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| Boilerpipe Text | News
Foreword
Preface
Audience
Navigating this book
Getting set up
Accessibility and inclusion
Contact us
Acknowledgments
License
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!
References
Published with bookdown
Bayes Rules! An Introduction to Applied Bayesian Modeling
Alicia A. Johnson, Miles Q. Ott, Mine Dogucu
2021-12-01 | |||||||||||||||
| Markdown | - [](https://www.bayesrulesbook.com/)
- [News](https://www.bayesrulesbook.com/)
- [Foreword](https://www.bayesrulesbook.com/foreword)
- [Preface](https://www.bayesrulesbook.com/preface)
- [Audience](https://www.bayesrulesbook.com/preface#audience)
- [Navigating this book](https://www.bayesrulesbook.com/preface#navigating-this-book)
- [Getting set up](https://www.bayesrulesbook.com/preface#setup)
- [Accessibility and inclusion](https://www.bayesrulesbook.com/preface#accessibility-and-inclusion)
- [Contact us](https://www.bayesrulesbook.com/preface#contact-us)
- [Acknowledgments](https://www.bayesrulesbook.com/preface#acknowledgments)
- [License](https://www.bayesrulesbook.com/preface#license)
- [About the Authors](https://www.bayesrulesbook.com/about-the-authors)
- **I Bayesian Foundations**
- [**1** The Big (Bayesian) Picture](https://www.bayesrulesbook.com/chapter-1)
- [**1\.1** Thinking like a Bayesian](https://www.bayesrulesbook.com/chapter-1#thinking-like-a-bayesian)
- [**1\.1.1** Quiz yourself](https://www.bayesrulesbook.com/chapter-1#quiz-yourself-1)
- [**1\.1.2** The meaning of probability](https://www.bayesrulesbook.com/chapter-1#bayes-v-freq-probability)
- [**1\.1.3** The Bayesian balancing act](https://www.bayesrulesbook.com/chapter-1#the-bayesian-balancing-act)
- [**1\.1.4** Asking questions](https://www.bayesrulesbook.com/chapter-1#asking-questions)
- [**1\.2** A quick history lesson](https://www.bayesrulesbook.com/chapter-1#a-quick-history-lesson)
- [**1\.3** A look ahead](https://www.bayesrulesbook.com/chapter-1#a-look-ahead)
- [**1\.3.1** Unit 1: Bayesian foundations](https://www.bayesrulesbook.com/chapter-1#ch1-unit-1)
- [**1\.3.2** Unit 2: Posterior simulation & analysis](https://www.bayesrulesbook.com/chapter-1#unit-2-posterior-simulation-analysis)
- [**1\.3.3** Unit 3: Bayesian regression & classification](https://www.bayesrulesbook.com/chapter-1#unit-3-bayesian-regression-classification)
- [**1\.3.4** Unit 4: Hierarchical Bayesian models](https://www.bayesrulesbook.com/chapter-1#unit-4-hierarchical-bayesian-models)
- [**1\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-1#chapter-summary)
- [**1\.5** Exercises](https://www.bayesrulesbook.com/chapter-1#exercises)
- [**2** Bayes’ Rule](https://www.bayesrulesbook.com/chapter-2)
- [**2\.1** Building a Bayesian model for events](https://www.bayesrulesbook.com/chapter-2#building-a-bayesian-model-for-events)
- [**2\.1.1** Prior probability model](https://www.bayesrulesbook.com/chapter-2#prior-probability-model)
- [**2\.1.2** Conditional probability & likelihood](https://www.bayesrulesbook.com/chapter-2#conditional-probability-likelihood)
- [**2\.1.3** Normalizing constants](https://www.bayesrulesbook.com/chapter-2#normalizing-constants)
- [**2\.1.4** Posterior probability model via Bayes’ Rule\!](https://www.bayesrulesbook.com/chapter-2#posterior-probability-model-via-bayes-rule)
- [**2\.1.5** Posterior simulation](https://www.bayesrulesbook.com/chapter-2#posterior-simulation)
- [**2\.2** Example: Pop vs soda vs coke](https://www.bayesrulesbook.com/chapter-2#michelle-simple)
- [**2\.3** Building a Bayesian model for random variables](https://www.bayesrulesbook.com/chapter-2#cousin-cole)
- [**2\.3.1** Prior probability model](https://www.bayesrulesbook.com/chapter-2#prior-probability-model-1)
- [**2\.3.2** The Binomial data model](https://www.bayesrulesbook.com/chapter-2#chapter-2-binomial-likelihood)
- [**2\.3.3** The Binomial likelihood function](https://www.bayesrulesbook.com/chapter-2#the-binomial-likelihood-function)
- [**2\.3.4** Normalizing constant](https://www.bayesrulesbook.com/chapter-2#normalizing-constant)
- [**2\.3.5** Posterior probability model](https://www.bayesrulesbook.com/chapter-2#posterior-probability-model)
- [**2\.3.6** Posterior shortcut](https://www.bayesrulesbook.com/chapter-2#shortcut-2)
- [**2\.3.7** Posterior simulation](https://www.bayesrulesbook.com/chapter-2#chapter-2-simulation)
- [**2\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-2#chapter-summary-1)
- [**2\.5** Exercises](https://www.bayesrulesbook.com/chapter-2#exercises-1)
- [**2\.5.1** Building up to Bayes’ Rule](https://www.bayesrulesbook.com/chapter-2#building-up-to-bayes-rule)
- [**2\.5.2** Practice Bayes’ Rule for events](https://www.bayesrulesbook.com/chapter-2#practice-bayes-rule-for-events)
- [**2\.5.3** Practice Bayes’ Rule for random variables](https://www.bayesrulesbook.com/chapter-2#practice-bayes-rule-for-random-variables)
- [**2\.5.4** Simulation exercises](https://www.bayesrulesbook.com/chapter-2#simulation-exercises)
- [**3** The Beta-Binomial Bayesian Model](https://www.bayesrulesbook.com/chapter-3)
- [**3\.1** The Beta prior model](https://www.bayesrulesbook.com/chapter-3#ch3-prior)
- [**3\.1.1** Beta foundations](https://www.bayesrulesbook.com/chapter-3#beta-foundations)
- [**3\.1.2** Tuning the Beta prior](https://www.bayesrulesbook.com/chapter-3#tuning-the-beta-prior)
- [**3\.2** The Binomial data model & likelihood function](https://www.bayesrulesbook.com/chapter-3#the-binomial-data-model-likelihood-function)
- [**3\.3** The Beta posterior model](https://www.bayesrulesbook.com/chapter-3#the-beta-posterior-model)
- [**3\.4** The Beta-Binomial model](https://www.bayesrulesbook.com/chapter-3#ch3-bbmodel)
- [**3\.5** Simulating the Beta-Binomial](https://www.bayesrulesbook.com/chapter-3#chapter-3-simulation)
- [**3\.6** Example: Milgram’s behavioral study of obedience](https://www.bayesrulesbook.com/chapter-3#milgram-3)
- [**3\.6.1** A Bayesian analysis](https://www.bayesrulesbook.com/chapter-3#a-bayesian-analysis)
- [**3\.6.2** The role of ethics in statistics and data science](https://www.bayesrulesbook.com/chapter-3#the-role-of-ethics-in-statistics-and-data-science)
- [**3\.7** Chapter summary](https://www.bayesrulesbook.com/chapter-3#chapter-summary-2)
- [**3\.8** Exercises](https://www.bayesrulesbook.com/chapter-3#exercises-2)
- [**3\.8.1** Practice: Beta prior models](https://www.bayesrulesbook.com/chapter-3#practice-beta-prior-models)
- [**3\.8.2** Practice: Beta-Binomial models](https://www.bayesrulesbook.com/chapter-3#practice-beta-binomial-models)
- [**4** Balance and Sequentiality in Bayesian Analyses](https://www.bayesrulesbook.com/chapter-4)
- [**4\.1** Different priors, different posteriors](https://www.bayesrulesbook.com/chapter-4#ch4-priors)
- [**4\.2** Different data, different posteriors](https://www.bayesrulesbook.com/chapter-4#ch4-data)
- [**4\.3** Striking a balance between the prior & data](https://www.bayesrulesbook.com/chapter-4#striking-a-balance-between-the-prior-data)
- [**4\.3.1** Connecting observations to concepts](https://www.bayesrulesbook.com/chapter-4#connecting-observations-to-concepts)
- [**4\.3.2** Connecting concepts to theory](https://www.bayesrulesbook.com/chapter-4#ch4-concepts)
- [**4\.4** Sequential analysis: Evolving with data](https://www.bayesrulesbook.com/chapter-4#ch4-sequential)
- [**4\.5** Proving data order invariance](https://www.bayesrulesbook.com/chapter-4#proving-data-order-invariance)
- [**4\.6** Don’t be stubborn](https://www.bayesrulesbook.com/chapter-4#dont-be-stubborn)
- [**4\.7** A note on subjectivity](https://www.bayesrulesbook.com/chapter-4#a-note-on-subjectivity)
- [**4\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-4#chapter-summary-3)
- [**4\.9** Exercises](https://www.bayesrulesbook.com/chapter-4#exercises-3)
- [**4\.9.1** Review exercises](https://www.bayesrulesbook.com/chapter-4#review-exercises)
- [**4\.9.2** Practice: Different priors, different posteriors](https://www.bayesrulesbook.com/chapter-4#practice-different-priors-different-posteriors)
- [**4\.9.3** Practice: Balancing the data & prior](https://www.bayesrulesbook.com/chapter-4#practice-balancing-the-data-prior)
- [**4\.9.4** Practice: Sequentiality](https://www.bayesrulesbook.com/chapter-4#practice-sequentiality)
- [**5** Conjugate Families](https://www.bayesrulesbook.com/chapter-5)
- [**5\.1** Revisiting choice of prior](https://www.bayesrulesbook.com/chapter-5#revisiting-choice-of-prior)
- [**5\.2** Gamma-Poisson conjugate family](https://www.bayesrulesbook.com/chapter-5#gamma-poisson-conjugate-family)
- [**5\.2.1** The Poisson data model](https://www.bayesrulesbook.com/chapter-5#the-poisson-data-model)
