I am trying to implement HMC-NUTS algorithm using a log-Pareto-tailed Normal (LPTN) distribution as my likelihood distribution. The LPTN distribution is specified as follows:

where g(.| phi) is the standard normal distribution. Note that alpha and beta are already pre-determined values so the parameters we want to make inference on are the location (mu) and scale (sigma) of the distribution.

I think I need to specify my own distribution class with some kind of truncation properties but I find it difficult because the truncation cutoff point depends on the inference parameters. Does anybody have any tips or suggestions on how to do this?

Thank you in advance.