Bayesian Inference. Theory, Methods, Computations

✍ Scribed by Silvelyn Zwanzig, Rauf Ahmad

• Publisher: CRC Press
• Year: 2024
• Tongue: English
• Leaves: 347
• Category: Library

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
1. Introduction
2. Bayesian Modelling
2.1. Statistical Model
2.2. Bayes Model
2.3. Advantages
2.3.1. Sequential Analysis
2.3.2. Big Data
2.3.3. Hierarchical Models
2.4. List of Problems
3. Choice of Prior
3.1. Subjective Priors
3.2. Conjugate Priors
3.3. Non-informative Priors
3.3.1. Laplace Prior
3.3.2. Jeffreys Prior
3.3.3. Reference Priors
3.4. List of Problems
4. Decision Theory
4.1. Basics of Decision Theory
4.2. Bayesian Decision Theory
4.3. Common Bayes Decision Rules
4.3.1. Quadratic Loss
4.3.2. Absolute Error Loss
4.3.3. Prediction
4.3.4. The 0–1 Loss
4.3.5. Intrinsic Losses
4.4. The Minimax Criterion
4.5. Bridges
4.6. List of Problems
5. Asymptotic Theory
5.1. Consistency
5.2. Schwartz’ Theorem
5.3. List of Problems
6. Normal Linear Models
6.1. Univariate Linear Models
6.2. Bayes Linear Models
6.2.1. Conjugate Prior: Parameter θ = β, σ2 Known
6.2.2. Conjugate Prior: Parameter θ = (β, σ2)
6.2.3. Jeffreys Prior
6.3. Linear Mixed Models
6.3.1. Bayes Linear Mixed Model, Marginal Model
6.3.2. Bayes Hierarchical Linear Mixed Model
6.4. Multivariate Linear Models
6.5. Bayes Multivariate Linear Models
6.5.1. Conjugate Prior
6.5.2. Jeffreys Prior
6.6. List of Problems
7. Estimation
7.1. Maximum a Posteriori (MAP) Estimator
7.1.1. Regularized Estimators
7.2. Bayes Rules
7.2.1. Estimation in Univariate Linear Models
7.2.2. Estimation in Multivariate Linear Models
7.3. Credible Sets
7.3.1. Credible Sets in Linear Models
7.4. Prediction
7.4.1. Prediction in Linear Models
7.5. List of Problems
8. Testing and Model Comparison
8.1. Bayes Rule
8.2. Bayes Factor
8.2.1. Point Null Hypothesis
8.2.2. Bayes Factor in Linear Model
8.2.3. Improper Prior
8.3. Bayes Information
8.3.1. Bayesian Information Criterion (BIC)
8.3.2. Deviance Information Criterion (DIC)
8.4. List of Problems
9. Computational Techniques
9.1. Deterministic Methods
9.1.1. Brute-Force
9.1.2. Laplace Approximation
9.2. Independent Monte Carlo Methods
9.2.1. Importance Sampling (IS)
9.3. Sampling from the Posterior
9.3.1. Sampling Importance Resampling (SIR)
9.3.2. Rejection Algorithm
9.4. Markov Chain Monte Carlo (MCMC)
9.4.1. Metropolis–Hastings Algorithms
9.4.2. Gibbs Sampling
9.5. Approximative Bayesian Computation (ABC)
9.6. Variational Inference (VI)
9.7. List of Problems
10. Solutions
10.1. Solutions for Chapter 2
10.2. Solutions for Chapter 3
10.3. Solutions for Chapter 4
10.4. Solutions for Chapter 5
10.5. Solutions for Chapter 6
10.6. Solutions for Chapter 7
10.7. Solutions for Chapter 8
10.8. Solutions for Chapter 9
11. Appendix
11.1. Discrete Distributions
11.2. Continuous Distributions
11.3. Multivariate Distributions
11.4. Matrix-Variate Distributions
Bibliography
Index