I'm looking for resources (books, articles, sites, etc.) to learn about Bayesian inference, but specifically applied to astronomy and astrophysics.
I just finished a book titled, Bayesian Models for Astrophysical Data: using R/JAGS and Python/Stan, Cambridge University Press. I am lead author (Joseph Hilbe), with two astrophysicists, Rafael de Souza and Emille Ishida. It is in production now and should be out early next year.
In the meantime you should look at Andreon and Weaver (2015), Bayesian Methods for the Physical Sciences: Learning examples in Astronomy and Physics, Springer. It is part of the Springer Series in Astrostatistics books.
Why not just scour the ADS and see what's relevant to your case?
Perhaps a review by Trotta (2008) may be a good starting point? I can't post more than two links (or comment on your post for that matter), but do check the references within.
In particular, if you're interested in model selection, Liddle (2004, 2007) papers are good examples of its use in cosmology. For commonly used numerical methods, the use of the nested sampling algorithm [Skilling (2004)] became commonplace for evidence calculation after its implementation MultiNest [Feroz et al. (2008, 2009, 2013)] gained quite a bit of popularity in cosmology and particle physics.