- [**5\.2.2** Potential priors](https://www.bayesrulesbook.com/chapter-5#potential-priors)
- [**5\.2.3** Gamma prior](https://www.bayesrulesbook.com/chapter-5#gamma-prior)
- [**5\.2.4** Gamma-Poisson conjugacy](https://www.bayesrulesbook.com/chapter-5#gamma-poisson-conjugacy)
- [**5\.3** Normal-Normal conjugate family](https://www.bayesrulesbook.com/chapter-5#normal-normal-conjugate-family)
- [**5\.3.1** The Normal data model](https://www.bayesrulesbook.com/chapter-5#the-normal-data-model)
- [**5\.3.2** Normal prior](https://www.bayesrulesbook.com/chapter-5#normal-prior)
- [**5\.3.3** Normal-Normal conjugacy](https://www.bayesrulesbook.com/chapter-5#ch5-n-n-section)
- [**5\.3.4** Optional: Proving Normal-Normal conjugacy](https://www.bayesrulesbook.com/chapter-5#ch5-optional)
- [**5\.4** Why no simulation in this chapter?](https://www.bayesrulesbook.com/chapter-5#why-no-simulation-in-this-chapter)
- [**5\.5** Critiques of conjugate family models](https://www.bayesrulesbook.com/chapter-5#critiques-of-conjugate-family-models)
- [**5\.6** Chapter summary](https://www.bayesrulesbook.com/chapter-5#chapter-summary-4)
- [**5\.7** Exercises](https://www.bayesrulesbook.com/chapter-5#exercises-4)
- [**5\.7.1** Practice: Gamma-Poisson](https://www.bayesrulesbook.com/chapter-5#practice-gamma-poisson)
- [**5\.7.2** Practice: Normal-Normal](https://www.bayesrulesbook.com/chapter-5#practice-normal-normal)
- [**5\.7.3** General practice exercises](https://www.bayesrulesbook.com/chapter-5#general-practice-exercises)
- **II Posterior Simulation & Analysis**
- [**6** Approximating the Posterior](https://www.bayesrulesbook.com/chapter-6)
- [**6\.1** Grid approximation](https://www.bayesrulesbook.com/chapter-6#grid-approximation)
- [**6\.1.1** A Beta-Binomial example](https://www.bayesrulesbook.com/chapter-6#beta-binomial-grid)
- [**6\.1.2** A Gamma-Poisson example](https://www.bayesrulesbook.com/chapter-6#Gamma-Poisson-grid)
- [**6\.1.3** Limitations](https://www.bayesrulesbook.com/chapter-6#limitations)
- [**6\.2** Markov chains via rstan](https://www.bayesrulesbook.com/chapter-6#markov-chains-via-rstan)
- [**6\.2.1** A Beta-Binomial example](https://www.bayesrulesbook.com/chapter-6#a-beta-binomial-example)
- [**6\.2.2** A Gamma-Poisson example](https://www.bayesrulesbook.com/chapter-6#a-gamma-poisson-example)
- [**6\.3** Markov chain diagnostics](https://www.bayesrulesbook.com/chapter-6#diagnostics)
- [**6\.3.1** Examining trace plots](https://www.bayesrulesbook.com/chapter-6#chapter-6-trace)
- [**6\.3.2** Comparing parallel chains](https://www.bayesrulesbook.com/chapter-6#chapter-6-parallel)
- [**6\.3.3** Calculating effective sample size & autocorrelation](https://www.bayesrulesbook.com/chapter-6#calculating-effective-sample-size-autocorrelation)
- [**6\.3.4** Calculating R-hat](https://www.bayesrulesbook.com/chapter-6#calculating-r-hat)
- [**6\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-6#chapter-summary-5)
- [**6\.5** Exercises](https://www.bayesrulesbook.com/chapter-6#exercises-5)
- [**6\.5.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-6#conceptual-exercises)
- [**6\.5.2** Practice: Grid approximation](https://www.bayesrulesbook.com/chapter-6#practice-grid-approximation)
- [**6\.5.3** Practice: MCMC](https://www.bayesrulesbook.com/chapter-6#practice-mcmc)
- [**7** MCMC under the Hood](https://www.bayesrulesbook.com/chapter-7)
- [**7\.1** The big idea](https://www.bayesrulesbook.com/chapter-7#the-big-idea)
- [**7\.2** The Metropolis-Hastings algorithm](https://www.bayesrulesbook.com/chapter-7#the-metropolis-hastings-algorithm)
- [**7\.3** Implementing the Metropolis-Hastings](https://www.bayesrulesbook.com/chapter-7#ch7-implementation)
- [**7\.4** Tuning the Metropolis-Hastings algorithm](https://www.bayesrulesbook.com/chapter-7#tuning-the-metropolis-hastings-algorithm)
- [**7\.5** A Beta-Binomial example](https://www.bayesrulesbook.com/chapter-7#a-beta-binomial-example-1)
- [**7\.6** Why the algorithm works](https://www.bayesrulesbook.com/chapter-7#why-the-algorithm-works)
- [**7\.7** Variations on the theme](https://www.bayesrulesbook.com/chapter-7#variations-on-the-theme)
- [**7\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-7#chapter-summary-6)
- [**7\.9** Exercises](https://www.bayesrulesbook.com/chapter-7#exercises-6)
- [**7\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-7#conceptual-exercises-1)
- [**7\.9.2** Practice: Normal-Normal simulation](https://www.bayesrulesbook.com/chapter-7#practice-normal-normal-simulation)
- [**7\.9.3** Practice: Simulating more Bayesian models](https://www.bayesrulesbook.com/chapter-7#practice-simulating-more-bayesian-models)
- [**8** Posterior Inference & Prediction](https://www.bayesrulesbook.com/chapter-8)
- [**8\.1** Posterior estimation](https://www.bayesrulesbook.com/chapter-8#chapter-8-estimation)
- [**8\.2** Posterior hypothesis testing](https://www.bayesrulesbook.com/chapter-8#posterior-hypothesis-testing)
- [**8\.2.1** One-sided tests](https://www.bayesrulesbook.com/chapter-8#one-sided-tests)
- [**8\.2.2** Two-sided tests](https://www.bayesrulesbook.com/chapter-8#two-sided-tests)
- [**8\.3** Posterior prediction](https://www.bayesrulesbook.com/chapter-8#ch8-post-pred)
- [**8\.4** Posterior analysis with MCMC](https://www.bayesrulesbook.com/chapter-8#posterior-analysis-with-mcmc)
- [**8\.4.1** Posterior simulation](https://www.bayesrulesbook.com/chapter-8#posterior-simulation-1)
- [**8\.4.2** Posterior estimation & hypothesis testing](https://www.bayesrulesbook.com/chapter-8#posterior-estimation-hypothesis-testing)
- [**8\.4.3** Posterior prediction](https://www.bayesrulesbook.com/chapter-8#ch8-pred)
- [**8\.5** Bayesian benefits](https://www.bayesrulesbook.com/chapter-8#benefits-chapter-8)
- [**8\.6** Chapter summary](https://www.bayesrulesbook.com/chapter-8#chapter-summary-7)
- [**8\.7** Exercises](https://www.bayesrulesbook.com/chapter-8#exercises-7)
- [**8\.7.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-8#conceptual-exercises-2)
- [**8\.7.2** Practice exercises](https://www.bayesrulesbook.com/chapter-8#practice-exercises)
- [**8\.7.3** Applied exercises](https://www.bayesrulesbook.com/chapter-8#applied-exercises)
- **III Bayesian Regression & Classification**
- [**9** Simple Normal Regression](https://www.bayesrulesbook.com/chapter-9)
- [**9\.1** Building the regression model](https://www.bayesrulesbook.com/chapter-9#building-the-regression-model)
- [**9\.1.1** Specifying the data model](https://www.bayesrulesbook.com/chapter-9#specifying-the-data-model)
- [**9\.1.2** Specifying the priors](https://www.bayesrulesbook.com/chapter-9#specifying-the-priors)
- [**9\.1.3** Putting it all together](https://www.bayesrulesbook.com/chapter-9#putting-it-all-together)
- [**9\.2** Tuning prior models for regression parameters](https://www.bayesrulesbook.com/chapter-9#tuning-prior-models-for-regression-parameters)
- [**9\.3** Posterior simulation](https://www.bayesrulesbook.com/chapter-9#ch9-post-sim)
- [**9\.3.1** Simulation via rstanarm](https://www.bayesrulesbook.com/chapter-9#simulation-via-rstanarm)
- [**9\.3.2** Optional: Simulation via rstan](https://www.bayesrulesbook.com/chapter-9#optional-simulation-via-rstan)
- [**9\.4** Interpreting the posterior](https://www.bayesrulesbook.com/chapter-9#interpreting-the-posterior)
- [**9\.5** Posterior prediction](https://www.bayesrulesbook.com/chapter-9#chapter-9-prediction)
- [**9\.5.1** Building a posterior predictive model](https://www.bayesrulesbook.com/chapter-9#building-a-posterior-predictive-model)
- [**9\.5.2** Posterior prediction with rstanarm](https://www.bayesrulesbook.com/chapter-9#posterior-prediction-with-rstanarm)
- [**9\.6** Sequential regression modeling](https://www.bayesrulesbook.com/chapter-9#sequential-regression-modeling)
- [**9\.7** Using default rstanarm priors](https://www.bayesrulesbook.com/chapter-9#ch9-default)
- [**9\.8** You’re not done yet\!](https://www.bayesrulesbook.com/chapter-9#youre-not-done-yet)
- [**9\.9** Chapter summary](https://www.bayesrulesbook.com/chapter-9#chapter-summary-8)
- [**9\.10** Exercises](https://www.bayesrulesbook.com/chapter-9#exercises-8)
- [**9\.10.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-9#conceptual-exercises-3)
- [**9\.10.2** Applied exercises](https://www.bayesrulesbook.com/chapter-9#applied-exercises-1)
- [**10** Evaluating Regression Models](https://www.bayesrulesbook.com/chapter-10)
- [**10\.1** Is the model fair?](https://www.bayesrulesbook.com/chapter-10#is-the-model-fair)
- [**10\.2** How wrong is the model?](https://www.bayesrulesbook.com/chapter-10#chapter-10-wrong-model)
