Variational Message Passing and Local Constraint Manipulation in Factor Graphs.

Area of Science:

• Machine Learning
• Statistical Modeling
• Computational Science

Background:

• Accurate Bayesian model evidence evaluation is essential for model development but often computationally intractable.
• Variational Free Energy (VFE) minimization offers an alternative, providing an upper bound on negative model log-evidence (NLE).
• Constraint manipulation in variational inference can improve VFE tractability but may compromise NLE accuracy, presenting an engineering trade-off.

Purpose of the Study:

• To develop a unifying framework for understanding constraint manipulation in variational inference for factor graphs.
• To identify the Bethe Free Energy (BFE) as an approximation to VFE within this framework.
• To derive message passing algorithms from first principles via constrained BFE minimization.

Main Methods:

• Developed a unifying account of constraint manipulation for variational inference in factor graphs.
• Identified the Bethe Free Energy (BFE) as an approximation to the Variational Free Energy (VFE).
• Derived message passing algorithms by minimizing the constrained BFE.

Main Results:

• The proposed method enables BFE evaluation in factor graphs.
• Demonstrated that well-known message passing algorithms arise from minimizing the constrained BFE.
• The framework supports model scoring and the development of novel inference algorithms.

Conclusions:

• Constraint manipulation in variational inference for factor graphs can be unified under the BFE minimization framework.
• This approach offers a principled way to balance tractability and accuracy in model evidence estimation.
• The derived methods can improve model scoring and lead to more accurate evidence estimation algorithms.