Research Paper#Bayesian Inference, Dimension Reduction, Mutual Information🔬 ResearchAnalyzed: Jan 3, 2026 19:18
Bayesian Effective Dimension: A Mutual Information Approach
Published:Dec 28, 2025 19:17
•1 min read
•ArXiv
Analysis
This paper introduces the Bayesian effective dimension, a novel concept for understanding dimension reduction in high-dimensional Bayesian inference. It uses mutual information to quantify the number of statistically learnable directions in the parameter space, offering a unifying perspective on shrinkage priors, regularization, and approximate Bayesian methods. The paper's significance lies in providing a formal, quantitative measure of effective dimensionality, moving beyond informal notions like sparsity and intrinsic dimension. This allows for a better understanding of how these methods work and how they impact uncertainty quantification.
Key Takeaways
- •Introduces the Bayesian effective dimension as a measure of effective dimensionality.
- •Defines effective dimension using mutual information.
- •Provides a unifying perspective on dimension reduction techniques in Bayesian inference.
- •Offers insights into uncertainty quantification and the behavior of approximate posteriors.
- •Demonstrates connections with spectral complexity and effective rank in specific examples.
Reference
“The paper introduces the Bayesian effective dimension, a model- and prior-dependent quantity defined through the mutual information between parameters and data.”