- [**10\.2.1** Checking the model assumptions](https://www.bayesrulesbook.com/chapter-10#checking-the-model-assumptions)
- [**10\.2.2** Dealing with wrong models](https://www.bayesrulesbook.com/chapter-10#dealing-with-wrong-models)
- [**10\.3** How accurate are the posterior predictive models?](https://www.bayesrulesbook.com/chapter-10#chapter-10-train-test)
- [**10\.3.1** Posterior predictive summaries](https://www.bayesrulesbook.com/chapter-10#simple-summary-10)
- [**10\.3.2** Cross-validation](https://www.bayesrulesbook.com/chapter-10#cross-validation)
- [**10\.3.3** Expected log-predictive density](https://www.bayesrulesbook.com/chapter-10#chapter-10-elpd)
- [**10\.3.4** Improving posterior predictive accuracy](https://www.bayesrulesbook.com/chapter-10#improving-posterior-predictive-accuracy)
- [**10\.4** How good is the MCMC simulation vs how good is the model?](https://www.bayesrulesbook.com/chapter-10#how-good-is-the-mcmc-simulation-vs-how-good-is-the-model)
- [**10\.5** Chapter summary](https://www.bayesrulesbook.com/chapter-10#chapter-summary-9)
- [**10\.6** Exercises](https://www.bayesrulesbook.com/chapter-10#exercises-9)
- [**10\.6.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-10#conceptual-exercises-4)
- [**10\.6.2** Applied exercises](https://www.bayesrulesbook.com/chapter-10#applied-exercises-2)
- [**10\.6.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-10#open-ended-exercises)
- [**11** Extending the Normal Regression Model](https://www.bayesrulesbook.com/chapter-11)
- [**11\.1** Utilizing a categorical predictor](https://www.bayesrulesbook.com/chapter-11#utilizing-a-categorical-predictor)
- [**11\.1.1** Building the model](https://www.bayesrulesbook.com/chapter-11#building-the-model)
- [**11\.1.2** Simulating the posterior](https://www.bayesrulesbook.com/chapter-11#simulating-the-posterior)
- [**11\.2** Utilizing two predictors](https://www.bayesrulesbook.com/chapter-11#ch-11-2-predictors)
- [**11\.2.1** Building the model](https://www.bayesrulesbook.com/chapter-11#building-the-model-1)
- [**11\.2.2** Understanding the priors](https://www.bayesrulesbook.com/chapter-11#understanding-the-priors)
- [**11\.2.3** Simulating the posterior](https://www.bayesrulesbook.com/chapter-11#simulating-the-posterior-1)
- [**11\.2.4** Posterior prediction](https://www.bayesrulesbook.com/chapter-11#posterior-prediction)
- [**11\.3** Optional: Utilizing interaction terms](https://www.bayesrulesbook.com/chapter-11#optional-utilizing-interaction-terms)
- [**11\.3.1** Building the model](https://www.bayesrulesbook.com/chapter-11#building-the-model-2)
- [**11\.3.2** Simulating the posterior](https://www.bayesrulesbook.com/chapter-11#simulating-the-posterior-2)
- [**11\.3.3** Do you need an interaction term?](https://www.bayesrulesbook.com/chapter-11#do-you-need-an-interaction-term)
- [**11\.4** Dreaming bigger: Utilizing more than 2 predictors\!](https://www.bayesrulesbook.com/chapter-11#ch11-more)
- [**11\.5** Model evaluation & comparison](https://www.bayesrulesbook.com/chapter-11#ch11-model-comparison)
- [**11\.5.1** Evaluating predictive accuracy using visualizations](https://www.bayesrulesbook.com/chapter-11#evaluating-predictive-accuracy-using-visualizations)
- [**11\.5.2** Evaluating predictive accuracy using cross-validation](https://www.bayesrulesbook.com/chapter-11#evaluating-predictive-accuracy-using-cross-validation)
- [**11\.5.3** Evaluating predictive accuracy using ELPD](https://www.bayesrulesbook.com/chapter-11#evaluating-predictive-accuracy-using-elpd)
- [**11\.5.4** The bias-variance trade-off](https://www.bayesrulesbook.com/chapter-11#the-bias-variance-trade-off)
- [**11\.6** Chapter summary](https://www.bayesrulesbook.com/chapter-11#chapter-summary-10)
- [**11\.7** Exercises](https://www.bayesrulesbook.com/chapter-11#exercises-10)
- [**11\.7.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-11#conceptual-exercises-5)
- [**11\.7.2** Applied exercises](https://www.bayesrulesbook.com/chapter-11#applied-exercises-3)
- [**11\.7.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-11#open-ended-exercises-1)
- [**12** Poisson & Negative Binomial Regression](https://www.bayesrulesbook.com/chapter-12)
- [**12\.1** Building the Poisson regression model](https://www.bayesrulesbook.com/chapter-12#building-the-poisson-regression-model)
- [**12\.1.1** Specifying the data model](https://www.bayesrulesbook.com/chapter-12#specifying-the-data-model-1)
- [**12\.1.2** Specifying the priors](https://www.bayesrulesbook.com/chapter-12#ch12-prior-spec)
- [**12\.2** Simulating the posterior](https://www.bayesrulesbook.com/chapter-12#ch12-simulation)
- [**12\.3** Interpreting the posterior](https://www.bayesrulesbook.com/chapter-12#ch12-interp-pois)
- [**12\.4** Posterior prediction](https://www.bayesrulesbook.com/chapter-12#posterior-prediction-1)
- [**12\.5** Model evaluation](https://www.bayesrulesbook.com/chapter-12#model-evaluation)
- [**12\.6** Negative Binomial regression for overdispersed counts](https://www.bayesrulesbook.com/chapter-12#chapter-12-overdispersion)
- [**12\.7** Generalized linear models: Building on the theme](https://www.bayesrulesbook.com/chapter-12#ch12-glm)
- [**12\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-12#chapter-summary-11)
- [**12\.9** Exercises](https://www.bayesrulesbook.com/chapter-12#exercises-11)
- [**12\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-12#conceptual-exercises-6)
- [**12\.9.2** Applied exercises](https://www.bayesrulesbook.com/chapter-12#applied-exercises-4)
- [**13** Logistic Regression](https://www.bayesrulesbook.com/chapter-13)
- [**13\.1** Pause: Odds & probability](https://www.bayesrulesbook.com/chapter-13#pause-odds-probability)
- [**13\.2** Building the logistic regression model](https://www.bayesrulesbook.com/chapter-13#building-the-logistic-regression-model)
- [**13\.2.1** Specifying the data model](https://www.bayesrulesbook.com/chapter-13#specifying-the-data-model-2)
- [**13\.2.2** Specifying the priors](https://www.bayesrulesbook.com/chapter-13#specifying-the-priors-1)
- [**13\.3** Simulating the posterior](https://www.bayesrulesbook.com/chapter-13#ch13-post-sim-sec)
- [**13\.4** Prediction & classification](https://www.bayesrulesbook.com/chapter-13#chapter-13-prediction)
- [**13\.5** Model evaluation](https://www.bayesrulesbook.com/chapter-13#model-evaluation-1)
- [**13\.6** Extending the model](https://www.bayesrulesbook.com/chapter-13#extending-the-model)
- [**13\.7** Chapter summary](https://www.bayesrulesbook.com/chapter-13#chapter-summary-12)
- [**13\.8** Exercises](https://www.bayesrulesbook.com/chapter-13#exercises-12)
- [**13\.8.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-13#conceptual-exercises-7)
- [**13\.8.2** Applied exercises](https://www.bayesrulesbook.com/chapter-13#applied-exercises-5)
- [**13\.8.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-13#open-ended-exercises-2)
- [**14** Naive Bayes Classification](https://www.bayesrulesbook.com/chapter-14)
- [**14\.1** Classifying one penguin](https://www.bayesrulesbook.com/chapter-14#naive-one-case)
- [**14\.1.1** One categorical predictor](https://www.bayesrulesbook.com/chapter-14#one-categorical-predictor)
- [**14\.1.2** One quantitative predictor](https://www.bayesrulesbook.com/chapter-14#naive-one-quant)
- [**14\.1.3** Two predictors](https://www.bayesrulesbook.com/chapter-14#naive-two-quant)
- [**14\.2** Implementing & evaluating naive Bayes classification](https://www.bayesrulesbook.com/chapter-14#implementing-evaluating-naive-bayes-classification)
- [**14\.3** Naive Bayes vs logistic regression](https://www.bayesrulesbook.com/chapter-14#naive-bayes-vs-logistic-regression)
- [**14\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-14#chapter-summary-13)
- [**14\.5** Exercises](https://www.bayesrulesbook.com/chapter-14#exercises-13)
- [**14\.5.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-14#conceptual-exercises-8)
- [**14\.5.2** Applied exercises](https://www.bayesrulesbook.com/chapter-14#applied-exercises-6)
- [**14\.5.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-14#open-ended-exercises-3)
- **IV Hierarchical Bayesian models**
- [**15** Hierarchical Models are Exciting](https://www.bayesrulesbook.com/chapter-15)
- [**15\.1** Complete pooling](https://www.bayesrulesbook.com/chapter-15#ch15-complete)
- [**15\.2** No pooling](https://www.bayesrulesbook.com/chapter-15#no-pooling)
- [**15\.3** Hierarchical data](https://www.bayesrulesbook.com/chapter-15#hierarchical-data)
- [**15\.4** Partial pooling with hierarchical models](https://www.bayesrulesbook.com/chapter-15#partial-pooling-with-hierarchical-models)
