Bayesian Statistics for the Social Sciences (Methodology in the Social Sciences)

Cover
Half Title Page
Title Page
Copyright
Series Editor’s Note
Preface to the Second Edition
Contents
Part I. Foundations
	1. Probability Concepts and Bayes’ Theorem
		1.1 Relevant Probability Axioms
			1.1.1 The Kolmogorov Axioms of Probability
			1.1.2 The Rényi Axioms of Probability
		1.2 Frequentist Probability
		1.3 Epistemic Probability
			1.3.1 Coherence and the Dutch Book
			1.3.2 Calibrating Epistemic Probability Assessments
		1.4 Bayes’ Theorem
			1.4.1 The Monty Hall Problem
		1.5 Summary
	2. Statistical Elements of Bayes’ Theorem
		2.1 Bayes’ Theorem Revisited
		2.2 Hierarchical Models and Pooling
		2.3 The Assumption of Exchangeability
		2.4 The Prior Distribution
			2.4.1 Non-Informative Priors
			2.4.2 Jeffreys’ Prior
			2.4.3 Weakly Informative Priors
			2.4.4 Informative Priors
			2.4.5 An Aside: Cromwell’s Rule
		2.5 Likelihood
			2.5.1 The Law of Likelihood
		2.6 The Posterior Distribution
		2.7 The Bayesian Central Limit Theorem and Bayesian Shrinkage
		2.8 Summary
	3. Common Probability Distributions and Their Priors
		3.1 The Gaussian Distribution
			3.1.1 Mean Unknown, Variance Known: The Gaussian Prior
			3.1.2 The Uniform Distribution as a Non-Informative Prior
			3.1.3 Mean Known, Variance Unknown: The Inverse-Gamma Prior
			3.1.4 Mean Known, Variance Unknown: The Half-Cauchy Prior
			3.1.5 Jeffreys’ Prior for the Gaussian Distribution
		3.2 The Poisson Distribution
			3.2.1 The Gamma Prior
			3.2.2 Jeffreys’ Prior for the Poisson Distribution
		3.3 The Binomial Distribution
			3.3.1 The Beta Prior
			3.3.2 Jeffreys’ Prior for the Binomial Distribution
		3.4 The Multinomial Distribution
			3.4.1 The Dirichlet Prior
			3.4.2 Jeffreys’ Prior for the Multinomial Distribution
		3.5 The Inverse-Wishart Distribution
		3.6 The LKJ Prior for Correlation Matrices
		3.7 Summary
	4. Obtaining and Summarizing the Posterior Distribution
		4.1 Basic Ideas of Markov Chain Monte Carlo Sampling
		4.2 The Random Walk Metropolis-Hastings Algorithm
		4.3 The Gibbs Sampler
		4.4 Hamiltonian Monte Carlo
			4.4.1 No-U-Turn Sampler (NUTS)
		4.5 Convergence Diagnostics
			4.5.1 Trace Plots
			4.5.2 Posterior Density Plots
			4.5.3 Autocorrelation Plots
			4.5.4 Effective Sample Size
			4.5.5 Potential Scale Reduction Factor
			4.5.6 Possible Error Messages When Using HMC/NUTS
		4.6 Summarizing the Posterior Distribution
			4.6.1 Point Estimates of the Posterior Distribution
			4.6.2 Interval Summaries of the Posterior Distribution
		4.7 Introduction to Stan and Example
		4.8 An Alternative Algorithm: Variational Bayes
			4.8.1 Evidence Lower Bound (ELBO)
			4.8.2 Variational Bayes Diagnostics
		4.9 Summary
Part II. Bayesian Model Building
	5. Bayesian Linear and Generalized Models
		5.1 The Bayesian Linear Regression Model
			5.1.1 Non-Informative Priors in the Linear Regression Model
		5.2 Bayesian Generalized Linear Models
			5.2.1 The Link Function
		5.3 Bayesian Logistic Regression
		5.4 Bayesian Multinomial Regression
		5.5 Bayesian Poisson Regression
		5.6 Bayesian Negative Binomial Regression
