The Bayesian Posterior and Marginal Densities of the Hierarchical Gamma–Gamma, Gamma–Inverse Gamma, Inverse Gamma–Gamma, and Inverse Gamma–Inverse Gamma Models with Conjugate Priors

Positive, continuous, and right-skewed data are fit by a mixture of gamma inverse distributions. For 16 hierarchical models distributions, there only 8 them that have conjugate priors. We first discuss some common typical problems for the eight do not Then, we calculate Bayesian posterior densities marginal After that, relations among analytical densities. Furthermore, find random variables beta Moreover, variable generations distributions using R software. In addition, numerical simulations performed to illustrate four aspects: plots densities, from density, transformations moment estimators hyperparameters model, conclusions about properties closed form. Finally, our method real example, in which original transformed density with different hyperparameters.