- [**15\.5** Chapter summary](https://www.bayesrulesbook.com/chapter-15#chapter-summary-14)
- [**15\.6** Exercises](https://www.bayesrulesbook.com/chapter-15#exercises-14)
- [**15\.6.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-15#conceptual-exercises-9)
- [**15\.6.2** Applied exercises](https://www.bayesrulesbook.com/chapter-15#applied-exercises-7)
- [**16** (Normal) Hierarchical Models without Predictors](https://www.bayesrulesbook.com/chapter-16)
- [**16\.1** Complete pooled model](https://www.bayesrulesbook.com/chapter-16#chapter-16-complete-pooling)
- [**16\.2** No pooled model](https://www.bayesrulesbook.com/chapter-16#chapter-16-no-pooling)
- [**16\.3** Building the hierarchical model](https://www.bayesrulesbook.com/chapter-16#hierarchical-building-16)
- [**16\.3.1** The hierarchy](https://www.bayesrulesbook.com/chapter-16#the-hierarchy)
- [**16\.3.2** Another way to think about it](https://www.bayesrulesbook.com/chapter-16#another-way-to-think-about-it)
- [**16\.3.3** Within- vs between-group variability](https://www.bayesrulesbook.com/chapter-16#ch16-within-btwn)
- [**16\.4** Posterior analysis](https://www.bayesrulesbook.com/chapter-16#ch-16-analyze)
- [**16\.4.1** Posterior simulation](https://www.bayesrulesbook.com/chapter-16#posterior-simulation-2)
- [**16\.4.2** Posterior analysis of global parameters](https://www.bayesrulesbook.com/chapter-16#posterior-analysis-of-global-parameters)
- [**16\.4.3** Posterior analysis of group-specific parameters](https://www.bayesrulesbook.com/chapter-16#posterior-analysis-of-group-specific-parameters)
- [**16\.5** Posterior prediction](https://www.bayesrulesbook.com/chapter-16#ch16-prediction)
- [**16\.6** Shrinkage & the bias-variance trade-off](https://www.bayesrulesbook.com/chapter-16#shrinkage-the-bias-variance-trade-off)
- [**16\.7** Not everything is hierarchical](https://www.bayesrulesbook.com/chapter-16#ch-16-nonhierarchical)
- [**16\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-16#chapter-summary-15)
- [**16\.9** Exercises](https://www.bayesrulesbook.com/chapter-16#exercises-15)
- [**16\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-16#conceptual-exercises-10)
- [**16\.9.2** Applied exercises](https://www.bayesrulesbook.com/chapter-16#applied-exercises-8)
- [**17** (Normal) Hierarchical Models with Predictors](https://www.bayesrulesbook.com/chapter-17)
- [**17\.1** First steps: Complete pooling](https://www.bayesrulesbook.com/chapter-17#first-steps-complete-pooling)
- [**17\.2** Hierarchical model with varying intercepts](https://www.bayesrulesbook.com/chapter-17#hierarchical-model-with-varying-intercepts)
- [**17\.2.1** Model building](https://www.bayesrulesbook.com/chapter-17#model-building)
- [**17\.2.2** Another way to think about it](https://www.bayesrulesbook.com/chapter-17#another-way-to-think-about-it-1)
- [**17\.2.3** Tuning the prior](https://www.bayesrulesbook.com/chapter-17#tuning-the-prior)
- [**17\.2.4** Posterior simulation & analysis](https://www.bayesrulesbook.com/chapter-17#chapter-17-random-intercepts)
- [**17\.3** Hierarchical model with varying intercepts & slopes](https://www.bayesrulesbook.com/chapter-17#hierarchical-model-with-varying-intercepts-slopes)
- [**17\.3.1** Model building](https://www.bayesrulesbook.com/chapter-17#model-building-1)
- [**17\.3.2** Optional: The decomposition of covariance model](https://www.bayesrulesbook.com/chapter-17#optional-the-decomposition-of-covariance-model)
- [**17\.3.3** Posterior simulation & analysis](https://www.bayesrulesbook.com/chapter-17#ch-17-rand-int-slope-sim)
- [**17\.4** Model evaluation & selection](https://www.bayesrulesbook.com/chapter-17#model-evaluation-selection)
- [**17\.5** Posterior prediction](https://www.bayesrulesbook.com/chapter-17#posterior-prediction-2)
- [**17\.6** Details: Longitudinal data](https://www.bayesrulesbook.com/chapter-17#details-longitudinal-data)
- [**17\.7** Example: Danceability](https://www.bayesrulesbook.com/chapter-17#example-danceability)
- [**17\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-17#chapter-summary-16)
- [**17\.9** Exercises](https://www.bayesrulesbook.com/chapter-17#exercises-16)
- [**17\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-17#conceptual-exercises-11)
- [**17\.9.2** Applied exercises](https://www.bayesrulesbook.com/chapter-17#applied-exercises-9)
- [**17\.9.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-17#open-ended-exercises-4)
- [**18** Non-Normal Hierarchical Regression & Classification](https://www.bayesrulesbook.com/chapter-18)
- [**18\.1** Hierarchical logistic regression](https://www.bayesrulesbook.com/chapter-18#hierarchical-logistic-regression)
- [**18\.1.1** Model building & simulation](https://www.bayesrulesbook.com/chapter-18#model-building-simulation)
- [**18\.1.2** Posterior analysis](https://www.bayesrulesbook.com/chapter-18#posterior-analysis)
- [**18\.1.3** Posterior classification](https://www.bayesrulesbook.com/chapter-18#posterior-classification)
- [**18\.1.4** Model evaluation](https://www.bayesrulesbook.com/chapter-18#model-evaluation-2)
- [**18\.2** Hierarchical Poisson & Negative Binomial regression](https://www.bayesrulesbook.com/chapter-18#hierarchical-poisson-negative-binomial-regression)
- [**18\.2.1** Model building & simulation](https://www.bayesrulesbook.com/chapter-18#ch-18-rat-sim)
- [**18\.2.2** Posterior analysis](https://www.bayesrulesbook.com/chapter-18#posterior-analysis-1)
- [**18\.2.3** Model evaluation](https://www.bayesrulesbook.com/chapter-18#model-evaluation-3)
- [**18\.3** Chapter summary](https://www.bayesrulesbook.com/chapter-18#chapter-summary-17)
- [**18\.4** Exercises](https://www.bayesrulesbook.com/chapter-18#exercises-17)
- [**18\.4.1** Applied & conceptual exercises](https://www.bayesrulesbook.com/chapter-18#applied-conceptual-exercises)
- [**18\.4.2** Open-ended exercises](https://www.bayesrulesbook.com/chapter-18#open-ended-exercises-5)
- [**19** Adding More Layers](https://www.bayesrulesbook.com/chapter-19)
- [**19\.1** Group-level predictors](https://www.bayesrulesbook.com/chapter-19#group-level-predictors)
- [**19\.1.1** A model using only individual-level predictors](https://www.bayesrulesbook.com/chapter-19#a-model-using-only-individual-level-predictors)
- [**19\.1.2** Incorporating group-level predictors](https://www.bayesrulesbook.com/chapter-19#incorporating-group-level-predictors)
- [**19\.1.3** Posterior simulation & global analysis](https://www.bayesrulesbook.com/chapter-19#posterior-simulation-global-analysis)
- [**19\.1.4** Posterior group-level analysis](https://www.bayesrulesbook.com/chapter-19#posterior-group-level-analysis)
- [**19\.1.5** We’re just scratching the surface\!](https://www.bayesrulesbook.com/chapter-19#were-just-scratching-the-surface)
- [**19\.2** Incorporating two (or more!) grouping variables](https://www.bayesrulesbook.com/chapter-19#incorporating-two-or-more-grouping-variables)
- [**19\.2.1** Data with two grouping variables](https://www.bayesrulesbook.com/chapter-19#data-with-two-grouping-variables)
- [**19\.2.2** Building a model with two grouping variables](https://www.bayesrulesbook.com/chapter-19#building-a-model-with-two-grouping-variables)
- [**19\.2.3** Simulating models with two grouping variables](https://www.bayesrulesbook.com/chapter-19#simulating-models-with-two-grouping-variables)
- [**19\.2.4** Examining the group-specific parameters](https://www.bayesrulesbook.com/chapter-19#examining-the-group-specific-parameters)
- [**19\.2.5** We’re just scratching the surface\!](https://www.bayesrulesbook.com/chapter-19#were-just-scratching-the-surface-1)
- [**19\.3** Exercises](https://www.bayesrulesbook.com/chapter-19#exercises-18)
- [**19\.3.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-19#conceptual-exercises-12)
- [**19\.3.2** Applied exercises](https://www.bayesrulesbook.com/chapter-19#applied-exercises-10)
- [**19\.4** Goodbye\!](https://www.bayesrulesbook.com/chapter-19#goodbye)
- [References](https://www.bayesrulesbook.com/references)
- [Published with bookdown](https://bookdown.org/)
# [Bayes Rules! An Introduction to Applied Bayesian Modeling](https://www.bayesrulesbook.com/)
# Bayes Rules! An Introduction to Applied Bayesian Modeling
*Alicia A. Johnson, Miles Q. Ott, Mine Dogucu*
*2021-12-01*
# News
**In addition to this online version**, *Bayes Rules\!* is now [available in print by CRC Press](https://www.routledge.com/Bayes-Rules-An-Introduction-to-Applied-Bayesian-Modeling/Johnson-Ott-Dogucu/p/book/9780367255398?utm_source=website&utm_medium=shared_link&utm_campaign=B023819_jm1_5ll_7pp_d676_bayesrulesauthorshare%0A%0A). The book is 20% off with the discount code BRD20.
