The name Bayesian statistics is attributed to Thomas Bayes (1702–1761), a Presbyterian minister, and amateur mathematician, who for the first time derived what we now know as Bayes’ theorem, which was published (posthumously) in 1763. However, one of the first people to really develop Bayesian methods was Pierre-Simon Laplace (1749–1827), so perhaps it would be a bit more correct to talk about Laplacian Statistics. Nevertheless, we will honor Stigler’s law of eponymy and also stick to tradition and keep talking about Bayesian approaches for the rest of this book. From the pioneering days of Bayes and Laplace (and many others) to the present day, a lot has happened - new ideas were developed, many of which were motivated and or being enabled by computers. The intent of this book is to provide a modern perspective on the subject, from the fundamentals in order to build a solid foundation into the application of a modern Bayesian workflow and tooling.
We write this book to help beginner Bayesian practitioners to become intermediate modelers. We do not claim this will automatically happen after you finish reading this book, but we hope the book can guide you in a fruitful direction specially if you read it thoroughly, do the exercises, apply the ideas in the book to your own problems and continue to learn from others.
Specifically stated this book targets the Bayesian practitioners who are interested in applying Bayesian models to solve data analysis problems. Often times a distinction is made between academia and industry. This book makes no such distinction, as it will be equally useful for a student in a university as it is for a machine learning engineer at a company.
It is our intent that upon completion of this book you will not only be familiar with Bayesian Inference but also feel comfortable performing Exploratory Analysis of Bayesian Models, including model comparison, diagnostics, evaluation and communication of the results. It is also our intent to teach all this from a modern and computational perspective. For us, Bayesian statistics is better understood and applied if we take a computational approach, this means, for example, that we care more about empirically checking how our assumptions are violated than trying to prove assumptions to be right. This also means we use many visualizations (if we do not do more is to avoid having a 1000 pages book). Other implications of the modeling approach will become clear as we progress through the pages.
Finally, as stated in the book’s title, we use the Python programming language in this book. More specifically, we will mainly focus on PyMC3  and TensorFlow Probability (TFP) , as the main probabilistic programming languages (PPLs) for model building and inference, and use ArviZ as the main library for exploratory analysis of Bayesian models . We do not intend to give an exhaustive survey and comparison of all Python PPLs in this book, as there are many choices, and they rapidly evolve. We instead focus on the practical aspects of Bayesian analysis. Programming languages and libraries are merely bridges to get where we want to go.
Even though our programming language of choice for this book is Python, with few selected libraries, the statistical and modeling concepts we cover are language and library agnostic and available in many computer programming languages such as R, Julia, and Scala among others. A motivated reader with knowledge of these languages but not Python can still benefit from reading the book, especially if they find the suitable packages that support, or code, the equivalent functionality in their language of choice to gain hands on practice. Furthermore, the authors encourage others to translate the code examples in this work to other languages or frameworks. Please get in touch if you like to do so.
As we write this book to help beginners to become intermediate practitioners, we assume prior exposure, but not mastery, of the basic ideas from Bayesian statistics such as priors, likelihoods and posteriors as well as some basic statistical concepts like random variables, probability distributions, expectations. For those of you that are a little bit rusty, we provide a whole section inside Chapter 11, with a refresher about basic statistical concepts. A couple of good books explaining these concepts in more depth are Understanding Advanced Statistical Methods  and Introduction to Probability . The latter is a little bit more theoretical, but both keep application in mind.
If you have a good understanding of statistics, either by practice or formal training, but you have never been exposed to Bayesian statistics, you may still use this book as an introduction to the subject, the pace at the start (mostly the first two chapters) will be a bit rapid, and may require a couple read-throughs.
We expect you to be familiar with some mathematical concepts like integrals, derivatives, and properties of logarithms. The level of writing will be the one generally taught at a technical high school or maybe the first year of college in science, technology, engineering, and mathematics careers. For those who need a refresher of such mathematical concepts we recommend the series of videos from 3Blue1Brown . We will not ask you to solve many mathematical exercises instead, we will primarily ask you to use code and an interactive computing environment to understand and solve problems. Mathematical formulas throughout the text are used only when they help to provide a better understanding of Bayesian statistical modeling.
This book assumes that the reader comes with some knowledge of scientific computer programming. Using the Python language we will also use a number of specialized packages, in particular Probabilistic Programming Languages. It will help, but is not necessary, to have fit at least one model in a Probabilistic Programming language prior to reading this book. For a reference on how to setup the computation environment needed for this book, read environment installation.
How to read this book¶
We will use toy models to understand important concepts without the data obscuring the main concepts and then use real datasets to approximate real practical problems such as sampling issues, reparametrization, prior/posterior calibration, etc. We encourage you to run these models in an interactive code environment while reading the book.
We strongly encourage you to read and use the online documentation for the various libraries. While we do our best to keep this book self-contained, there is an extensive amount of documentation on these tools online and referring it will aid in both learning this book, as well as utilizing the tools on your own.
Chapter 1 offers a refresher or a quick introduction to the basic and central notions in Bayesian inference. The concepts from this chapter are revisited and applied in the rest of the book.
