I used 'Anglican' which is based on Clojure, and I think that is not good for me. Bad documents and a too small community to find help. Also, I still can't get familiar with the Scheme-based languages. So I want to change the language to something based on Python.
Maybe Pyro or PyMC could be the case, but I totally have no idea about both of those.
(Updated for 2022)
Pyro is built on PyTorch. It has full MCMC, HMC and NUTS support. It has excellent documentation and few if any drawbacks that I'm aware of.
PyMC was built on Theano which is now a largely dead framework, but has been revived by a project called Aesara. PyMC3 is now simply called PyMC, and it still exists and is actively maintained. Its reliance on an obscure tensor library besides PyTorch/Tensorflow likely make it less appealing for widescale adoption--but as I note below, probabilistic programming is not really a widescale thing so this matters much, much less in the context of this question than it would for a deep learning framework.
There still is something called Tensorflow Probability, with the same great documentation we've all come to expect from Tensorflow (yes that's a joke). My personal opinion as a nerd on the internet is that Tensorflow is a beast of a library that was built predicated on the very Googley assumption that it would be both possible and cost-effective to employ multiple full teams to support this code in production, which isn't realistic for most organizations let alone individual researchers.
That said, they're all pretty much the same thing, so try them all, try whatever the guy next to you uses, or just flip a coin. The best library is generally the one you actually use to make working code, not the one that someone on StackOverflow says is the best. As for which one is more popular, probabilistic programming itself is very specialized so you're not going to find a lot of support with anything.