Abstract
The use of hierarchical Bayesian models in statistical practice is extensive, yet it is dangerous to implement the Gibbs sampler without checking that the posterior is proper. Formal approaches to objective Bayesian analysis, such as the Jeffreys-rule approach or reference prior approach, are only implementable in simple hierarchical settings. In this paper, we consider a 4-level multivariate normal hierarchical model. We demonstrate the posterior using our recommended prior which is proper in the 4-level normal hierarchical models. A primary advantage of the recommended prior over other proposed objective priors is that it can be used at any level of a hierarchical model.
Highlights
Bayesian hierarchical models have a wide range of modern applications including engineering [1], astrophysics [2], economics [3], environmental sciences [4], climatology [5], survival analysis [6], and genetics [7]
As Hobert and Casella [8] pointed, without proper precaution, simple noninformative priors can be misused, sometimes unknowingly, and lead to other difficulties, such as the nonconvergence of the Gibbs sampler. erefore, it is hazardous to skip the demonstration at the risk of making the inference from an improper posterior distribution. ere are many examples of this in the statistical and other literatures
We demonstrate that the posterior using the recommended prior is still proper in the 4-level normal hierarchical model
Summary
Bayesian hierarchical models have a wide range of modern applications including engineering [1], astrophysics [2], economics [3], environmental sciences [4], climatology [5], survival analysis [6], and genetics [7]. Nonhierarchical Jeffreys-rule priors for variances or covariance matrices result in improper posterior distributions if they are used at higher levels of a hierarchical model (see [12]). Michalak and Morris [15] pointed out that except in the simplest models when improper priors are used, it can be daunting and time-consuming to verify that the resulting posterior distribution is proper. We follow the story and consider the posterior propriety of the recommended prior in a 4-level normal hierarchical model. We demonstrate that the posterior using the recommended prior is still proper in the 4-level normal hierarchical model.
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