		5.7 Summary
	6. Model Evaluation and Comparison
		6.1 The Classical Approach to Hypothesis Testing and Its Limitations
		6.2 Model Assessment
			6.2.1 Prior Predictive Checking
			6.2.2 Posterior Predictive Checking
		6.3 Model Comparison
			6.3.1 Bayes Factors
			6.3.2 Criticisms of Bayes Factors and the BIC
			6.3.3 The Deviance Information Criterion (DIC)
			6.3.4 Widely Applicable Information Criterion (WAIC)
			6.3.5 Leave-One-Out Cross-Validation
			6.3.6 A Comparison of the WAIC and LOO
		6.4 Summary
	7. Bayesian Multilevel Modeling
		7.1 Revisiting Exchangeability
		7.2 Bayesian Random Effects Analysis of Variance
		7.3 Bayesian Intercepts as Outcomes Model
		7.4 Bayesian Intercepts and Slopes as Outcomes Model
		7.5 Summary
	8. Bayesian Latent Variable Modeling
		8.1 Bayesian Estimation for the CFA
			8.1.1 Priors for CFA Model Parameters
		8.2 Bayesian Latent Class Analysis
			8.2.1 The Problem of Label-Switching and a Possible Solution
			8.2.2 Comparison of VB to the EM Algorithm
		8.3 Summary
Part III. Advanced Topics and Methods
	9. Missing Data from a Bayesian Perspective
		9.1 A Nomenclature for Missing Data
		9.2 Ad Hoc Deletion Methods for Handling Missing Data
			9.2.1 Listwise Deletion
			9.2.2 Pairwise Deletion
		9.3 Single Imputation Methods
			9.3.1 Mean Imputation
			9.3.2 Regression Imputation
			9.3.3 Stochastic Regression Imputation
			9.3.4 Hot Deck Imputation
			9.3.5 Predictive Mean Matching
		9.4 Bayesian Methods of Multiple Imputation
			9.4.1 Data Augmentation
			9.4.2 Chained Equations
			9.4.3 EM Bootstrap: A Hybrid Bayesian/Frequentist Method
			9.4.4 Bayesian Bootstrap Predictive Mean Matching
			9.4.5 Accounting for Imputation Model Uncertainty
		9.5 Summary
	10. Bayesian Variable Selection and Sparsity
		10.1 Introduction
		10.2 The Ridge Prior
		10.3 The Lasso Prior
		10.4 The Horseshoe Prior
		10.5 Regularized Horseshoe Prior
		10.6 Comparison of Regularization Methods
			10.6.1 An Aside: The Spike-and-Slab Prior
		10.7 Summary
	11. Model Uncertainty
		11.1 Introduction
		11.2 Elements of Predictive Modeling
			11.2.1 Fixing Notation and Concepts
			11.2.2 Utility Functions for Evaluating Predictions
		11.3 Bayesian Model Averaging
			11.3.1 Statistical Specification of BMA
			11.3.2 Computational Considerations
			11.3.3 Markov Chain Monte Carlo Model Composition
			11.3.4 Parameter and Model Priors
			11.3.5 Evaluating BMA Results: Revisiting Scoring Rules
		11.4 True Models, Belief Models, and M-Frameworks
			11.4.1 Model Averaging in the M-Closed Framework
			11.4.2 Model Averaging in the M-Complete Framework
			11.4.3 Model Averaging in the M-Open Framework
		11.5 Bayesian Stacking
			11.5.1 Choice of Stacking Weights
		11.6 Summary
	12. Closing Thoughts
		12.1 A Bayesian Workflow for the Social Sciences
		12.2 Summarizing the Bayesian Advantage
			12.2.1 Coherence
			12.2.2 Conditioning on Observed Data
			12.2.3 Quantifying Evidence
			12.2.4 Validity
			12.2.5 Flexibility in Handling Complex Data Structures
			12.2.6 Formally Quantifying Uncertainty
List of Abbreviations and Acronyms
References
Author Index
Subject Index
About the Author