Reading list for learning Bayes' theory

If you agree on the statement that science is based on the previous evidence and that evidence could affect the researchers' thoughts toward the next researches, it is not surprising that every study will involve some subjective judgement while conducting the researches. This is not the terrible part. The worse part of the judgement is that this could affect the estimation implicitly. In other words, we did not consider this judgement during building the model. Bayes theorem is what we need to include all the possible information including researcher's judgement into the model and then use the data we collected to update the model. This theorem is powerful but not valued as a essential tool for researches especially in the field of social science. The massive usage of probability and mathematics could be the causes. In this post, I would like to introduce some references that are extremely useful for the stepping stone of learning Bayes theorem. You don't have to read every book in the section. Instead, I encourage you to find every book in each section and try to find one  that best fits you.  For a reading list for programming in Bayesian estimation, read my another post here.

• Mathematics

1. Calculus and its applications
2. Fundamental Methods of Mathematical Economics

• Probability theory

1. Introduction to probability (by Blitzstein) 
2. Introduction to probability

• Bayes' Theory 

Bayes' theory would be a game changer for statistical estimations. It is not just a method for estimations but also a philosophy which fits our common sense toward the estimation.  Sooner or latter, most statistics textbook could be revised because of statistical concepts from Bayes' theory. Other estimation could be just a special case of Bayes' theory and this makes Bayes' Theory demand more backgrounds of math and probability. Therefore, before I provide the book list of Bayes' theory. I have to warn you that you would not even survive the chapter one if you do not have any concepts of calculus and probability. Thus, you should read the references I mentioned above, and then you can have fun and feel the power of Bayes' Theory through the following books I suggest.

1. Bayesian Analysis for the Social Sciences: (PPT of this book could be found in author's website) This book provides many useful plots to give you the most direct learning of Bayes' theory.  The author also offers clear proofs in the appendix. For readers who only want to realize the basics, you should at least read introduction and chapter one of this book. 
2. Bayesian Ideas and Data Analysis: This book has the goodness of explanation of the reasons for using different priors. Best of all, it provides detailed Winbugs and R codes in their website. 
3. Bayesian Econometrics: This book is very popular in the field of economics. I especially like that the author introduces the Bayes' theory by closely relating it to linear models which we are familiar with most. Since the author is one of the leading researcher in Bayesian efficiency estimation (see my another post), you will also benefit a lot from reading this book before you jump in that field. This author's another book,Bayesian Econometric Methods, is also worth reading. It also focus on the linear model but provides more detailed proofs and clearer math notation. Other short introduction of Bayesian methods in econometrics could be found in chapter 13, Microeconometrics Methods and Applications.    
4. Bayesian Data Analysis: Very well-known and wildly used for an Bayesian analysis. The first two edition uses Winbugs for demonstrating their Bayesian results while the latest edition use Stan instead. This book is not appropriate for beginners but it provides wide topics. Therefore, you should consider it as a very good model source. The best of it, the math notation is very close to the first book I recommended, so you save lots time when you finish reading the first one.   
5. Bayesian Inference: with ecological applications

This article will continuously be updated and I welcome anyone of you to post your experience here.