  It is often of interest to combine available estimates of a similar quantity
from multiple data sources. When the corresponding variances of each estimate
are also available, a model should take into account the uncertainty of the
estimates themselves as well as the uncertainty in the estimation of variances.
In addition, if there exists a strong association between estimates and their
variances, the correlation between these two quantities should also be
considered. In this paper, we propose a bivariate hierarchical Bayesian model
that jointly models the estimates and their estimated variances assuming a
correlation between these two measures. We conduct simulations to explore the
performance of the proposed bivariate Bayesian model and compare it to other
commonly used methods under different correlation scenarios. The proposed
bivariate Bayesian model has a wide range of applications. We illustrate its
application in three very different areas: PET brain imaging studies,
meta-analysis, and small area estimation.
