[nerdponx]

I think Bayesian Data Analysis is the natural progression step.

Not sure if there is a more recent book that's updated to use modern Stan examples, but the Stan user guide itself has developed into a very useful resource on its own. It contains a large number of example models and builds up concepts incrementally. The writing style is also generally easy to follow.

[jvans]

I found that book impenetrable. I'm sure it's the most rigorous textbook on the subject but it is not explained in an intuitive or friendly way.

I will check out the stan guide though, thanks!

[t-vi]

It knows nothing of the modern stuff (because MacKay died too early), but skipping the first parts of David MacKay: Information Theory, Inference, and Learning Algorithms you get a very accessible course in (200x) Bayesian Inference that should cover most of what you need for diving into PPL applications.

http://www.inference.org.uk/mackay/itila/book.html

[nerdponx]

In my case, I used it in an actual course on Bayesian inference. Looking back over the material it doesn't seem particularly complicated for anyone with a solid probability background, but maybe the concepts are hard if you aren't seeing them presented nicely in a lecture setting.