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I am frequentist by training and practice, but I'd like to learn more about Bayesian statistics. I know the basics, but I would be at a loss if I had to, for example, replace my normal ANOVA hypothesis testing approach with a Bayesian alternative.

What book would you recommend to learn practical Bayesian approaches? Preferably using R.

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Peter D. Huff. A First Course in Bayesian Statistical Methods. Springer (2010)

Also

Andrew Gelman et. al. Bayesian Data Analysis (3rd ed.). CRC (2013)

The Gelman book isn't constrained to R but also uses Stan, a probabilistic programming language similar to BUGS or JAGS. I believe earlier editions of the book used BUGS instead of Stan, which is probably very similar.

And finally:

John Kruschke. Doing Bayesian Data Analysis: A tutorial with R and BUGS. Academic Press (2011)

More BUGS than R, but probably the most pragmatic of the three books I've suggested. Don't let the cover deter you, this is a perfectly respectable text.

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    $\begingroup$ I have to voice that the puppies on the cover of Doing Bayesian Data Analysis are respectably cute! And yes, it's a great text. $\endgroup$
    – Penguin_Knight
    Commented Nov 14, 2013 at 22:47
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    $\begingroup$ All great texts, but very different. If you don't know the basics of Bayesian Data analysis, and isn't a math stat student I think Kruschke's text is by far the most accessible. Gelman et al is, for sure, more comprehensive but not necessarily more comprehensible depending on you background knowledge. After having read Kruschke, a great second book is the extremely pragmatical Bugs Book mrc-bsu.cam.ac.uk/bugs/thebugsbook . $\endgroup$
    – Rasmus Bååth
    Commented Nov 16, 2013 at 21:49
  • $\begingroup$ @RasmusBååth +1, last link doesn't work. $\endgroup$
    – Patrick Coulombe
    Commented Feb 15, 2015 at 18:24
  • $\begingroup$ Hmm, seems to have changed, here's a new link to the BUGS book that works: mrc-bsu.cam.ac.uk/software/bugs/the-bugs-project-the-bugs-book $\endgroup$
    – Rasmus Bååth
    Commented Feb 15, 2015 at 21:47
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Fortuitous timing, as Bayesian Data Analysis, 3rd ed was just released. It's a good general-purpose text, with an emphasis on hierarchical methods, a section on advanced computation (that is, Markov chain Monte Carlo), and an appendix on Gelman's Bayesian inference tool, rstan.

The text focuses on statistics rather than programming, though, so perhaps this answer does not fit your R needs. That said, I've been able to recreate the text examples in R simply based on his clear prose descriptions.

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Both are introductory, but useful imho:

Bayesian Computation With R, by Jim Albert

Applied Bayesian Statistics, With R and OpenBUGS Examples, by Mary Kathryn Cowles

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    $\begingroup$ Giving full titles and authors names could be helpful in case the links go dead in the future (and even if they don't). $\endgroup$
    – Richard Hardy
    Commented Aug 28, 2018 at 19:11

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