Hierarchical bayesian modeling with elicited prior information

Abstract

Conditionally independent hierarchical models (CIHMs) have proven useful in problems where modest numbers of observations are made on a collection of k similar experimental units. In CIHMs, the observation vector Y _i for the i-th unit has a distribution indexed by a unit-specific parameter vector θ_i,. The unit-specific parameter vectors are i.i.d. with distributions indexed by a common hyperparameter vector λ

This paper presents a general methodology for subjective Bayesian inference in CIHMs. An analyst (e.g., a statistician) specifies a pre-elicitation prior distribution for the parameter vector ξ at the final stage of the hierarchy. “Data”, in the form of elicited probability statements, are then collected from a substantive expert and are used to obtain an estimate , yielding a fitted hyperprior for λ. This hyperprior can then be updated subsequent to observing the sample data in order to conduct Bayesian posterior inference about the first and second-stage parameter vectors and λ.

The approach is illustrated with a CIHM for binomial observations applied to baseball data that were originally analyzed by Efron and Morris (1975) and Morris (1983) using parametric empirical Bayes (PEB) methods. Here, prior probability statements are elicted from a baseball expert, and a fully Bayesian analysis is conducted for the first-stage parameter, yielding approximate posterior probability intervals. These intervals are found to be substantially shorter than the corresponding PEB intervals, yet retain the stated probability of coverage.

Keywords:

• expert judgement
• subjective probability
• hyperparameters
• asymptotic posterior approximations