I am looking to run some experiments in Julia after growing tired of jumping in and out of Stan and suffering from some performance issues that I think Julia could help eliminate. I am looking to do MCMC to learn parameters or form posterior predictives based on transformations of a standard Bayesian update. I.e. in Stan where one would usually write out:
mu ~ ... sigma ~ ... target += normal_lpdf(y | mu, sigma)
I would like to update using some transformation of the lpdf, e.g.
target += w * normal_lpdf(y | mu, sigma) - exp(normal_lpdf(y | mu / 10, sigma ^ 1/2))
Which is not a real example but just to illustrate the kind of thing I am doing. Is this possible in Julia, so far I am having difficulty wrapping my head around how I might achieve this using any of the packages I mentioned in the title. I am much more proficient in Stan, R and Python which is probably the reason for this so I apologise if this question is trivial. But so far all I can see is that in AdvancedHMC I could set my target to be equal to something imported from Distributions.jl and update with that as the target. Is there some way to easily define a custom acceptable target through transformations of log pdfs from Distributions, or perhaps using straight mathematical formulae for the distributions?
Thanks and please let me know if I can provide any more information.