The 19 Densities of the Hierarchical Bayes Model with Two Conditional Levels

There are 19 densities involved in the hierarchical Bayes model with two conditional levels, in which the 3 densities, that is, the likelihood function, the rst level prior density, and the second level prior density, are known densities. We have written the 16 unknown densities in terms of the 3 known densities in a theorem which is very handy for practitioners and researchers interested in the hierarchical Bayes model with two conditional levels. Finally, we apply the theorem to a specific hierarchical normal Bayes model with two conditional levels and obtain the functional forms of the 16 unknown densities. Moreover, we gure out the exact distributions of the 16 densities, which are one-, two-, or three-dimensional normal distributions.