MATLAB Control Systems Engineering

An engaging, sophisticated, and fun introduction to the field of Bayesian statistics, Bayes Rules!: An Introduction to Applied Bayesian Modeling brings the power of modern Bayesian thinking, modeling, and computing to a broad audience. In particular, the book is an ideal resource for advanced undergraduate statistics students and practitioners with comparable experience. the book assumes that readers are familiar with the content covered in a typical undergraduate-level introductory statistics course. Readers will also, ideally, have some experience with undergraduate-level probability, calculus, and the R statistical software. Readers without this background will still be able to follow along so long as they are eager to pick up these tools on the fly as all R code is provided.Bayes Rules! empowers readers to weave Bayesian approaches into their everyday practice. Discussions and applications are data driven. A natural progression from fundamental to multivariable, hierarchical models emphasizes a practical and generalizable model building process. The evaluation of these Bayesian models reflects the fact that a data analysis does not exist in a vacuum. Features • Utilizes data-driven examples and exercises. • Emphasizes the iterative model building and evaluation process. • Surveys an interconnected range of multivariable regression and classification models. • Presents fundamental Markov chain Monte Carlo simulation. • Integrates R code, including RStan modeling tools and the bayesrules package. • Encourages readers to tap into their intuition and learn by doing. • Provides a friendly and inclusive introduction to technical Bayesian concepts. • Supports Bayesian applications with foundational Bayesian theory. Cover Half Title Series Page Title Page Copyright Page Dedication Contents Foreword Preface About the Authors I. Bayesian Foundations 1. The Big (Bayesian) Picture 1.1. Thinking like a Bayesian 1.1.1. Quiz yourself 1.1.2. The meaning of probability 1.1.3. The Bayesian balancing act 1.1.4. Asking questions 1.2. A quick history lesson 1.3. A look ahead 1.3.1. Unit 1: Bayesian foundations 1.3.2. Unit 2: Posterior simulation & analysis 1.3.3. Unit 3: Bayesian regression & classification 1.3.4. Unit 4: Hierarchical Bayesian models 1.4. Chapter summary 1.5. Exercises 2. Bayes’ Rule 2.1. Building a Bayesian model for events 2.1.1. Prior probability model 2.1.2. Conditional probability & likelihood 2.1.3. Normalizing constants 2.1.4. Posterior probability model via Bayes’ Rule! 2.1.5. Posterior simulation 2.2. Example: Pop vs soda vs coke 2.3. Building a Bayesian model for random variables 2.3.1. Prior probability model 2.3.2. The Binomial data model 2.3.3. The Binomial likelihood function 2.3.4. Normalizing constant 2.3.5. Posterior probability model 2.3.6. Posterior shortcut 2.3.7. Posterior simulation 2.4. Chapter summary 2.5. Exercises 2.5.1. Building up to Bayes’ Rule 2.5.2. Practice Bayes’ Rule for events 2.5.3. Practice Bayes’ Rule for random variables 2.5.4. Simulation exercises 3. The Beta-Binomial Bayesian Model 3.1. The Beta prior model 3.1.1. Beta foundations 3.1.2. Tuning the Beta prior 3.2. The Binomial data model & likelihood function 3.3. The Beta posterior model 3.4. The Beta-Binomial model 3.5. Simulating the Beta-Binomial 3.6. Example: Milgram’s behavioral study of obedience 3.6.1. A Bayesian analysis 3.6.2. The role of ethics in statistics and data science 3.7. Chapter summary 3.8. Exercises 3.8.1. Practice: Beta prior models 3.8.2. Practice: Beta-Binomial models 4. Balance and Sequentiality in Bayesian Analyses 4.1. Different priors, different posteriors 4.2. Different data, different posteriors 4.3. Striking a balance between the prior & data 4.3.1. Connecting observations to concepts 4.3.2. Connecting concepts to theory 4.4. Sequential analysis: Evolving with data 4.5. Proving data order invariance 4.6. Don’t be stubborn 4.7. A note on subjectivity 4.8. Chapter summary 4.9. Exercises 4.9.1. Review exercises 4.9.2. Practice: Different priors, different posteriors 4.9.3. Practice: Balancing the data & prior 4.9.4. Practice: Sequentiality 5. Conjugate Families 5.1. Revisiting choice of prior 5.2. Gamma-Poisson conjugate family 5.2.1. The Poisson data model 5.2.2. Potential priors 5.2.3. Gamma prior 5.2.4. Gamma-Poisson conjugacy 5.3. Normal-Normal conjugate family 5.3.1. The Normal data model 5.3.2. Normal prior 5.3.3. Normal-Normal conjugacy 5.3.4. Optional: Proving Normal-Normal conjugacy 5.4. Why no simulation in this chapter? 