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Wow this is really interesting and might be really helpful for a problem I'm working on now. I'm wondering though, what is the advantage of using a whole new programming language rather than say, a Python library?

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There are packages that are just Python libraries, like PyMC3: http://pymc-devs.github.io/pymc3/ It allows you to specify a Probabilistic Program with a few lines of Python code (x ~ Normal(0, 1) becomes x = pymc3.Normal('x', 0, 1)) and then run automatic inference on whatever model you dream up, it's very powerful. I always found the approach of inventing a new language to specify Probabilistic Programs like Stan does a bit odd. There, you specify the model as a string and can't interact with it or debug if something goes wrong. Disclaimer: I'm a PyMC3 developer.

>I'm wondering though, what is the advantage of using a whole new programming language rather than say, a Python library?

The idea is that if you can write the program, you can sample from the generative model. Most libraries like Stan don't really provide a "Turing-complete" probabilistic language.

Daniel Roy, IIRC, studies precisely why or when you want this kind of "completeness", as well as other topics in PPL theory.

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