Bayesian Hierarchical Models and the Maximum Entropy Principle

2026-03-10

AI Summary

This study elucidates the implicit informational assumptions embedded in Bayesian hierarchical models, with a focus on the nature of constraints arising from dependencies among parameters. By integrating the principle of maximum entropy, probability integral transforms, and marginalization analysis, the work rigorously demonstrates that when hyperpriors are specified as maximum entropy distributions, the induced marginal priors retain the maximum entropy structure, with constraints acting directly on the marginal distributions of functions of unknown quantities. This result clarifies the informational content encoded in hierarchical priors, deepens the understanding of their semantic meaning and structural constraints, and provides theoretical foundations for enhancing the interpretability and principled design of Bayesian models.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Bayesian Learning
• Knowledge Representation and Reasoning: Reasoning with Beliefs

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Bayesian hierarchical models are frequently used in practical data analysis contexts. One interpretation of these models is that they provide an indirect way of assigning a prior for unknown parameters, through the introduction of hyperparameters. The resulting marginal prior for the parameters (integrating over the hyperparameters) is usually dependent, so that learning one parameter provides some information about the others. In this contribution, I will demonstrate that, when the prior given the hyperparameters is a canonical distribution (a maximum entropy distribution with moment constraints), the dependent marginal prior also has a maximum entropy property, with a different constraint. This constraint is on the marginal distribution of some function of the unknown quantities. The results shed light on what information is actually being assumed when we assign a hierarchical model.

Problem

• Bayesian hierarchical models
• maximum entropy principle
• prior distribution
• hyperparameters
• marginal prior

Innovation

• Bayesian hierarchical models
• maximum entropy principle
• canonical distributions
• marginal prior
• hyperparameters