Cameron Davidson-Pilon has seen many fields of applied mathematics, from evolutionary dynamics of genes and diseases to stochastic modeling of financial prices. His main contributions to the open-source community include Bayesian Methods for Hackers and lifelines. Cameron was raised in Guelph, Ontario, but was educated at the University of Waterloo and Independent University of Moscow. He currently lives in Ottawa, Ontario, working with the online commerce leader Shopify.

Foreword xiii

Preface xv

Acknowledgments xvii

About the Author xix

Chapter 1: The Philosophy of Bayesian Inference 1

1.1 Introduction 1

1.2 Our Bayesian Framework 5

1.3 Probability Distributions 8

1.4 Using Computers to Perform Bayesian Inference for Us 12

1.5 Conclusion 20

1.6 Appendix 20

1.7 Exercises 24

1.8 References 25

Chapter 2: A Little More on PyMC 27

2.1 Introduction 27

2.2 Modeling Approaches 33

2.3 Is Our Model Appropriate? 61

2.4 Conclusion 68

2.5 Appendix 68

2.6 Exercises 69

2.7 References 69

Chapter 3: Opening the Black Box of MCMC 71

3.1 The Bayesian Landscape 71

3.2 Diagnosing Convergence 92

3.3 Useful Tips for MCMC 98

3.4 Conclusion 99

3.5 Reference 99

Chapter 4: The Greatest Theorem Never Told 101

4.1 Introduction 101

4.2 The Law of Large Numbers 101

4.3 The Disorder of Small Numbers 107

4.4 Conclusion 122

4.5 Appendix 122

4.6 Exercises 123

4.7 References 125

Chapter 5: Would You Rather Lose an Arm or a Leg? 127

5.1 Introduction 127

5.2 Loss Functions 127

5.3 Machine Learning via Bayesian Methods 139

5.4 Conclusion 156

5.5 References 156

Chapter 6: Getting Our Priorities Straight 157

6.1 Introduction 157

6.2 Subjective versus Objective Priors 157

6.3 Useful Priors to Know About 161

6.4 Example: Bayesian Multi-Armed Bandits 164

6.5 Eliciting Prior Distributions from Domain Experts 176

6.6 Conjugate Priors 185

6.7 Jeffreys Priors 185

6.8 Effect of the Prior as N Increases 187

6.9 Conclusion 189

6.10 Appendix 190

6.11 References 193

Chapter 7: Bayesian A/B Testing 195

7.1 Introduction 195

7.2 Conversion Testing Recap 195

7.3 Adding a Linear Loss Function 198

7.4 Going Beyond Conversions: t-test 204

7.5 Estimating the Increase 207

7.6 Conclusion 211

7.7 References 212

Glossary 213

Index 217