book

Bayesian Methods for Hackers: Probabilistic Programming and Bayesian Inference

by Cameron Davidson-Pilon

October 2015

Beginner to intermediate

300 pages

7h 19m

English

Addison-Wesley Professional

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1.1 Introduction1.1.1 The Bayesian State of Mind1.1.2 Bayesian Inference in Practice1.1.3 Are Frequentist Methods Incorrect?1.1.4 A Note on “Big Data”1.2 Our Bayesian Framework1.2.1 Example: Mandatory Coin-Flip1.2.2 Example: Librarian or Farmer?1.3 Probability Distributions1.3.1 Discrete Case1.3.2 Continuous Case1.3.3 But What Is λ?1.4 Using Computers to Perform Bayesian Inference for Us1.4.1 Example: Inferring Behavior from Text-Message Data1.4.2 Introducing Our First Hammer: PyMC1.4.3 Interpretation1.4.4 What Good Are Samples from the Posterior, Anyway?1.5 Conclusion1.6 Appendix1.6.1 Determining Statistically if the Two λs Are Indeed Different?1.6.2 Extending to Two Switchpoints1.7 Exercises1.7.1 Answers1.8 References

2.1 Introduction2.1.1 Parent and Child Relationships2.1.2 PyMC Variables2.1.3 Including Observations in the Model2.1.4 Finally. . .2.2 Modeling Approaches2.2.1 Same Story, Different Ending2.2.2 Example: Bayesian A/B Testing2.2.3 A Simple Case2.2.4 A and B Together2.2.5 Example: An Algorithm for Human Deceit2.2.6 The Binomial Distribution2.2.7 Example: Cheating Among Students2.2.8 Alternative PyMC Model2.2.9 More PyMC Tricks2.2.10 Example: Challenger Space Shuttle Disaster2.2.11 The Normal Distribution2.2.12 What Happened the Day of the Challenger Disaster?2.3 Is Our Model Appropriate?2.3.1 Separation Plots2.4 Conclusion2.5 Appendix2.6 Exercises2.6.1 Answers2.7 References
3.1 The Bayesian Landscape3.1.1 Exploring the Landscape Using MCMC3.1.2 Algorithms to Perform MCMC3.1.3 Other Approximation Solutions to the Posterior3.1.4 Example: Unsupervised Clustering Using a Mixture Model3.1.5 Don’t Mix Posterior Samples3.1.6 Using MAP to Improve Convergence3.2 Diagnosing Convergence3.2.1 Autocorrelation3.2.2 Thinning3.2.3 pymc.Matplot.plot()3.3 Useful Tips for MCMC3.3.1 Intelligent Starting Values3.3.2 Priors3.3.3 The Folk Theorem of Statistical Computing3.4 Conclusion3.5 References
4.1 Introduction4.2 The Law of Large Numbers4.2.1 Intuition4.2.2 Example: Convergence of Poisson Random Variables4.2.3 How Do We Compute Var(Z)?4.2.4 Expected Values and Probabilities4.2.5 What Does All This Have to Do with Bayesian Statistics?4.3 The Disorder of Small Numbers4.3.1 Example: Aggregated Geographic Data4.3.2 Example: Kaggle’s U.S. Census Return Rate Challenge4.3.3 Example: How to Sort Reddit Comments4.3.4 Sorting!4.3.5 But This Is Too Slow for Real-Time!4.3.6 Extension to Starred Rating Systems4.4 Conclusion4.5 Appendix4.5.1 Derivation of Sorting Comments Formula4.6 Exercises4.6.1 Answers4.7 References
5.1 Introduction5.2 Loss Functions5.2.1 Loss Functions in the Real World5.2.2 Example: Optimizing for the Showcase on The Price Is Right5.3 Machine Learning via Bayesian Methods5.3.1 Example: Financial Prediction5.3.2 Example: Kaggle Contest on Observing Dark Worlds5.3.3 The Data5.3.4 Priors5.3.5 Training and PyMC Implementation5.4 Conclusion5.5 References
6.1 Introduction6.2 Subjective versus Objective Priors6.2.1 Objective Priors6.2.2 Subjective Priors6.2.3 Decisions, Decisions . . .6.2.4 Empirical Bayes6.3 Useful Priors to Know About6.3.1 The Gamma Distribution6.3.2 The Wishart Distribution6.3.3 The Beta Distribution6.4 Example: Bayesian Multi-Armed Bandits6.4.1 Applications6.4.2 A Proposed Solution6.4.3 A Measure of Good6.4.4 Extending the Algorithm6.5 Eliciting Prior Distributions from Domain Experts6.5.1 Trial Roulette Method6.5.2 Example: Stock Returns6.5.3 Pro Tips for the Wishart Distribution6.6 Conjugate Priors6.7 Jeffreys Priors6.8 Effect of the Prior as N Increases6.9 Conclusion6.10 Appendix6.10.1 Bayesian Perspective of Penalized Linear Regressions6.10.2 Picking a Degenerate Prior6.11 References
7.1 Introduction7.2 Conversion Testing Recap7.3 Adding a Linear Loss Function7.3.1 Expected Revenue Analysis7.3.2 Extending to an A/B Experiment7.4 Going Beyond Conversions: t-test7.4.1 The Setup of the t-test7.5 Estimating the Increase7.5.1 Creating Point Estimates7.6 Conclusion7.7 References

