Friendly reminder: Markov chain Monte Carlo estimators are not guaranteed to give you meaningful answers unless a whole bunch of conditions hold.
Firstly you need a central limit theorem to hold, which requires that the sampler and the target distribution play nicely with each other. In most cases this requires something called geometric ergodicity.
You can check for obstructions to a central limit theorem by running an ensemble of chains and looking for large Rhat values or even better use Hamiltonian Monte Carlo and check for divergences and a low energy fraction of missing information (E-FMI).
BUT even if a central limit theorem holds for well-behaved functions it doesn't imply that it will hold for _all_ functions! Typically each function has to be square integrable with respect to the target distribution, i.e. both its mean and variance have to be finite.
Finally the variance in the Markov chain Monte Carlo central limit theorem is moderated by the effective sample size for a given function, which we need to estimate from the Markov chain itself. If the chain is too autocorrelated then you won't be able to do this accurately.
Markov chain Monte Carlo is powerful in general but any implementation is susceptible to problems. It's up to you to do your due diligence and verify that the sampler is exploring well enough and you're looking at sufficiently well-behaved expectation values.
If _anyone_, even people who are considered experts in applied fields, tells you differently then walk away as quickly as you can.
Oh, yeah, you can also identify obstructions to a central limit theorem using that slick new SBC algorithm that all the young kids are talking about, arxiv.org/abs/1804.06788.