5.5. Critiques of conjugate family models 5.6. Chapter summary 5.7. Exercises 5.7.1. Practice: Gamma-Poisson 5.7.2. Practice: Normal-Normal 5.7.3. General practice exercises II. Posterior Simulation & Analysis 6. Approximating the Posterior 6.1. Grid approximation 6.1.1. A Beta-Binomial example 6.1.2. A Gamma-Poisson example 6.1.3. Limitations 6.2. Markov chains via rstan 6.2.1. A Beta-Binomial example 6.2.2. A Gamma-Poisson example 6.3. Markov chain diagnostics 6.3.1. Examining trace plots 6.3.2. Comparing parallel chains 6.3.3. Calculating effective sample size & autocorrelation 6.3.4. Calculating R-hat 6.4. Chapter summary 6.5. Exercises 6.5.1. Conceptual exercises 6.5.2. Practice: Grid approximation 6.5.3. Practice: MCMC 7. MCMC under the Hood 7.1. The big idea 7.2. The Metropolis-Hastings algorithm 7.3. Implementing the Metropolis-Hastings 7.4. Tuning the Metropolis-Hastings algorithm 7.5. A Beta-Binomial example 7.6. Why the algorithm works 7.7. Variations on the theme 7.8. Chapter summary 7.9. Exercises 7.9.1. Conceptual exercises 7.9.2. Practice: Normal-Normal simulation 7.9.3. Practice: Simulating more Bayesian models 8. Posterior Inference & Prediction 8.1. Posterior estimation 8.2. Posterior hypothesis testing 8.2.1. One-sided tests 8.2.2. Two-sided tests 8.3. Posterior prediction 8.4. Posterior analysis with MCMC 8.4.1. Posterior simulation 8.4.2. Posterior estimation & hypothesis testing 8.4.3. Posterior prediction 8.5. Bayesian benefits 8.6. Chapter summary 8.7. Exercises 8.7.1. Conceptual exercises 8.7.2. Practice exercises 8.7.3. Applied exercises III. Bayesian Regression & Classification 9. Simple Normal Regression 9.1. Building the regression model 9.1.1. Specifying the data model 9.1.2. Specifying the priors 9.1.3. Putting it all together 9.2. Tuning prior models for regression parameters 9.3. Posterior simulation 9.3.1. Simulation via rstanarm 9.3.2. Optional: Simulation via rstan 9.4. Interpreting the posterior 9.5. Posterior prediction 9.5.1. Building a posterior predictive model 9.5.2. Posterior prediction with rstanarm 9.6. Sequential regression modeling 9.7. Using default rstanarm priors 9.8. You’re not done yet! 9.9. Chapter summary 9.10. Exercises 9.10.1. Conceptual exercises 9.10.2. Applied exercises 10. Evaluating Regression Models 10.1. Is the model fair? 10.2. How wrong is the model? 10.2.1. Checking the model assumptions 10.2.2. Dealing with wrong models 10.3. How accurate are the posterior predictive models? 10.3.1. Posterior predictive summaries 10.3.2. Cross-validation 10.3.3. Expected log-predictive density 10.3.4. Improving posterior predictive accuracy 10.4. How good is the MCMC simulation vs how good is the model? 10.5. Chapter summary 10.6. Exercises 10.6.1. Conceptual exercises 10.6.2. Applied exercises 10.6.3. Open-ended exercises 11. Extending the Normal Regression Model 11.1. Utilizing a categorical predictor 11.1.1. Building the model 11.1.2. Simulating the posterior 11.2. Utilizing two predictors 11.2.1. Building the model 11.2.2. Understanding the priors 11.2.3. Simulating the posterior 11.2.4. Posterior prediction 11.3. Optional: Utilizing interaction terms 11.3.1. Building the model 11.3.2. Simulating the posterior 11.3.3. Do you need an interaction term? 11.4. Dreaming bigger: Utilizing more than 2 predictors! 11.5. Model evaluation & comparison 11.5.1. Evaluating predictive accuracy using visualizations 11.5.2. Evaluating predictive accuracy using cross-validation 11.5.3. Evaluating predictive accuracy using ELPD 11.5.4. The bias-variance trade-off 11.6. Chapter summary 11.7. Exercises 11.7.1. Conceptual exercises 11.7.2. Applied exercises 11.7.3. Open-ended exercises 12. Poisson & Negative Binomial Regression 12.1. Building the Poisson regression model 12.1.1. Specifying the data model 12.1.2. Specifying the priors 12.2. Simulating the posterior 12.3. Interpreting the posterior 12.4. Posterior prediction 12.5. Model evaluation 12.6. Negative Binomial regression for overdispersed counts 12.7. Generalized linear models: Building on the theme 12.8. Chapter summary 12.9. Exercises 12.9.1. Conceptual exercises 12.9.2. Applied exercises 13. Logistic Regression 13.1. Pause: Odds & probability 13.2. Building the logistic regression model 13.2.1. Specifying the data model 