I’m learning HMC and Stan. I just finished reading Betancourt 2017 (A Conceptual Introduction to Hamiltonian Monte Carlo) and realized that HMC excels at exploring the typical set: posterior x volume. This worries me because I have always thought that MCMC, including HMC, is a way to sample from the posterior distribution, and therefore we can plug the MCMC sample into a function to calculate the average as approximate population mean of the function (i.e. Monte Carlo). If the sample are actually drawn from the typical set rather than from the posterior whose distributions are different (see Fig 3 in Betancourt 2017), what am I calculating by doing Monte Carlo using this sample (i.e. plugging sample into a function and then calculate the average)?
I guess I’m missing something in my understanding of HMC or Stan. Any help is greatly appreciated. Thanks!