Uncertainty Quantification with R: Bayesian Methods

✍ Scribed by Eduardo Souza de Cursi

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 493
• Series: s: IInternational Series in Operations Research & Management Science
• Category: Library

✦ Synopsis

This book is a rigorous but practical presentation of the Bayesian techniques of uncertainty quantification, with applications in R. This volume includes mathematical arguments at the level necessary to make the presentation rigorous and the assumptions clearly established, while maintaining a focus on practical applications of Bayesian uncertainty quantification methods. Practical aspects of applied probability are also discussed, making the content accessible to students. The introduction of R allows the reader to solve more complex problems involving a more significant number of variables. Users will be able to use examples laid out in the text to solve medium-sized problems.
The list of topics covered in this volume includes basic Bayesian probabilities, entropy, Bayesian estimation and decision, sequential Bayesian estimation, and numerical methods. Blending theoretical rigor and practical applications, this volume will be of interest to professionals, researchers, graduate and undergraduate students interested in the use of Bayesian uncertainty quantification techniques within the framework of operations research and mathematical programming, for applications in management and planning.

✦ Table of Contents

Introduction
Contents
Chapter 1: Basic Bayesian Probabilities
1.1 A Historical Perspective
1.2 Probabilities
1.3 Representing a Probability Space in R
1.4 Independent Events and Bayes´ Formula
1.5 Representation of Product Spaces
1.6 Random Variables
1.7 The Exponential Family
Classical Discrete Distributions of the Exponential Family
Classical Continuous Distributions
1.8 Distributions That Are Not in the Exponential Family
1.9 Random Vectors
1.10 Independent Random Variables
1.11 Central Limit and Levy´s Theorem
1.12 Exchangeability
1.13 Stochastic Processes
References
Chapter 2: Beliefs
2.1 Beliefs
2.2 Working with Beliefs in R
2.3 Commonality
2.4 Retrieving Mass Functions from Other Representations
2.5 Couples, Product Frames, and Marginality
2.6 Contour Functions
2.7 Combination of Beliefs
2.8 Conditional Beliefs, Plausibilities, and Frames
2.9 Connections Between Probabilities and Beliefs
References
Chapter 3: Information and Entropy
3.1 Entropy
3.2 Joint and Conditional Entropy
3.3 Mutual Information and Relative Entropy
3.4 Surprise Quantification
References
Chapter 4: Maximum Entropy
4.1 The Principle of Maximum Entropy
4.2 Karhunen-Loève Expansions for the Numerical Generation of Stochastic Processes
4.3 Numerical Construction of Karhunen-Loève Expansions
References
Chapter 5: Bayesian Inference
5.1 Bayesian Estimation
5.2 Jeffreys´ Priors
5.3 Using UQ in Bayesian Estimation
5.4 Bayesian Regression
5.5 Model Selection and Hypothesis Testing
5.6 The Expectation-Maximization Algorithm
References
Chapter 6: Sequential Bayesian Estimation
6.1 Monte-Carlo Approach
6.2 Metropolis-Hastings Algorithm
6.3 Kalman Filtering
6.4 Particle Filtering
6.5 Determination of the Distribution of the Noise
6.6 Bayesian Optimization
References
Index