Chapter 2 offers an introduction to Exploratory Analysis of Bayesian models. Namely introduces many of the concepts that are part of the Bayesian workflow but are not inference itself. We apply and revisit the concepts from this chapter in the rest of the book.
Chapter 3 is the first chapter dedicated to a specific model architecture. It offers an introduction to Linear Regression models and establishes the basic groundwork for the next 5 chapters. Chapter 3 also fully introduces the primary probabilistic programming languages used in the book, PyMC3 and TensorFlow Probability.
Chapter 4 extends Linear Regression models and discusses more advanced topics like robust regression, hierarchical models and model reparametrization. This chapter uses PyMC3 and TensorFlow Probability.
Chapter 5 introduces basis functions and in particular splines as an extension to linear models that allows us to build more flexible models. This chapter uses PyMC3.
Chapter 6 focuses on time series models, from modeling time series as a regression to more complex model like ARIMA and linear Gaussian State Space model. This chapter uses TensorFlow Probability.
Chapter 7 offers an introduction to Bayesian additive regression trees a non-parametric model. We discuss the interpretability of this model and variable importance. This Chapter use PyMC3.
Chapter 8 brings the attention to the Approximate Bayesian Computation (ABC) framework, which is useful for problems where we do not have an explicit formulation for the likelihood. This chapter uses PyMC3.
Chapter 9 gives an overview of end-to-end Bayesian workflows. It showcases both an observational study in a business setting and an experimental study in a research setting. This chapter uses PyMC3.
Chapter 10 provides a deep dive on Probabilistic Programming Languages. Various different Probabilistic Programming languages are shown in this chapter.
Chapter 11 serves as a support when reading other chapters, as the topics inside it are loosely related to each other, and you may not want to read linearly.
Text in this book will be emphasized with bold or italics. Bold
text will highlight new concepts or emphasis of a concept. Italic
text will indicate a colloquial or non-rigorous expression. When a
specific code is mentioned they are also highlighted:
Blocks of code in the book are marked by a shaded box with the lines numbers on the left. And are referenced using the chapter number followed by the number of the Code Block. For example:
for i in range(3): print(i**2)
0 1 4
Every time you see a code block look for a result. Often times it is a figure, a number, code output, or a table. Conversely most figures in the book have an associated code block, sometimes we omit code blocks in the book to save space, but you can still access them at the GitHub repository. The repository also includes additional material for some exercises. The notebooks in that repository may also include additional figures, code, or outputs not seen in the book, but that were used to develop the models seen in the book. Also included in GitHub are instructions for how to create a standard computation environment on whatever equipment you have.
We use boxes to provide a quick reference for statistical, mathematical, or (Python) Programming concepts that are important for you to know. We also provide references for you to continue learning about the topic.
Central Limit Theorem
In probability theory, the central limit theorem establishes that, in some situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution even if the original variables themselves are not normally distributed.
Let \(X_1, X_2, X_3, ...\) be i.i.d. with mean \(\mu\) and standard deviation \(\sigma\). As \(n \rightarrow \infty\), we got:
The book Introduction to Probability  is a good resource for learning many theoretical aspects of probability that are useful in practice.
In this book we use the following conventions when importing Python packages.
# Basic import numpy as np from scipy import stats import pandas as pd from patsy import bs, dmatrix import matplotlib.pyplot as plt # Exploratory Analysis of Bayesian Models import arviz as az # Probabilistic programming languages import bambi as bmb import pymc3 as pm import tensorflow_probability as tfp tfd = tfp.distributions # Computational Backend import theano import theano.tensor as tt import tensorflow as tf
We also use the ArviZ style
How to interact with this book¶
As our audience is not a Bayesian reader, but a Bayesian practitioner. We will be providing the materials to practice Bayesian inference and exploratory analysis of Bayesian models. As leveraging computation and code is a core skill required for modern Bayesian practitioners, we will provide you with examples that can be played around with to build intuition over many tries. Our expectation is that the code in this book is read, executed, modified by the reader, and executed again many times. We can only show so many examples in this book, but you can make an infinite amount of examples for yourself using your computer. This way you learn not only the statistical concepts, but how to use your computer to generate value from those concepts.
Computers will also remove you from the limitations of printed text, for example lack of colors, lack of animation, and side-by-side comparisons. Modern Bayesian practitioners leverage the flexibility afforded by monitors and quick computational “double checks” and we have specifically created our examples to allow for the same level of interactivity. We have included exercises to test your learning and extra practice at the end of each chapter as well. Exercises are labeled Easy (E), Medium (M), and Hard (H). Solutions are available on request.
We are grateful to our friends and colleagues that have been kind enough to provide their time and energy to read early drafts and propose and provide useful feedback that helps us to improve the book and also helps us to fix many bugs in the book. Thank you:
Oriol Abril-Pla, Alex Andorra, Paul Anzel, Dan Becker, Tomás Capretto, Allen Downey, Christopher Fonnesbeck, Meenal Jhajharia, Will Kurt, Asael Matamoros, Kevin Murphy, and Aki Vehtari.