For other information and resources related to the book, check out <https://bayes-rules.github.io/>. | |||||||||||||||
| Readable Markdown | - [](https://www.bayesrulesbook.com/)
- [News](https://www.bayesrulesbook.com/)
- [Foreword](https://www.bayesrulesbook.com/foreword)
- [Preface](https://www.bayesrulesbook.com/preface)
- [Audience](https://www.bayesrulesbook.com/preface#audience)
- [Navigating this book](https://www.bayesrulesbook.com/preface#navigating-this-book)
- [Getting set up](https://www.bayesrulesbook.com/preface#setup)
- [Accessibility and inclusion](https://www.bayesrulesbook.com/preface#accessibility-and-inclusion)
- [Contact us](https://www.bayesrulesbook.com/preface#contact-us)
- [Acknowledgments](https://www.bayesrulesbook.com/preface#acknowledgments)
- [License](https://www.bayesrulesbook.com/preface#license)
- [About the Authors](https://www.bayesrulesbook.com/about-the-authors)
- **I Bayesian Foundations**
- [**1** The Big (Bayesian) Picture](https://www.bayesrulesbook.com/chapter-1)
- [**1\.1** Thinking like a Bayesian](https://www.bayesrulesbook.com/chapter-1#thinking-like-a-bayesian)
- [**1\.1.1** Quiz yourself](https://www.bayesrulesbook.com/chapter-1#quiz-yourself-1)
- [**1\.1.2** The meaning of probability](https://www.bayesrulesbook.com/chapter-1#bayes-v-freq-probability)
- [**1\.1.3** The Bayesian balancing act](https://www.bayesrulesbook.com/chapter-1#the-bayesian-balancing-act)
- [**1\.1.4** Asking questions](https://www.bayesrulesbook.com/chapter-1#asking-questions)
- [**1\.2** A quick history lesson](https://www.bayesrulesbook.com/chapter-1#a-quick-history-lesson)
- [**1\.3** A look ahead](https://www.bayesrulesbook.com/chapter-1#a-look-ahead)
- [**1\.3.1** Unit 1: Bayesian foundations](https://www.bayesrulesbook.com/chapter-1#ch1-unit-1)
- [**1\.3.2** Unit 2: Posterior simulation & analysis](https://www.bayesrulesbook.com/chapter-1#unit-2-posterior-simulation-analysis)
- [**1\.3.3** Unit 3: Bayesian regression & classification](https://www.bayesrulesbook.com/chapter-1#unit-3-bayesian-regression-classification)
- [**1\.3.4** Unit 4: Hierarchical Bayesian models](https://www.bayesrulesbook.com/chapter-1#unit-4-hierarchical-bayesian-models)
- [**1\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-1#chapter-summary)
- [**1\.5** Exercises](https://www.bayesrulesbook.com/chapter-1#exercises)
- [**2** Bayes’ Rule](https://www.bayesrulesbook.com/chapter-2)
- [**2\.1** Building a Bayesian model for events](https://www.bayesrulesbook.com/chapter-2#building-a-bayesian-model-for-events)
- [**2\.1.1** Prior probability model](https://www.bayesrulesbook.com/chapter-2#prior-probability-model)
- [**2\.1.2** Conditional probability & likelihood](https://www.bayesrulesbook.com/chapter-2#conditional-probability-likelihood)
- [**2\.1.3** Normalizing constants](https://www.bayesrulesbook.com/chapter-2#normalizing-constants)
- [**2\.1.4** Posterior probability model via Bayes’ Rule\!](https://www.bayesrulesbook.com/chapter-2#posterior-probability-model-via-bayes-rule)
- [**2\.1.5** Posterior simulation](https://www.bayesrulesbook.com/chapter-2#posterior-simulation)
- [**2\.2** Example: Pop vs soda vs coke](https://www.bayesrulesbook.com/chapter-2#michelle-simple)
- [**2\.3** Building a Bayesian model for random variables](https://www.bayesrulesbook.com/chapter-2#cousin-cole)
- [**2\.3.1** Prior probability model](https://www.bayesrulesbook.com/chapter-2#prior-probability-model-1)
- [**2\.3.2** The Binomial data model](https://www.bayesrulesbook.com/chapter-2#chapter-2-binomial-likelihood)
- [**2\.3.3** The Binomial likelihood function](https://www.bayesrulesbook.com/chapter-2#the-binomial-likelihood-function)
- [**2\.3.4** Normalizing constant](https://www.bayesrulesbook.com/chapter-2#normalizing-constant)
- [**2\.3.5** Posterior probability model](https://www.bayesrulesbook.com/chapter-2#posterior-probability-model)
- [**2\.3.6** Posterior shortcut](https://www.bayesrulesbook.com/chapter-2#shortcut-2)
- [**2\.3.7** Posterior simulation](https://www.bayesrulesbook.com/chapter-2#chapter-2-simulation)
- [**2\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-2#chapter-summary-1)
- [**2\.5** Exercises](https://www.bayesrulesbook.com/chapter-2#exercises-1)
- [**2\.5.1** Building up to Bayes’ Rule](https://www.bayesrulesbook.com/chapter-2#building-up-to-bayes-rule)
- [**2\.5.2** Practice Bayes’ Rule for events](https://www.bayesrulesbook.com/chapter-2#practice-bayes-rule-for-events)
- [**2\.5.3** Practice Bayes’ Rule for random variables](https://www.bayesrulesbook.com/chapter-2#practice-bayes-rule-for-random-variables)
- [**2\.5.4** Simulation exercises](https://www.bayesrulesbook.com/chapter-2#simulation-exercises)
- [**3** The Beta-Binomial Bayesian Model](https://www.bayesrulesbook.com/chapter-3)
- [**3\.1** The Beta prior model](https://www.bayesrulesbook.com/chapter-3#ch3-prior)
- [**3\.1.1** Beta foundations](https://www.bayesrulesbook.com/chapter-3#beta-foundations)
- [**3\.1.2** Tuning the Beta prior](https://www.bayesrulesbook.com/chapter-3#tuning-the-beta-prior)
- [**3\.2** The Binomial data model & likelihood function](https://www.bayesrulesbook.com/chapter-3#the-binomial-data-model-likelihood-function)
- [**3\.3** The Beta posterior model](https://www.bayesrulesbook.com/chapter-3#the-beta-posterior-model)
- [**3\.4** The Beta-Binomial model](https://www.bayesrulesbook.com/chapter-3#ch3-bbmodel)
- [**3\.5** Simulating the Beta-Binomial](https://www.bayesrulesbook.com/chapter-3#chapter-3-simulation)
- [**3\.6** Example: Milgram’s behavioral study of obedience](https://www.bayesrulesbook.com/chapter-3#milgram-3)
- [**3\.6.1** A Bayesian analysis](https://www.bayesrulesbook.com/chapter-3#a-bayesian-analysis)
- [**3\.6.2** The role of ethics in statistics and data science](https://www.bayesrulesbook.com/chapter-3#the-role-of-ethics-in-statistics-and-data-science)
- [**3\.7** Chapter summary](https://www.bayesrulesbook.com/chapter-3#chapter-summary-2)
- [**3\.8** Exercises](https://www.bayesrulesbook.com/chapter-3#exercises-2)
- [**3\.8.1** Practice: Beta prior models](https://www.bayesrulesbook.com/chapter-3#practice-beta-prior-models)
- [**3\.8.2** Practice: Beta-Binomial models](https://www.bayesrulesbook.com/chapter-3#practice-beta-binomial-models)
- [**4** Balance and Sequentiality in Bayesian Analyses](https://www.bayesrulesbook.com/chapter-4)
- [**4\.1** Different priors, different posteriors](https://www.bayesrulesbook.com/chapter-4#ch4-priors)
- [**4\.2** Different data, different posteriors](https://www.bayesrulesbook.com/chapter-4#ch4-data)
- [**4\.3** Striking a balance between the prior & data](https://www.bayesrulesbook.com/chapter-4#striking-a-balance-between-the-prior-data)
- [**4\.3.1** Connecting observations to concepts](https://www.bayesrulesbook.com/chapter-4#connecting-observations-to-concepts)
- [**4\.3.2** Connecting concepts to theory](https://www.bayesrulesbook.com/chapter-4#ch4-concepts)
- [**4\.4** Sequential analysis: Evolving with data](https://www.bayesrulesbook.com/chapter-4#ch4-sequential)
- [**4\.5** Proving data order invariance](https://www.bayesrulesbook.com/chapter-4#proving-data-order-invariance)
- [**4\.6** Don’t be stubborn](https://www.bayesrulesbook.com/chapter-4#dont-be-stubborn)
- [**4\.7** A note on subjectivity](https://www.bayesrulesbook.com/chapter-4#a-note-on-subjectivity)
- [**4\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-4#chapter-summary-3)
- [**4\.9** Exercises](https://www.bayesrulesbook.com/chapter-4#exercises-3)
- [**4\.9.1** Review exercises](https://www.bayesrulesbook.com/chapter-4#review-exercises)
- [**4\.9.2** Practice: Different priors, different posteriors](https://www.bayesrulesbook.com/chapter-4#practice-different-priors-different-posteriors)
- [**4\.9.3** Practice: Balancing the data & prior](https://www.bayesrulesbook.com/chapter-4#practice-balancing-the-data-prior)
- [**4\.9.4** Practice: Sequentiality](https://www.bayesrulesbook.com/chapter-4#practice-sequentiality)
- [**5** Conjugate Families](https://www.bayesrulesbook.com/chapter-5)
- [**5\.1** Revisiting choice of prior](https://www.bayesrulesbook.com/chapter-5#revisiting-choice-of-prior)
- [**5\.2** Gamma-Poisson conjugate family](https://www.bayesrulesbook.com/chapter-5#gamma-poisson-conjugate-family)
- [**5\.2.1** The Poisson data model](https://www.bayesrulesbook.com/chapter-5#the-poisson-data-model)
- [**5\.2.2** Potential priors](https://www.bayesrulesbook.com/chapter-5#potential-priors)
