# Reference Bayesian analysis for hierarchical models

## TL;DR

This paper introduces a novel method for constructing invariant Jeffreys priors in hierarchical models using a Fisher information decomposition based on KL divergence, facilitating computation in complex multilevel Bayesian models.

## Contribution

It proposes a new approach to derive Jeffreys priors for hierarchical models by decomposing Fisher information via KL divergence, avoiding marginalization and enabling practical computation.

## Key findings

- The method provides an alternative way to compute Jeffreys priors for hyperparameters.
- It offers an upper bound for prior information, guiding prior informativeness.
- The approach is applicable in complex models like mixture, lasso, and Student-t models.

## Abstract

This paper proposes an alternative approach for constructing invariant Jeffreys prior distributions tailored for hierarchical or multilevel models. In particular, our proposal is based on a flexible decomposition of the Fisher information for hierarchical models which overcomes the marginalization step of the likelihood of model parameters. The Fisher information matrix for the hierarchical model is derived from the Hessian of the Kullback-Liebler (KL) divergence for the model in a neighborhood of the parameter value of interest. Properties of the KL divergence are used to prove the proposed decomposition. Our proposal takes advantage of the hierarchy and leads to an alternative way of computing Jeffreys priors for the hyperparameters and an upper bound for the prior information. While the Jeffreys prior gives the minimum information about parameters, the proposed bound gives an upper limit for the information put in any prior distribution. A prior with information above that limit may be considered too informative. From a practical point of view, the proposed prior may be evaluated computationally as part of a MCMC algorithm. This property might be essential for modeling setups with many levels in which analytic marginalization is not feasible. We illustrate the usefulness of our proposal with examples in mixture models, in model selection priors such as lasso and in the Student-t model.