## Bayesian Statistics for Experimental Scientists

by Chechile

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An introduction to the Bayesian approach to statistical inference that demonstrates its superiority to orthodox frequentist statistical analysis.

This book offers an introduction to the Bayesian approach to statistical inference, with a focus on nonparametric and distribution-free methods. It covers not only well-developed methods for doing Bayesian statistics but also novel tools that enable Bayesian statistical analyses for cases that previously did not have a full Bayesian solution. The book's premise is that there are fundamental problems with orthodox frequentist statistical analyses that distort the scientific process. Side-by-side comparisons of Bayesian and frequentist methods illustrate the mismatch between the needs of experimental scientists in making inferences from data and the properties of the standard tools of classical statistics.

The book first covers elementary probability theory, the binomial model, the multinomial model, and methods for comparing different experimental conditions or groups. It then turns its focus to distribution-free statistics that are based on having ranked data, examining data from experimental studies and rank-based correlative methods. Each chapter includes exercises that help readers achieve a more complete understanding of the material.

The book devotes considerable attention not only to the linkage of statistics to practices in experimental science but also to the theoretical foundations of statistics. Frequentist statistical practices often violate their own theoretical premises. The beauty of Bayesian statistics, readers will learn, is that it is an internally coherent system of scientific inference that can be proved from probability theory.

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Contents (pg. vii)
Preface (pg. xiii)
I: Introduction to Bayesian Analysis for Categorical Data (pg. 1)
I.1 Overview (pg. 3)
I.2 Statistics as a Tool for Building Evidence (pg. 3)
I.3 Broad Data Types (pg. 4)
I.4 Obtaining and Using R Software (pg. 6)
I.5 Organization of Part I (pg. 9)
1. Probability and Inference (pg. 13)
1.1 Overview (pg. 13)
1.2 Samples, Populations, and Statistical Inference (pg. 13)
1.3 Defining Probability (pg. 19)
1.4 Assigning Probability Values (pg. 34)
1.5 Conjunctive Events (pg. 45)
1.6 Probability Trees and Unlimited Games (pg. 54)
1.7 Exercises (pg. 57)
2. Binomial Model (pg. 63)
2.1 Overview (pg. 63)
2.2 Binomial Features and Examples (pg. 63)
2.3 Binomial Distribution (pg. 65)
2.4 Bayesian Inference—Discrete Approximation (pg. 74)
2.5 Bayesian Inference—Continuous Model (pg. 83)
2.6 Which Prior? (pg. 98)
2.7 Statistical Decisions and the Bayes Factor (pg. 108)
2.8 Comparison to the Frequentist Analysis (pg. 120)
2.9 Exercises (pg. 145)
3. Multinomial Data (pg. 153)
3.1 Overview (pg. 153)
3.2 Multinomial Distribution and Examples (pg. 153)
3.3 The Dirichlet Distribution (pg. 159)
3.4 Random Samples from a Dirichlet Distribution (pg. 167)
3.5 Multinomial Process Models (pg. 171)
3.6 Markov Chain Monte Carlo Estimation (pg. 179)
3.7 Population Parameter Mapping (pg. 192)
3.8 Exercises (pg. 201)
3.9 Appendix: Proofs of Selected Theorems (pg. 204)
4. Condition Effects: Categorical Data (pg. 209)
4.1 Overview (pg. 209)
4.2 The Importance of Comparison Conditions (pg. 210)
4.3 Related Contingency Tables (L=2 Conditions) (pg. 211)
4.4 Bayesian CR Analysis (L=2 Conditions) (pg. 222)
4.5 Multiple Comparisons for Bayesian Inference (pg. 226)
4.6 L≥2 Completely Randomized Conditions (pg. 248)
4.7 L 2 Randomized-Block Conditions (pg. 255)
4.8 2×2 Split-Plot or Mixed Designs (pg. 263)
4.9 Planning the Sample Size in Advance (pg. 265)
4.10 Overview of Bayesian Comparison Procedures (pg. 270)
4.11 Exercises (pg. 272)
II: Bayesian Analysis of Ordinal Information (pg. 279)
5. Median- and Sign-Based Methods (pg. 285)
5.1 Overview (pg. 285)
5.2 Median Test (pg. 285)
5.3 Sign Test for RB Research Designs (pg. 298)
5.4 Bayesian Nonparametric Split-Plot Analysis (pg. 305)
5.5 Exercises (pg. 312)
6. Wilcoxon Signed-Rank Procedure (pg. 317)
6.1 Overview (pg. 317)
6.2 Frequentist Wilcoxon Signed-Rank Analysis (pg. 317)
6.3 Bayesian Discrete Small-Sample Analysis (pg. 321)
6.4 Continuous Large-Sample Model (pg. 328)
6.5 Comparisons with Other Procedures (pg. 334)
6.6 Exercises (pg. 337)
6.7 Appendix: Discrete-Approximation Software (pg. 339)
7. Mann-Whitney Procedure (pg. 341)
7.1 Overview (pg. 341)
7.2 Frequentist Mann-Whitney Statistic (pg. 341)
7.3 Bayesian Mann-Whitney Analysis: Discrete Case (pg. 347)
7.4 Continuous Larger-Sample Approximation (pg. 355)
7.5 Planning and Bayes-Factor Relative Efficiency (pg. 361)
7.6 Comparisons to the Independent-Groups t Test (pg. 362)
7.7 Exercises (pg. 363)
7.8 Appendix: Programs and Documentation (pg. 365)
8. Distribution-Free Correlation (pg. 371)
8.1 Overview (pg. 371)
8.2 Introduction to Rank-Based Correlation (pg. 371)
8.3 The Kendall Tau with Tied Ranks (pg. 387)
8.4 Bayesian Analysis for the Kendall Tau (pg. 395)
8.5 Testing Theories with the Kendall Tau (pg. 414)
8.6 Exercises (pg. 426)
References (pg. 429)
Index (pg. 447)

#### Richard A. Chechile

Richard A. Chechile is Professor of Psychology and Cognitive and Brain Science at Tufts University. He is the author of Analyzing Memory: The Formation, Retention, and Measurement of Memory (MIT Press).

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