## Hyperprior Imprecision in Hierarchical Bayesian Modeling of Cluster Bernoulli
Observations

### by Calvin L. Williams and Ann-Janette Locke.

**Abstract:**
The focus of this article is on combining the techniques of hierarchical Bayesian modeling with methods of prior imprecision for the purpose of modeling bernoulli observations that are clustered in structure in a situation of little or no prior information. When data have a clustered structure, it is necessary to determine a model which will account not only for the variation between clusters of observations, but also for the variation within clusters. This can be achieved for the clustered bernoulli likelihood problem by using one of many hierarchical models. However, any such model that is used will require specification of prior distributions. In the absence of prior information, a usual approach is to specify noninformative priors (such as ones with a large variance), but the problem of being able to justify any specific choice of prior parametric information remains. A solution in the hierarchical modeling setting is to select imprecise hyperprior distributions, creating an imprecise hierarchical Bayesian model. These models allow the researcher to specify broad intervals into which hyperparameters fall instead of specific values for them. In turn, such great hyperprior imprecision results in very little posterior imprecision because the imprecision is introduced at the hyperprior level instead of in the first level of the prior. The result is a posterior that is dominated by the likelihood and practically unaffected by the prior specification{the desired result when there is lacking information. Several hierarchical Bayesian models with imprecise hyperpriors are developed and applied to a problem involving estimation of survival rates between two sets of 16 litters of rats. One of these sets has been treated with a chemical, and the goal is to estimate the survival rate for each set. There is lacking prior information in this application, and the data are clustered in structure, so this is an appropriate setting for hierarchical Bayesian modeling with hyperprior imprecision. Posterior estimates of the survival rates are obtained for each of the models that is developed. The posterior estimates for each model are shown to change very little over large intervals of hyperprior imprecision. The different models are compared based on their ability to allow the posterior be dominated by the likelihood.
**Key Words: **
Clustered Data; Gibbs Sampling; No Prior Information; Posterior Dominated By the Likelihood; Imprecise Prior

**Authors:**

Calvin L. Williams, calvinw@ces.clemson.edu

Ann-Janette Locke

**Editor:**
Kamel Ashour,ashoursamir@hotmail.com

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