algorithm in probabilistic graphical models.
• To provide principled rules for simplifying the structure of the approximating distribution in advance, without degrading approximation quality.
• To integrate the proposed method into broader classes of modern variational approximations used in machine learning.

Proposed method

• The method uses an approximating distribution that factorizes into cluster potentials, allowing representation via undirected graphs, directed acyclic graphs, or junction trees.
• Generalized mean field equations are derived to optimize the cluster potentials, extending standard mean field updates.
• The approach allows for structured variational approximations that interpolate between mean field and junction tree methods.
• Graphical structure simplification rules are derived from the generalized mean field equations to reduce complexity without affecting approximation accuracy.
• The framework is shown to be compatible with other contemporary variational inference techniques, such as expectation propagation and structured variational inference.

Experimental results

Research questions

• RQ1 How can variational inference be generalized beyond standard mean field assumptions to achieve better accuracy?
• RQ2 What is the relationship between mean field theory and the exact junction tree algorithm in terms of variational approximation quality?
• RQ3 Can structured approximations using cluster potentials improve the trade-off between accuracy and computational cost?
• RQ4 What criteria or rules can be used to simplify the structure of the approximating distribution without degrading performance?
• RQ5 How does the proposed method relate to or integrate with other popular variational inference techniques?

Key findings

• The proposed method successfully interpolates between mean field theory and the exact junction tree algorithm, offering a spectrum of approximation quality.
• Generalized mean field equations are derived that allow optimization of cluster potentials in a way that maintains consistency with the underlying graphical model structure.
• The method enables the use of structured approximations such as junction trees while retaining the computational efficiency of mean field methods.
• Simplification rules derived from the generalized equations allow for pruning of unnecessary cluster structures without loss of approximation quality.
• The approach is shown to be compatible with other modern variational inference frameworks, enhancing its applicability across different probabilistic models.