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jsalvatier comments on The Best Textbooks on Every Subject - Less Wrong
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Update see my comment for new thoughts
Topic: Introductory Bayesian Statistics (as distinct from more advanced Bayesian statistics)
Recommendation: Data Analysis: A Bayesian Tutorial by Skilling and Sivia
Why: Sivia's book is well suited for smart people who have not had little or no statistical training. It starts from the basics and covers a lot of important ground. I think it takes the right approach, first doing some simple examples where analytical solutions are available or it is feasible to integrate naively and numerically. Then it teaches into maximum likelihood estimation (MLE), how to do it and why it makes sense from a Bayesian perspective. I think MLE is a very very useful technique, especially so for engineers. I would overall recommend just Part I: The Essentials, I don't think the second half is so useful, except perhaps the MLE extensions chapter. There are better places to learn about MCMC approximation.
Why not other books?
Bayesian Data Analysis by Gelman - Geared more for people who have done statistics before.
Bayesian Statistics by Bolstad - Doesn't cover as much as Sivia's book, most notably doesn't cover MLE. Goes kinda slowly and spends a lot of time on comparing Bayesian statistics to Frequentist statistics.
The Bayesian Choice - more of a mathematical statistics book, not suited for beginners.
Brandon Reinhart used both Sivia's book and Bolstad's book and found (3rd message) Bolstad's book better for those with no stats experience:
Based on these comments, I think I was underestimating inferential distance, and I now change my recommendation. You should read Bolstad's book first (skipping the parts comparing bayesian and frequentist methods unless that's important to you) and then read Sivia's book. If you have experience with statistics you may start with Sivia's book.