13.2.2. Specifying the priors 13.3. Simulating the posterior 13.4. Prediction & classification 13.5. Model evaluation 13.6. Extending the model 13.7. Chapter summary 13.8. Exercises 13.8.1. Conceptual exercises 13.8.2. Applied exercises 13.8.3. Open-ended exercises 14. Naive Bayes Classification 14.1. Classifying one penguin 14.1.1. One categorical predictor 14.1.2. One quantitative predictor 14.1.3. Two predictors 14.2. Implementing & evaluating naive Bayes classification 14.3. Naive Bayes vs logistic regression 14.4. Chapter summary 14.5. Exercises 14.5.1. Conceptual exercises 14.5.2. Applied exercises 14.5.3. Open-ended exercises IV. Hierarchical Bayesian models 15. Hierarchical Models are Exciting 15.1. Complete pooling 15.2. No pooling 15.3. Hierarchical data 15.4. Partial pooling with hierarchical models 15.5. Chapter summary 15.6. Exercises 15.6.1. Conceptual exercises 15.6.2. Applied exercises 16. (Normal) Hierarchical Models without Predictors 16.1. Complete pooled model 16.2. No pooled model 16.3. Building the hierarchical model 16.3.1. The hierarchy 16.3.2. Another way to think about it 16.3.3. Within- vs between-group variability 16.4. Posterior analysis 16.4.1. Posterior simulation 16.4.2. Posterior analysis of global parameters 16.4.3. Posterior analysis of group-specific parameters 16.5. Posterior prediction 16.6. Shrinkage & the bias-variance trade-off 16.7. Not everything is hierarchical 16.8. Chapter summary 16.9. Exercises 16.9.1. Conceptual exercises 16.9.2. Applied exercises 17. (Normal) Hierarchical Models with Predictors 17.1. First steps: Complete pooling 17.2. Hierarchical model with varying intercepts 17.2.1. Model building 17.2.2. Another way to think about it 17.2.3. Tuning the prior 17.2.4. Posterior simulation & analysis 17.3. Hierarchical model with varying intercepts & slopes 17.3.1. Model building 17.3.2. Optional: The decomposition of covariance model 17.3.3. Posterior simulation & analysis 17.4. Model evaluation & selection 17.5. Posterior prediction 17.6. Details: Longitudinal data 17.7. Example: Danceability 17.8. Chapter summary 17.9. Exercises 17.9.1. Conceptual exercises 17.9.2. Applied exercises 17.9.3. Open-ended exercises 18. Non-Normal Hierarchical Regression & Classification 18.1. Hierarchical logistic regression 18.1.1. Model building & simulation 18.1.2. Posterior analysis 18.1.3. Posterior classification 18.1.4. Model evaluation 18.2. Hierarchical Poisson & Negative Binomial regression 18.2.1. Model building & simulation 18.2.2. Posterior analysis 18.2.3. Model evaluation 18.3. Chapter summary 18.4. Exercises 18.4.1. Applied & conceptual exercises 18.4.2. Open-ended exercises 19. Adding More Layers 19.1. Group-level predictors 19.1.1. A model using only individual-level predictors 19.1.2. Incorporating group-level predictors 19.1.3. Posterior simulation & global analysis 19.1.4. Posterior group-level analysis 19.1.5. We’re just scratching the surface! 19.2. Incorporating two (or more!) grouping variables 19.2.1. Data with two grouping variables 19.2.2. Building a model with two grouping variables 19.2.3. Simulating models with two grouping variables 19.2.4. Examining the group-specific parameters 19.2.5. We’re just scratching the surface! 19.3. Exercises 19.3.1. Conceptual exercises 19.3.2. Applied exercises 19.4. Goodbye! Bibliography Index "An engaging, sophisticated, and fun introduction to the field of Bayesian Statistics, Bayes Rules! An Introduction to Bayesian Modeling with R brings the power of modern Bayesian thinking, modeling, and computing to a broad audience. In particular, it is an ideal resource for advanced undergraduate Statistics students and practitioners with comparable experience. Bayes Rules! empowers readers to weave Bayesian approaches into their everyday practice. Discussions and applications are data driven. A natural progression from fundamental to multivariable, hierarchical models emphasizes a practical and generalizable model building process. The evaluation of these Bayesian models reflects the fact that a data analysis does not exist in a vacuum"-- Provided by publisher. This book brings the power of modern Bayesian thinking, modeling, and computing to a broad audience. In particular, it is an ideal resource for advanced undergraduate statistics students and practitioners with comparable experience. It empowers readers to weave Bayesian approaches into their everyday practice.