- [**5\.2.3** Gamma prior](https://www.bayesrulesbook.com/chapter-5#gamma-prior)
- [**5\.2.4** Gamma-Poisson conjugacy](https://www.bayesrulesbook.com/chapter-5#gamma-poisson-conjugacy)
- [**5\.3** Normal-Normal conjugate family](https://www.bayesrulesbook.com/chapter-5#normal-normal-conjugate-family)
- [**5\.3.1** The Normal data model](https://www.bayesrulesbook.com/chapter-5#the-normal-data-model)
- [**5\.3.2** Normal prior](https://www.bayesrulesbook.com/chapter-5#normal-prior)
- [**5\.3.3** Normal-Normal conjugacy](https://www.bayesrulesbook.com/chapter-5#ch5-n-n-section)
- [**5\.3.4** Optional: Proving Normal-Normal conjugacy](https://www.bayesrulesbook.com/chapter-5#ch5-optional)
- [**5\.4** Why no simulation in this chapter?](https://www.bayesrulesbook.com/chapter-5#why-no-simulation-in-this-chapter)
- [**5\.5** Critiques of conjugate family models](https://www.bayesrulesbook.com/chapter-5#critiques-of-conjugate-family-models)
- [**5\.6** Chapter summary](https://www.bayesrulesbook.com/chapter-5#chapter-summary-4)
- [**5\.7** Exercises](https://www.bayesrulesbook.com/chapter-5#exercises-4)
- [**5\.7.1** Practice: Gamma-Poisson](https://www.bayesrulesbook.com/chapter-5#practice-gamma-poisson)
- [**5\.7.2** Practice: Normal-Normal](https://www.bayesrulesbook.com/chapter-5#practice-normal-normal)
- [**5\.7.3** General practice exercises](https://www.bayesrulesbook.com/chapter-5#general-practice-exercises)
- **II Posterior Simulation & Analysis**
- [**6** Approximating the Posterior](https://www.bayesrulesbook.com/chapter-6)
- [**6\.1** Grid approximation](https://www.bayesrulesbook.com/chapter-6#grid-approximation)
- [**6\.1.1** A Beta-Binomial example](https://www.bayesrulesbook.com/chapter-6#beta-binomial-grid)
- [**6\.1.2** A Gamma-Poisson example](https://www.bayesrulesbook.com/chapter-6#Gamma-Poisson-grid)
- [**6\.1.3** Limitations](https://www.bayesrulesbook.com/chapter-6#limitations)
- [**6\.2** Markov chains via rstan](https://www.bayesrulesbook.com/chapter-6#markov-chains-via-rstan)
- [**6\.2.1** A Beta-Binomial example](https://www.bayesrulesbook.com/chapter-6#a-beta-binomial-example)
- [**6\.2.2** A Gamma-Poisson example](https://www.bayesrulesbook.com/chapter-6#a-gamma-poisson-example)
- [**6\.3** Markov chain diagnostics](https://www.bayesrulesbook.com/chapter-6#diagnostics)
- [**6\.3.1** Examining trace plots](https://www.bayesrulesbook.com/chapter-6#chapter-6-trace)
- [**6\.3.2** Comparing parallel chains](https://www.bayesrulesbook.com/chapter-6#chapter-6-parallel)
- [**6\.3.3** Calculating effective sample size & autocorrelation](https://www.bayesrulesbook.com/chapter-6#calculating-effective-sample-size-autocorrelation)
- [**6\.3.4** Calculating R-hat](https://www.bayesrulesbook.com/chapter-6#calculating-r-hat)
- [**6\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-6#chapter-summary-5)
- [**6\.5** Exercises](https://www.bayesrulesbook.com/chapter-6#exercises-5)
- [**6\.5.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-6#conceptual-exercises)
- [**6\.5.2** Practice: Grid approximation](https://www.bayesrulesbook.com/chapter-6#practice-grid-approximation)
- [**6\.5.3** Practice: MCMC](https://www.bayesrulesbook.com/chapter-6#practice-mcmc)
- [**7** MCMC under the Hood](https://www.bayesrulesbook.com/chapter-7)
- [**7\.1** The big idea](https://www.bayesrulesbook.com/chapter-7#the-big-idea)
- [**7\.2** The Metropolis-Hastings algorithm](https://www.bayesrulesbook.com/chapter-7#the-metropolis-hastings-algorithm)
- [**7\.3** Implementing the Metropolis-Hastings](https://www.bayesrulesbook.com/chapter-7#ch7-implementation)
- [**7\.4** Tuning the Metropolis-Hastings algorithm](https://www.bayesrulesbook.com/chapter-7#tuning-the-metropolis-hastings-algorithm)
- [**7\.5** A Beta-Binomial example](https://www.bayesrulesbook.com/chapter-7#a-beta-binomial-example-1)
- [**7\.6** Why the algorithm works](https://www.bayesrulesbook.com/chapter-7#why-the-algorithm-works)
- [**7\.7** Variations on the theme](https://www.bayesrulesbook.com/chapter-7#variations-on-the-theme)
- [**7\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-7#chapter-summary-6)
- [**7\.9** Exercises](https://www.bayesrulesbook.com/chapter-7#exercises-6)
- [**7\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-7#conceptual-exercises-1)
- [**7\.9.2** Practice: Normal-Normal simulation](https://www.bayesrulesbook.com/chapter-7#practice-normal-normal-simulation)
- [**7\.9.3** Practice: Simulating more Bayesian models](https://www.bayesrulesbook.com/chapter-7#practice-simulating-more-bayesian-models)
- [**8** Posterior Inference & Prediction](https://www.bayesrulesbook.com/chapter-8)
- [**8\.1** Posterior estimation](https://www.bayesrulesbook.com/chapter-8#chapter-8-estimation)
- [**8\.2** Posterior hypothesis testing](https://www.bayesrulesbook.com/chapter-8#posterior-hypothesis-testing)
- [**8\.2.1** One-sided tests](https://www.bayesrulesbook.com/chapter-8#one-sided-tests)
- [**8\.2.2** Two-sided tests](https://www.bayesrulesbook.com/chapter-8#two-sided-tests)
- [**8\.3** Posterior prediction](https://www.bayesrulesbook.com/chapter-8#ch8-post-pred)
- [**8\.4** Posterior analysis with MCMC](https://www.bayesrulesbook.com/chapter-8#posterior-analysis-with-mcmc)
- [**8\.4.1** Posterior simulation](https://www.bayesrulesbook.com/chapter-8#posterior-simulation-1)
- [**8\.4.2** Posterior estimation & hypothesis testing](https://www.bayesrulesbook.com/chapter-8#posterior-estimation-hypothesis-testing)
- [**8\.4.3** Posterior prediction](https://www.bayesrulesbook.com/chapter-8#ch8-pred)
- [**8\.5** Bayesian benefits](https://www.bayesrulesbook.com/chapter-8#benefits-chapter-8)
- [**8\.6** Chapter summary](https://www.bayesrulesbook.com/chapter-8#chapter-summary-7)
- [**8\.7** Exercises](https://www.bayesrulesbook.com/chapter-8#exercises-7)
- [**8\.7.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-8#conceptual-exercises-2)
- [**8\.7.2** Practice exercises](https://www.bayesrulesbook.com/chapter-8#practice-exercises)
- [**8\.7.3** Applied exercises](https://www.bayesrulesbook.com/chapter-8#applied-exercises)
- **III Bayesian Regression & Classification**
- [**9** Simple Normal Regression](https://www.bayesrulesbook.com/chapter-9)
- [**9\.1** Building the regression model](https://www.bayesrulesbook.com/chapter-9#building-the-regression-model)
- [**9\.1.1** Specifying the data model](https://www.bayesrulesbook.com/chapter-9#specifying-the-data-model)
- [**9\.1.2** Specifying the priors](https://www.bayesrulesbook.com/chapter-9#specifying-the-priors)
- [**9\.1.3** Putting it all together](https://www.bayesrulesbook.com/chapter-9#putting-it-all-together)
- [**9\.2** Tuning prior models for regression parameters](https://www.bayesrulesbook.com/chapter-9#tuning-prior-models-for-regression-parameters)
- [**9\.3** Posterior simulation](https://www.bayesrulesbook.com/chapter-9#ch9-post-sim)
- [**9\.3.1** Simulation via rstanarm](https://www.bayesrulesbook.com/chapter-9#simulation-via-rstanarm)
- [**9\.3.2** Optional: Simulation via rstan](https://www.bayesrulesbook.com/chapter-9#optional-simulation-via-rstan)
- [**9\.4** Interpreting the posterior](https://www.bayesrulesbook.com/chapter-9#interpreting-the-posterior)
- [**9\.5** Posterior prediction](https://www.bayesrulesbook.com/chapter-9#chapter-9-prediction)
- [**9\.5.1** Building a posterior predictive model](https://www.bayesrulesbook.com/chapter-9#building-a-posterior-predictive-model)
- [**9\.5.2** Posterior prediction with rstanarm](https://www.bayesrulesbook.com/chapter-9#posterior-prediction-with-rstanarm)
- [**9\.6** Sequential regression modeling](https://www.bayesrulesbook.com/chapter-9#sequential-regression-modeling)
- [**9\.7** Using default rstanarm priors](https://www.bayesrulesbook.com/chapter-9#ch9-default)
- [**9\.8** You’re not done yet\!](https://www.bayesrulesbook.com/chapter-9#youre-not-done-yet)
- [**9\.9** Chapter summary](https://www.bayesrulesbook.com/chapter-9#chapter-summary-8)
- [**9\.10** Exercises](https://www.bayesrulesbook.com/chapter-9#exercises-8)
- [**9\.10.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-9#conceptual-exercises-3)
- [**9\.10.2** Applied exercises](https://www.bayesrulesbook.com/chapter-9#applied-exercises-1)
- [**10** Evaluating Regression Models](https://www.bayesrulesbook.com/chapter-10)
- [**10\.1** Is the model fair?](https://www.bayesrulesbook.com/chapter-10#is-the-model-fair)
- [**10\.2** How wrong is the model?](https://www.bayesrulesbook.com/chapter-10#chapter-10-wrong-model)
