Bayesian Hierarchical Models and the Maximum Entropy Principle

A new theoretical paper by Brendon J. Brewer, accepted for the 44th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (MaxEnt 2025), establishes a formal connection between two foundational concepts in statistics and machine learning. The 6-page work, published on arXiv, proves that when a Bayesian hierarchical model uses a canonical distribution—a maximum entropy distribution with moment constraints—as its conditional prior, the resulting marginal prior (after integrating over hyperparameters) also possesses a maximum entropy property. This property holds under a transformed constraint on the marginal distribution of some function of the parameters.

This result provides crucial insight for practitioners who routinely use hierarchical models in fields from astrophysics to bioinformatics. It explicitly reveals what information is being encoded when a modeler chooses a particular hierarchical structure. The paper sheds light on why these models, where learning about one parameter provides information about others, work so effectively in practice. By framing the dependent priors that emerge from hierarchical modeling as solutions to a maximum entropy problem, Brewer gives data scientists a clearer principle for model construction and interpretation.