Bayesian Hierarchical Models and the Maximum Entropy Principle
This work provides theoretical insight into the information encoded in hierarchical models, which is incremental for statisticians and practitioners in data analysis.
The paper demonstrates that in Bayesian hierarchical models with canonical priors, the resulting dependent marginal prior also satisfies a maximum entropy principle under a different constraint, clarifying the underlying information assumptions.
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.