Bayesian graph selection consistency under model misspecification.

Area of Science:

• Statistics
• Machine Learning
• Computational Statistics

Background:

• Gaussian graphical models are widely used for inferring variable dependencies.
• Bayesian methods offer simultaneous learning of covariance and graph structures.
• Decomposability is often imposed for computational efficiency in Bayesian structure learning.

Purpose of the Study:

• To investigate if Bayesian decomposable structure learning can recover non-decomposable graphs.
• To determine conditions under which the posterior distribution selects a meaningful, close graph in high dimensions.
• To address the open problem of posterior concentration on non-decomposable true graphs.

Main Methods:

• Utilizing a hyper-inverse Wishart prior for the covariance matrix.
• Employing a suitable complexity prior on the graph space.
• Analyzing posterior distribution concentration under specific conditions on the precision matrix and graph.

Main Results:

• Demonstrated strong selection consistency in high-dimensional settings (p = O(n)) for alpha < 1/3, without assuming sparsity.
• Showed that the posterior distribution concentrates on minimal triangulations of the true graph when it is non-decomposable.
• Identified conditions ensuring the posterior selects a meaningful decomposable graph close to the true non-decomposable graph.

Conclusions:

• The Bayesian approach with decomposable priors can effectively learn non-decomposable graph structures.
• Minimal triangulations are key to understanding posterior concentration for non-decomposable graphs.
• This work provides theoretical guarantees for Bayesian structure learning in high-dimensional scenarios.