Bayesian statistics has been developing for more than 250 years. During this time, it has enjoyed as much recognition and appreciation as it has faced disdain and contempt. Throughout the last few decades, it has gained more and more attention from people in statistics and almost all the other sciences, engineering, and even outside the boundaries of the academic world. This revival has been possible due to theoretical and computational advancements developed mostly throughout the second half of the 20th century. Indeed, modern Bayesian statistics is mostly computational statistics. The necessity for flexible and transparent models and a more intuitive interpretation of statistical models and analysis has only contributed to the trend.
In this book we will adopt a pragmatic approach to Bayesian statistics and we will not care too much about other statistical paradigms and their relationships with Bayesian statistics. The aim of this book is to learn how to do Bayesian data analysis; philosophical discussions are interesting, but they have already been undertaken elsewhere in a richer way that is simply outside the scope of these pages.
We will take a modeling approach to statistics, learn how to think in terms of probabilistic models, and apply Bayes' theorem to derive the logical consequences of our models and data. The approach will also be computational; models will be coded using PyMC3, a library for Bayesian statistics that hides most of the mathematical details and computations from the user, and ArviZ, a Python package for exploratory analysis of Bayesian models.
Bayesian methods are theoretically grounded in probability theory, and so it's no wonder that many books about Bayesian statistics are full of mathematical formulas requiring a certain level of mathematical sophistication. Learning the mathematical foundations of statistics could certainly help you build better models and gain intuition about problems, models, and results. Nevertheless, libraries such as PyMC3 allow us to learn and do Bayesian statistics with only a modest amount of mathematical knowledge, as you will be able to verify yourself throughout this book.