Bayesian Inference: Theory, Methods, Computations

✍ Scribed by Silvelyn Zwanzig, Rauf Ahmad

• Publisher: Chapman and Hall/CRC
• Year: 2024
• Tongue: English
• Leaves: 347
• Edition: 1
• Category: Library

✦ Synopsis

Bayesian Inference: Theory, Methods, Computations provides a comprehensive coverage of the fundamentals of Bayesian inference from all important perspectives, namely theory, methods and computations.

All theoretical results are presented as formal theorems, corollaries, lemmas etc., furnished with detailed proofs. The theoretical ideas are explained in simple and easily comprehensible forms, supplemented with several examples. A clear reasoning on the validity, usefulness, and pragmatic approach of the Bayesian methods is provided. A large number of examples and exercises, and solutions to all exercises, are provided to help students understand the concepts through ample practice.

The book is primarily aimed at first or second semester master students, where parts of the book can also be used at Ph.D. level or by research community at large. The emphasis is on exact cases. However, to gain further insight into the core concepts, an entire chapter is dedicated to computer intensive techniques. Selected chapters and sections of the book can be used for a one-semester course on Bayesian statistics.

Key Features:

• Explains basic ideas of Bayesian statistical inference in an easily comprehensible form
• Illustrates main ideas through sketches and plots
• Contains large number of examples and exercises
• Provides solutions to all exercises
• Includes R codes

Silvelyn Zwanzig is a Professor for Mathematical Statistics at Uppsala University. She studied Mathematics at the Humboldt University of Berlin. Before coming to Sweden, she was Assistant Professor at the University of Hamburg in Germany. She received her Ph.D. in Mathematics at the Academy of Sciences of the GDR. She has taught Statistics to undergraduate and graduate students since 1991. Her research interests include theoretical statistics and computer-intensive methods.

Rauf Ahmad is Associate Professor at the Department of Statistics, Uppsala University. He did his Ph.D. at the University of Göttingen, Germany. Before joining Uppsala University, he worked at the Division of Mathematical Statistics, Department of Mathematics, Linköping University, and at Biometry Division, Swedish University of Agricultural Sciences, Uppsala. He has taught Statistics to undergraduate and graduate students since 1995. His research interests include high-dimensional inference, mathematical statistics, and U-statistics.

✦ 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