# Multi-Goal Prior Selection: A Way to Reconcile Bayesian and Classical Approaches for Random Effects Models

## TL;DR

This paper introduces a multi-goal prior for Bayesian hyperparameter selection in random effects models, aligning Bayesian and classical solutions and establishing analytical equivalence of posterior variances with bootstrap estimates.

## Contribution

It proposes a novel multi-goal prior that reconciles Bayesian and classical approaches in hierarchical models, with theoretical and practical implications.

## Key findings

- Multi-goal prior produces Bayesian solutions matching classical methods.
- Analytical equivalence between posterior variances and bootstrap MSE estimates.
- Enhanced understanding of hyperparameter selection in random effects models.

## Abstract

The two-level normal hierarchical model has played an important role in statistical theory and applications. In this paper, we first introduce a general adjusted maximum likelihood method for estimating the unknown variance component of the model and the associated empirical best linear unbiased predictor of the random effects. We then discuss a new idea for selecting prior for the hyperparameters. The prior, called a multi-goal prior, produces Bayesian solutions for hyperparmeters and random effects that match (in the higher order asymptotic sense) the corresponding classical solution in linear mixed model with respect to several properties. Moreover, we establish for the first time an analytical equivalence of the posterior variances under the proposed multi-goal prior and the corresponding parametric bootstrap second-order mean squared error estimates in the context of a random effects model.