An Introductory Handbook of Bayesian Thinking

1st Edition - April 17, 2024
Author: Stephen C. Loftus
Language: English
Paperback ISBN:
9 7 8 - 0 - 3 2 3 - 9 5 4 5 9 - 4
eBook ISBN:
9 7 8 - 0 - 4 4 3 - 2 9 1 1 1 - 1

An Introductory Handbook of Bayesian Thinking brings Bayesian thinking and methods to a wide audience beyond the mathematical sciences. Appropriate for students with some backgr… Read more

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An Introductory Handbook of Bayesian Thinking brings Bayesian thinking and methods to a wide audience beyond the mathematical sciences. Appropriate for students with some background in calculus and introductory statistics, particularly for nonstatisticians with a sufficient mathematical background, the text provides a gentle introduction to Bayesian ideas with a wide array of supporting examples from a variety of fields.

Cover image
Title page
Table of Contents
Copyright
Dedication
About the author
Stephen Loftus
Preface
Purpose of this book
Format of this book
Expectations of the reader
My history with Bayes
Part 1: An introduction to probability and R
Chapter 1: Probability and random variables
1.1. Introduction
1.2. Probability and random events
1.3. Sets and combining events
1.4. The axioms of probability
1.5. Calculating probabilities
1.6. Random variables
1.7. Conclusion
References
Chapter 2: Probability distributions, expected value, and variance
2.1. Introduction
2.2. Probability distributions
2.3. Cumulative distribution functions
2.4. Probability mass functions
2.5. Probability distribution functions
2.6. Expected values
2.7. Variance
2.8. Conclusion
References
Chapter 3: Common probability distributions
3.1. Introduction
3.2. Discrete probability distributions
3.3. Continuous distributions
3.4. Conclusion
References
Chapter 4: Conditional probability and Bayes' Rule
4.1. Introduction
4.2. Conditional probability
4.3. Bayes' Rule
4.4. Conclusion
References
Chapter 5: Finding and using the distributions of data
5.1. Introduction
5.2. Joint distributions
5.3. Likelihood functions
5.4. Estimating parameters using the likelihood
5.5. Conclusion
References
Chapter 6: Marginal and conditional distributions
6.1. Introduction
6.2. Marginal distributions
6.3. Conditional distributions
6.4. Conditional distributions and independence
6.5. Conclusion
References
Chapter 7: The Bayesian switch
7.1. Introduction
7.2. The goal of statistics
7.3. The random parameter
7.4. Finding the posterior distribution
7.5. Why Bayes?
7.6. Conclusion
References
Chapter 8: A brief review of R
8.1. Introduction
8.2. Calculations
8.3. Data types and storing data
8.4. Subsetting data
8.5. Libraries and loading data
8.6. Data summaries
8.7. Creating graphs
8.8. Probability distributions
8.9. For loops
8.10. Custom functions
8.11. Practice problems
8.12. Conclusion
References
Part 2: Basic Bayesian models
Chapter 9: Single-parameter inference
9.1. Introduction
9.2. Posterior distribution review
9.3. Finding our posterior
9.4. Posterior distributions calculation: fir seedlings
9.5. Conjugate prior distributions
9.6. Point summaries for the posterior
9.7. Practice problems
9.8. Conclusion
References
Chapter 10: Two-parameter inference
10.1. Introduction
10.2. Likelihoods of multiple parameters
10.3. Finding our posterior for multiple parameters
10.4. Monte Carlo integration
10.5. Gibbs sampling
10.6. Full conditionals: soil nitrogen example
10.7. Practice problems
10.8. Conclusion
References
Chapter 11: Gibbs Sampling in R
11.1. Introduction
11.2. Setup of Gibbs Samplers
11.3. Visualizing convergence and the posterior
11.4. Finding posterior summaries
11.5. Practice problems
11.6. Conclusion
References
Chapter 12: Bayesian linear regression
12.1. Introduction
12.2. Likelihood for regression data
12.3. Priors for parameters
12.4. Gibbs sampler for regression model
12.5. Extending to multiple regression
12.6. Practice problems
12.7. Conclusion
References
Chapter 13: Bayesian binary regression
13.1. Introduction
13.2. Generalized linear models
13.3. Regression for binary data
13.4. Data augmentation: likelihoods and priors
13.5. Gibbs sampler for data augmentation
13.6. An extension to ordinal regression
13.7. Practice problems
13.8. Conclusion
References
Chapter 14: Probabilistic clustering
14.1. Introduction
14.2. Univariate clustering with two groups: likelihoods and priors
14.3. Univariate clustering with two groups: Gibbs Sampler
14.4. Univariate clustering: beyond two groups
14.5. Practice problems
14.6. Conclusion
References
Chapter 15: Nonconjugate prior models
15.1. Introduction
15.2. When priors don't match
15.3. Metropolis-Hastings algorithm
15.4. Coding up Metropolis-Hastings
15.5. Metropolis-Hastings: calcium concentration in urine
15.6. Practice problems
15.7. Conclusion
References
Chapter 16: Models for count data
16.1. Introduction
16.2. Models for overdispersed data
16.3. Zero-inflated Poisson models
16.4. Poisson regression
16.5. Conclusion
References
Chapter 17: Testing hypotheses with Bayes
17.1. Introduction
17.2. Hypotheses and models
17.3. Calculating posterior model probabilities
17.4. Models with unknown parameters
17.5. Bayes Factors
17.6. Practice problems
17.7. Conclusion
References
Chapter 18: Bayesian inference beyond this book
18.1. The world of statistical challenges
18.2. The answers Bayesian methodology provide
18.3. Bringing everything back to basics
References
Appendix A: Matrix-form of Bayesian linear regression
A.1. Introduction
A.2. Multivariate normal likelihood
A.3. Extension to data augmentation
Appendix B: Multivariate clustering
B.1. Introduction
B.2. Multivariate normal likelihood
B.3. The Wishart distribution
B.4. Gibbs Sampler for bivariate clustering
B.5. Gibbs Sampler for multivariate clustering
Appendix C: List of probability distributions
Discrete distributions
Continuous distributions
Appendix D: Solutions to practice problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
References
References
Index

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An Introductory Handbook of Bayesian Thinking

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Stephen C. Loftus