In fully Bayesian analyses, prior distributions are specified before observing data. Prior elicitation methods transfigure prior information into quantifiable prior distributions. Recently, methods that leverage copulas have been proposed to accommodate more flexible dependence structures when eliciting multivariate priors. The resulting priors have been framed as suitable candidates for Bayesian analysis. We prove that under broad conditions, the posterior cannot retain many of these flexible prior dependence structures as data are observed. However, these flexible copula-based priors are useful for design purposes. Because correctly specifying the dependence structure a priori can be difficult, we consider how the choice of prior copula impacts the posterior distribution in terms of convergence of the posterior mode. We also make recommendations regarding prior dependence specification for posterior analyses that streamline the prior elicitation process.
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