Texts for Your Shelf
The following are three texts I recommend in my consulting to folks who are interested in doing Bayesian data analyis. They can be seen as introduction, intermediate, and advanced respectively.
Kruschke, J. (2014) Doing Bayesian Data Analysis. A very introductory text, but might be good for those not too confident in statistics generally speaking. And who doesn’t like puppies? Second edition has Stan examples.
McElreath, R. (2015). Statistical Rethinking. A good modeling book in general, by one who has contributed a lot to helping others learn Stan. Comes with its own R package too.
Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., and Rubin, D. B. (2013). Bayesian Data Analysis.
Stan Specific Resources
More resources here
Statisticat, LLC. Bayesian Inference. A quick overview from the original author of the LaplacesDemon package.
Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. 3rd ed.
Gelman, Andrew, and Jennifer Hill. 2006. Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge University Press.
Gelman, Andrew, Jessica Hwang, and Aki Vehtari. 2014. “Understanding Predictive Information Criteria for Bayesian Models.” Statistics and Computing 24 (6). Springer: 997–1016.
Gelman, Andrew, and Iain Pardoe. 2006. “Bayesian Measures of Explained Variance and Pooling in Multilevel (Hierarchical) Models.” Technometrics 48 (2). Taylor & Francis: 241–51.
Gill, Jeff. 2008. Bayesian Methods : A Social and Behavioral Sciences Approach. Second. Boca Raton: Chapman & Hall/CRC.
Jackman, Simon. 2009. Bayesian Analysis for the Social Sciences. Chichester, UK: Wiley.
Kruschke, John. 2010. Doing Bayesian Data Analysis: A Tutorial Introduction with R. Academic Press.
Lunn, David, Chris Jackson, Nicky Best, Andrew Thomas, and David Spiegelhalter. 2012. The BUGS Book: A Practical Introduction to Bayesian Analysis. Boca Raton, FL: Chapman; Hall/CRC.
Lynch, Scott M. 2007. Introduction to Applied Bayesian Statistics and Estimation for Social Scientists. New York: Springer.
McElreath, Richard. 2016. Statistical Rethinking: A Bayesian Course with Examples in R and Stan. Vol. 122. CRC Press.
McGrayne, Sharon Bertsch. 2012. The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy. Reprint. New Haven Conn.: Yale University Press.
Vehtari, Aki, and Andrew Gelman. 2014. “WAIC and Cross-Validation in Stan.”