- [**10\.2.1** Checking the model assumptions](https://www.bayesrulesbook.com/chapter-10#checking-the-model-assumptions)
- [**10\.2.2** Dealing with wrong models](https://www.bayesrulesbook.com/chapter-10#dealing-with-wrong-models)
- [**10\.3** How accurate are the posterior predictive models?](https://www.bayesrulesbook.com/chapter-10#chapter-10-train-test)
- [**10\.3.1** Posterior predictive summaries](https://www.bayesrulesbook.com/chapter-10#simple-summary-10)
- [**10\.3.2** Cross-validation](https://www.bayesrulesbook.com/chapter-10#cross-validation)
- [**10\.3.3** Expected log-predictive density](https://www.bayesrulesbook.com/chapter-10#chapter-10-elpd)
- [**10\.3.4** Improving posterior predictive accuracy](https://www.bayesrulesbook.com/chapter-10#improving-posterior-predictive-accuracy)
- [**10\.4** How good is the MCMC simulation vs how good is the model?](https://www.bayesrulesbook.com/chapter-10#how-good-is-the-mcmc-simulation-vs-how-good-is-the-model)
- [**10\.5** Chapter summary](https://www.bayesrulesbook.com/chapter-10#chapter-summary-9)
- [**10\.6** Exercises](https://www.bayesrulesbook.com/chapter-10#exercises-9)
- [**10\.6.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-10#conceptual-exercises-4)
- [**10\.6.2** Applied exercises](https://www.bayesrulesbook.com/chapter-10#applied-exercises-2)
- [**10\.6.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-10#open-ended-exercises)
- [**11** Extending the Normal Regression Model](https://www.bayesrulesbook.com/chapter-11)
- [**11\.1** Utilizing a categorical predictor](https://www.bayesrulesbook.com/chapter-11#utilizing-a-categorical-predictor)
- [**11\.1.1** Building the model](https://www.bayesrulesbook.com/chapter-11#building-the-model)
- [**11\.1.2** Simulating the posterior](https://www.bayesrulesbook.com/chapter-11#simulating-the-posterior)
- [**11\.2** Utilizing two predictors](https://www.bayesrulesbook.com/chapter-11#ch-11-2-predictors)
- [**11\.2.1** Building the model](https://www.bayesrulesbook.com/chapter-11#building-the-model-1)
- [**11\.2.2** Understanding the priors](https://www.bayesrulesbook.com/chapter-11#understanding-the-priors)
- [**11\.2.3** Simulating the posterior](https://www.bayesrulesbook.com/chapter-11#simulating-the-posterior-1)
- [**11\.2.4** Posterior prediction](https://www.bayesrulesbook.com/chapter-11#posterior-prediction)
- [**11\.3** Optional: Utilizing interaction terms](https://www.bayesrulesbook.com/chapter-11#optional-utilizing-interaction-terms)
- [**11\.3.1** Building the model](https://www.bayesrulesbook.com/chapter-11#building-the-model-2)
- [**11\.3.2** Simulating the posterior](https://www.bayesrulesbook.com/chapter-11#simulating-the-posterior-2)
- [**11\.3.3** Do you need an interaction term?](https://www.bayesrulesbook.com/chapter-11#do-you-need-an-interaction-term)
- [**11\.4** Dreaming bigger: Utilizing more than 2 predictors\!](https://www.bayesrulesbook.com/chapter-11#ch11-more)
- [**11\.5** Model evaluation & comparison](https://www.bayesrulesbook.com/chapter-11#ch11-model-comparison)
- [**11\.5.1** Evaluating predictive accuracy using visualizations](https://www.bayesrulesbook.com/chapter-11#evaluating-predictive-accuracy-using-visualizations)
- [**11\.5.2** Evaluating predictive accuracy using cross-validation](https://www.bayesrulesbook.com/chapter-11#evaluating-predictive-accuracy-using-cross-validation)
- [**11\.5.3** Evaluating predictive accuracy using ELPD](https://www.bayesrulesbook.com/chapter-11#evaluating-predictive-accuracy-using-elpd)
- [**11\.5.4** The bias-variance trade-off](https://www.bayesrulesbook.com/chapter-11#the-bias-variance-trade-off)
- [**11\.6** Chapter summary](https://www.bayesrulesbook.com/chapter-11#chapter-summary-10)
- [**11\.7** Exercises](https://www.bayesrulesbook.com/chapter-11#exercises-10)
- [**11\.7.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-11#conceptual-exercises-5)
- [**11\.7.2** Applied exercises](https://www.bayesrulesbook.com/chapter-11#applied-exercises-3)
- [**11\.7.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-11#open-ended-exercises-1)
- [**12** Poisson & Negative Binomial Regression](https://www.bayesrulesbook.com/chapter-12)
- [**12\.1** Building the Poisson regression model](https://www.bayesrulesbook.com/chapter-12#building-the-poisson-regression-model)
- [**12\.1.1** Specifying the data model](https://www.bayesrulesbook.com/chapter-12#specifying-the-data-model-1)
- [**12\.1.2** Specifying the priors](https://www.bayesrulesbook.com/chapter-12#ch12-prior-spec)
- [**12\.2** Simulating the posterior](https://www.bayesrulesbook.com/chapter-12#ch12-simulation)
- [**12\.3** Interpreting the posterior](https://www.bayesrulesbook.com/chapter-12#ch12-interp-pois)
- [**12\.4** Posterior prediction](https://www.bayesrulesbook.com/chapter-12#posterior-prediction-1)
- [**12\.5** Model evaluation](https://www.bayesrulesbook.com/chapter-12#model-evaluation)
- [**12\.6** Negative Binomial regression for overdispersed counts](https://www.bayesrulesbook.com/chapter-12#chapter-12-overdispersion)
- [**12\.7** Generalized linear models: Building on the theme](https://www.bayesrulesbook.com/chapter-12#ch12-glm)
- [**12\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-12#chapter-summary-11)
- [**12\.9** Exercises](https://www.bayesrulesbook.com/chapter-12#exercises-11)
- [**12\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-12#conceptual-exercises-6)
- [**12\.9.2** Applied exercises](https://www.bayesrulesbook.com/chapter-12#applied-exercises-4)
- [**13** Logistic Regression](https://www.bayesrulesbook.com/chapter-13)
- [**13\.1** Pause: Odds & probability](https://www.bayesrulesbook.com/chapter-13#pause-odds-probability)
- [**13\.2** Building the logistic regression model](https://www.bayesrulesbook.com/chapter-13#building-the-logistic-regression-model)
- [**13\.2.1** Specifying the data model](https://www.bayesrulesbook.com/chapter-13#specifying-the-data-model-2)
- [**13\.2.2** Specifying the priors](https://www.bayesrulesbook.com/chapter-13#specifying-the-priors-1)
- [**13\.3** Simulating the posterior](https://www.bayesrulesbook.com/chapter-13#ch13-post-sim-sec)
- [**13\.4** Prediction & classification](https://www.bayesrulesbook.com/chapter-13#chapter-13-prediction)
- [**13\.5** Model evaluation](https://www.bayesrulesbook.com/chapter-13#model-evaluation-1)
- [**13\.6** Extending the model](https://www.bayesrulesbook.com/chapter-13#extending-the-model)
- [**13\.7** Chapter summary](https://www.bayesrulesbook.com/chapter-13#chapter-summary-12)
- [**13\.8** Exercises](https://www.bayesrulesbook.com/chapter-13#exercises-12)
- [**13\.8.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-13#conceptual-exercises-7)
- [**13\.8.2** Applied exercises](https://www.bayesrulesbook.com/chapter-13#applied-exercises-5)
- [**13\.8.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-13#open-ended-exercises-2)
- [**14** Naive Bayes Classification](https://www.bayesrulesbook.com/chapter-14)
- [**14\.1** Classifying one penguin](https://www.bayesrulesbook.com/chapter-14#naive-one-case)
- [**14\.1.1** One categorical predictor](https://www.bayesrulesbook.com/chapter-14#one-categorical-predictor)
- [**14\.1.2** One quantitative predictor](https://www.bayesrulesbook.com/chapter-14#naive-one-quant)
- [**14\.1.3** Two predictors](https://www.bayesrulesbook.com/chapter-14#naive-two-quant)
- [**14\.2** Implementing & evaluating naive Bayes classification](https://www.bayesrulesbook.com/chapter-14#implementing-evaluating-naive-bayes-classification)
- [**14\.3** Naive Bayes vs logistic regression](https://www.bayesrulesbook.com/chapter-14#naive-bayes-vs-logistic-regression)
- [**14\.4** Chapter summary](https://www.bayesrulesbook.com/chapter-14#chapter-summary-13)
- [**14\.5** Exercises](https://www.bayesrulesbook.com/chapter-14#exercises-13)
- [**14\.5.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-14#conceptual-exercises-8)
- [**14\.5.2** Applied exercises](https://www.bayesrulesbook.com/chapter-14#applied-exercises-6)
- [**14\.5.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-14#open-ended-exercises-3)
- **IV Hierarchical Bayesian models**
- [**15** Hierarchical Models are Exciting](https://www.bayesrulesbook.com/chapter-15)
- [**15\.1** Complete pooling](https://www.bayesrulesbook.com/chapter-15#ch15-complete)
- [**15\.2** No pooling](https://www.bayesrulesbook.com/chapter-15#no-pooling)
