In the application of Bayesian methods, the choice of a prior density for defined model parameters is often a practical issue. In scientific and medical applications, it requires careful justification. In many settings, the properties of the statistical likelihood function and the overall statistical model, within which the model parameters are defined, may be useful in justifying the chosen prior density. In some settings this is extended to the matching of Bayesian and frequentist results. In this paper, prior density selections reflecting specific model-related and matching-related properties are reviewed. It is shown that if these considerations are extended to include the statistical information-based matching of likelihood and posterior density local second-order derivatives in a central region, typically plus or minus one standard error about the mode of the likelihood, they provide new matching-related guidance, both inclusionary and exclusionary, regarding the selection and form of prior densities. Multivariate extensions are also briefly considered.
The use of approximate Bayesian inference for model and variable selection for hierarchical Bayesian models is introduced, based on the application of Markov chain Monte Carlo methods on the model space.
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