Overview

Master Bayesian Inference through Practical Examples and Computation–Without Advanced Mathematical Analysis

Bayesian methods of inference are deeply natural and extremely powerful. However, most discussions of Bayesian inference rely on intensely complex mathematical analyses and artificial examples, making it inaccessible to anyone without a strong mathematical background. Now, though, Cameron Davidson-Pilon introduces Bayesian inference from a computational perspective, bridging theory to practice–freeing you to get results using computing power.

Bayesian Methods for Hackers illuminates Bayesian inference through probabilistic programming with the powerful PyMC language and the closely related Python tools NumPy, SciPy, and Matplotlib. Using this approach, you can reach effective solutions in small increments, without extensive mathematical intervention.

Davidson-Pilon begins by introducing the concepts underlying Bayesian inference, comparing it with other techniques and guiding you through building and training your first Bayesian model. Next, he introduces PyMC through a series of detailed examples and intuitive explanations that have been refined after extensive user feedback. You’ll learn how to use the Markov Chain Monte Carlo algorithm, choose appropriate sample sizes and priors, work with loss functions, and apply Bayesian inference in domains ranging from finance to marketing. Once you’ve mastered these techniques, you’ll constantly turn to this guide for the working PyMC code you need to jumpstart future projects.

Coverage includes

• Learning the Bayesian “state of mind” and its practical implications

• Understanding how computers perform Bayesian inference

• Using the PyMC Python library to program Bayesian analyses

• Building and debugging models with PyMC

• Testing your model’s “goodness of fit”

• Opening the “black box” of the Markov Chain Monte Carlo algorithm to see how and why it works

• Leveraging the power of the “Law of Large Numbers”

• Mastering key concepts, such as clustering, convergence, autocorrelation, and thinning

• Using loss functions to measure an estimate’s weaknesses based on your goals and desired outcomes

• Selecting appropriate priors and understanding how their influence changes with dataset size

• Overcoming the “exploration versus exploitation” dilemma: deciding when “pretty good” is good enough

• Using Bayesian inference to improve A/B testing

• Solving data science problems when only small amounts of data are available

Cameron Davidson-Pilon has worked in many areas of applied mathematics, from the evolutionary dynamics of genes and diseases to stochastic modeling of financial prices. His contributions to the open source community include lifelines, an implementation of survival analysis in Python. Educated at the University of Waterloo and at the Independent University of Moscow, he currently works with the online commerce leader Shopify.