- [**15\.3** Hierarchical data](https://www.bayesrulesbook.com/chapter-15#hierarchical-data)
- [**15\.4** Partial pooling with hierarchical models](https://www.bayesrulesbook.com/chapter-15#partial-pooling-with-hierarchical-models)
- [**15\.5** Chapter summary](https://www.bayesrulesbook.com/chapter-15#chapter-summary-14)
- [**15\.6** Exercises](https://www.bayesrulesbook.com/chapter-15#exercises-14)
- [**15\.6.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-15#conceptual-exercises-9)
- [**15\.6.2** Applied exercises](https://www.bayesrulesbook.com/chapter-15#applied-exercises-7)
- [**16** (Normal) Hierarchical Models without Predictors](https://www.bayesrulesbook.com/chapter-16)
- [**16\.1** Complete pooled model](https://www.bayesrulesbook.com/chapter-16#chapter-16-complete-pooling)
- [**16\.2** No pooled model](https://www.bayesrulesbook.com/chapter-16#chapter-16-no-pooling)
- [**16\.3** Building the hierarchical model](https://www.bayesrulesbook.com/chapter-16#hierarchical-building-16)
- [**16\.3.1** The hierarchy](https://www.bayesrulesbook.com/chapter-16#the-hierarchy)
- [**16\.3.2** Another way to think about it](https://www.bayesrulesbook.com/chapter-16#another-way-to-think-about-it)
- [**16\.3.3** Within- vs between-group variability](https://www.bayesrulesbook.com/chapter-16#ch16-within-btwn)
- [**16\.4** Posterior analysis](https://www.bayesrulesbook.com/chapter-16#ch-16-analyze)
- [**16\.4.1** Posterior simulation](https://www.bayesrulesbook.com/chapter-16#posterior-simulation-2)
- [**16\.4.2** Posterior analysis of global parameters](https://www.bayesrulesbook.com/chapter-16#posterior-analysis-of-global-parameters)
- [**16\.4.3** Posterior analysis of group-specific parameters](https://www.bayesrulesbook.com/chapter-16#posterior-analysis-of-group-specific-parameters)
- [**16\.5** Posterior prediction](https://www.bayesrulesbook.com/chapter-16#ch16-prediction)
- [**16\.6** Shrinkage & the bias-variance trade-off](https://www.bayesrulesbook.com/chapter-16#shrinkage-the-bias-variance-trade-off)
- [**16\.7** Not everything is hierarchical](https://www.bayesrulesbook.com/chapter-16#ch-16-nonhierarchical)
- [**16\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-16#chapter-summary-15)
- [**16\.9** Exercises](https://www.bayesrulesbook.com/chapter-16#exercises-15)
- [**16\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-16#conceptual-exercises-10)
- [**16\.9.2** Applied exercises](https://www.bayesrulesbook.com/chapter-16#applied-exercises-8)
- [**17** (Normal) Hierarchical Models with Predictors](https://www.bayesrulesbook.com/chapter-17)
- [**17\.1** First steps: Complete pooling](https://www.bayesrulesbook.com/chapter-17#first-steps-complete-pooling)
- [**17\.2** Hierarchical model with varying intercepts](https://www.bayesrulesbook.com/chapter-17#hierarchical-model-with-varying-intercepts)
- [**17\.2.1** Model building](https://www.bayesrulesbook.com/chapter-17#model-building)
- [**17\.2.2** Another way to think about it](https://www.bayesrulesbook.com/chapter-17#another-way-to-think-about-it-1)
- [**17\.2.3** Tuning the prior](https://www.bayesrulesbook.com/chapter-17#tuning-the-prior)
- [**17\.2.4** Posterior simulation & analysis](https://www.bayesrulesbook.com/chapter-17#chapter-17-random-intercepts)
- [**17\.3** Hierarchical model with varying intercepts & slopes](https://www.bayesrulesbook.com/chapter-17#hierarchical-model-with-varying-intercepts-slopes)
- [**17\.3.1** Model building](https://www.bayesrulesbook.com/chapter-17#model-building-1)
- [**17\.3.2** Optional: The decomposition of covariance model](https://www.bayesrulesbook.com/chapter-17#optional-the-decomposition-of-covariance-model)
- [**17\.3.3** Posterior simulation & analysis](https://www.bayesrulesbook.com/chapter-17#ch-17-rand-int-slope-sim)
- [**17\.4** Model evaluation & selection](https://www.bayesrulesbook.com/chapter-17#model-evaluation-selection)
- [**17\.5** Posterior prediction](https://www.bayesrulesbook.com/chapter-17#posterior-prediction-2)
- [**17\.6** Details: Longitudinal data](https://www.bayesrulesbook.com/chapter-17#details-longitudinal-data)
- [**17\.7** Example: Danceability](https://www.bayesrulesbook.com/chapter-17#example-danceability)
- [**17\.8** Chapter summary](https://www.bayesrulesbook.com/chapter-17#chapter-summary-16)
- [**17\.9** Exercises](https://www.bayesrulesbook.com/chapter-17#exercises-16)
- [**17\.9.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-17#conceptual-exercises-11)
- [**17\.9.2** Applied exercises](https://www.bayesrulesbook.com/chapter-17#applied-exercises-9)
- [**17\.9.3** Open-ended exercises](https://www.bayesrulesbook.com/chapter-17#open-ended-exercises-4)
- [**18** Non-Normal Hierarchical Regression & Classification](https://www.bayesrulesbook.com/chapter-18)
- [**18\.1** Hierarchical logistic regression](https://www.bayesrulesbook.com/chapter-18#hierarchical-logistic-regression)
- [**18\.1.1** Model building & simulation](https://www.bayesrulesbook.com/chapter-18#model-building-simulation)
- [**18\.1.2** Posterior analysis](https://www.bayesrulesbook.com/chapter-18#posterior-analysis)
- [**18\.1.3** Posterior classification](https://www.bayesrulesbook.com/chapter-18#posterior-classification)
- [**18\.1.4** Model evaluation](https://www.bayesrulesbook.com/chapter-18#model-evaluation-2)
- [**18\.2** Hierarchical Poisson & Negative Binomial regression](https://www.bayesrulesbook.com/chapter-18#hierarchical-poisson-negative-binomial-regression)
- [**18\.2.1** Model building & simulation](https://www.bayesrulesbook.com/chapter-18#ch-18-rat-sim)
- [**18\.2.2** Posterior analysis](https://www.bayesrulesbook.com/chapter-18#posterior-analysis-1)
- [**18\.2.3** Model evaluation](https://www.bayesrulesbook.com/chapter-18#model-evaluation-3)
- [**18\.3** Chapter summary](https://www.bayesrulesbook.com/chapter-18#chapter-summary-17)
- [**18\.4** Exercises](https://www.bayesrulesbook.com/chapter-18#exercises-17)
- [**18\.4.1** Applied & conceptual exercises](https://www.bayesrulesbook.com/chapter-18#applied-conceptual-exercises)
- [**18\.4.2** Open-ended exercises](https://www.bayesrulesbook.com/chapter-18#open-ended-exercises-5)
- [**19** Adding More Layers](https://www.bayesrulesbook.com/chapter-19)
- [**19\.1** Group-level predictors](https://www.bayesrulesbook.com/chapter-19#group-level-predictors)
- [**19\.1.1** A model using only individual-level predictors](https://www.bayesrulesbook.com/chapter-19#a-model-using-only-individual-level-predictors)
- [**19\.1.2** Incorporating group-level predictors](https://www.bayesrulesbook.com/chapter-19#incorporating-group-level-predictors)
- [**19\.1.3** Posterior simulation & global analysis](https://www.bayesrulesbook.com/chapter-19#posterior-simulation-global-analysis)
- [**19\.1.4** Posterior group-level analysis](https://www.bayesrulesbook.com/chapter-19#posterior-group-level-analysis)
- [**19\.1.5** We’re just scratching the surface\!](https://www.bayesrulesbook.com/chapter-19#were-just-scratching-the-surface)
- [**19\.2** Incorporating two (or more!) grouping variables](https://www.bayesrulesbook.com/chapter-19#incorporating-two-or-more-grouping-variables)
- [**19\.2.1** Data with two grouping variables](https://www.bayesrulesbook.com/chapter-19#data-with-two-grouping-variables)
- [**19\.2.2** Building a model with two grouping variables](https://www.bayesrulesbook.com/chapter-19#building-a-model-with-two-grouping-variables)
- [**19\.2.3** Simulating models with two grouping variables](https://www.bayesrulesbook.com/chapter-19#simulating-models-with-two-grouping-variables)
- [**19\.2.4** Examining the group-specific parameters](https://www.bayesrulesbook.com/chapter-19#examining-the-group-specific-parameters)
- [**19\.2.5** We’re just scratching the surface\!](https://www.bayesrulesbook.com/chapter-19#were-just-scratching-the-surface-1)
- [**19\.3** Exercises](https://www.bayesrulesbook.com/chapter-19#exercises-18)
- [**19\.3.1** Conceptual exercises](https://www.bayesrulesbook.com/chapter-19#conceptual-exercises-12)
- [**19\.3.2** Applied exercises](https://www.bayesrulesbook.com/chapter-19#applied-exercises-10)
- [**19\.4** Goodbye\!](https://www.bayesrulesbook.com/chapter-19#goodbye)
- [References](https://www.bayesrulesbook.com/references)
- [Published with bookdown](https://bookdown.org/)
## Bayes Rules! An Introduction to Applied Bayesian Modeling
*Alicia A. Johnson, Miles Q. Ott, Mine Dogucu*
*2021-12-01* | |||||||||||||||
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