I have a complex pdf based on hierarchical Bayesian formalism where x depends on the priors w'and w'', and I consider hyper-prior for the latter's as w=Php(zeta,beta) where Php stands for the hyper prior pdf which I take to be gamma distribution. The resulting joint pdf is in the attached picture. I want to sample both of w', w'', and x. My approach is to use the Hamiltonian Monte Carlo sampler to benefit from its mixing speed. I thought before that I can sample w' and w'' from the gamma distribution, then sample x based on the exact algorithm by Piniski but one colleague adviced that this may not be correct. If his claim is true, can one body suggest what is the appropriate way to sample x (which is the main parameter I am searching for) and also w' and w''. Thanks

Exact HMC algorithm: https://arxiv.org/abs/1208.4118

The joint posterior enter image description here



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