By Osvaldo A. Martin, Ravin Kumar, Junpeng Lao

Bayesian Modeling and Computation in Python

Book Description

Bayesian Modeling and Computation in Python aims to help beginner Bayesian practitioners to become intermediate modelers. It uses a hands on approach with PyMC3, Tensorflow Probability, ArviZ and other libraries focusing on the practice of applied statistics with references to the underlying mathematical theory.

The book starts with a refresher of the Bayesian Inference concepts. The second chapter introduces modern methods for Exploratory Analysis of Bayesian Models. With an understanding of these two fundamentals the subsequent chapters talk through various models including linear regressions, splines, time series, Bayesian additive regression trees. The final chapters include Approximate Bayesian Computation, end to end case studies showing how to apply Bayesian modelling in different settings, and a chapter about the internals of probabilistic programming languages. Finally the last chapter serves as a reference for the rest of the book by getting closer into mathematical aspects or by extending the discussion of certain topics.

This book is written by contributors of PyMC3, ArviZ, Bambi, and Tensorflow Probability among other libraries.

Foreword
Preface
Symbols
Chapter 1 Bayesian Inference
Chapter 2 Exploratory Analysis of Bayesian Models
Chapter 3 Linear Models and Probabilistic Programming Languages
Chapter 4 Extending Linear Models
Chapter 5 Splines
Chapter 6 Time Series
Chapter 7 Bayesian Additive Regression Trees
Chapter 8 Approximate Bayesian Computation
Chapter 9 End to End Bayesian Workflows
Chapter 10 Probabilistic Programming Languages
Chapter 11 Appendiceal Topics
Glossary
Bibliography
Index

By Vladimir Rakocevic

Fixed Point Results in W-Distance Spaces

Book Description

Fixed Point Results in W-Distance Spaces is a self-contained and comprehensive reference for advanced fixed-point theory and can serve as a useful guide for related research. The book can be used as a teaching resource for advanced courses on fixed-point theory, which is a modern and important field in mathematics. It would be especially valuable for graduate and postgraduate courses and seminars.

Features

Written in a concise and fluent style, covers a broad range of topics and includes related topics from research.
Suitable for researchers and postgraduates.
Contains brand new results not published elsewhere.

1. Introduction. 1.1. Metric Spaces. 1.2. Banach Contraction Principle. 1.3. Kannan Contraction. 1.4. ?iri?s Quasi-Contraction. 2. Some Basic Properties of W-Distances. 2.1. Definition and Examples. 2.2. Basic Properties of W-Distances. 2.3. More Results on W-Distances. 3. Fixed Point Results in the Framework of W-Distances. 3.1. Basic Fixed Point Results. 3.2. Banach Contraction Principle. 3.3 Rakotch’s Theorem. 3.4 Meir and Keeler’s Theorem. 3.5. Kannan Mappings. 3.6. ?iri?s Quasi-Contraction. 3.7. Fisher Quasi-Contraction. 4. Some Common Fixed Point Results using W-Distances. 4.1. Some Results of Ume and Kim. 4.2. Das and Naik Contraction. 4.3. Common Coupled Fixed Point Results. 4.4. Some of Mohanta’s Results. 4.5. Second Fisher theorem. 5. Best Proximity Points and Various (φ, ψ, p)-Contractive Mappings. 5.1 Best Proximity Points Involving Simulation Functions. 5.2. Best Proximity Points with R-Functions. 5.3. (φ, ψ, p)-Contractive Mappings. 5.4. (φ, ψ, p)-Weakly Contractive Mappings. 5.5. Generalized Weak Contraction Mappings. 5.6. W? ?-Kannan Contractions. 6. Miscellaneous Complements. 6.1. Multivalued Mappings. 6.2. ?iri?s Type Contractions at a Point. 6.3. Extension of a Result by Ri. 6.4. Weaker Meir-Keeler Function. 6.5. Contractive Mappings of Integral Type. 6.6 Ekeland’s Variational Principle. 6.7 Some Generalizations and Comments. Bibliography. Index.

By Steven J. Rosenberg

An Invitation to Abstract Algebra

Book Description

Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics.

To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.

The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois’ achievement in understanding how we can?and cannot?represent the roots of polynomials.

The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory.

The presentation includes the following features:

Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters.
The text can be used for a one, two, or three-term course.
Each new topic is motivated with a question.
A collection of projects appears in Chapter 23.
Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks?period. This book is offered as a manual to a new way of thinking. The author’s aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.

Preface
Symbols
1.Review of Sets, Functions, and Proofs
2.Introduction: A Number Game
3.Groups
4.Subgroups
5.Symmetry
6.Free Groups
7.Group Homomorphisms
8.Lagrange’s Theorem
9.Special Types of Homomorphisms
10.Making Groups
11.Rings
12.Results on Commutative Rings
13.Vector Spaces
14.Polynomial Rings
15.Field Theory
16.Galois Theory
17.Direct Sums and Direct Products
18.The Structure of Finite Abelian Groups
19.Group Actions
20.Learning from Z
21.The Problems of the Ancients
22.Solvability of Polynomial Equations by Radicals
23.Projects
Bibliography
Index

Edited By Mahlet G. Tadesse, Marina Vannucci

Handbook of Bayesian Variable Selection

Book Description

Bayesian variable selection has experienced substantial developments over the past 30 years with the proliferation of large data sets. Identifying relevant variables to include in a model allows simpler interpretation, avoids overfitting and multicollinearity, and can provide insights into the mechanisms underlying an observed phenomenon. Variable selection is especially important when the number of potential predictors is substantially larger than the sample size and sparsity can reasonably be assumed.

The Handbook of Bayesian Variable Selection provides a comprehensive review of theoretical, methodological and computational aspects of Bayesian methods for variable selection. The topics covered include spike-and-slab priors, continuous shrinkage priors, Bayes factors, Bayesian model averaging, partitioning methods, as well as variable selection in decision trees and edge selection in graphical models. The handbook targets graduate students and established researchers who seek to understand the latest developments in the field. It also provides a valuable reference for all interested in applying existing methods and/or pursuing methodological extensions.

Features:

? Provides a comprehensive review of methods and applications of Bayesian variable selection.
? Divided into four parts: Spike-and-Slab Priors; Continuous Shrinkage Priors; Extensions to various Modeling; Other Approaches to Bayesian Variable Selection.
? Covers theoretical and methodological aspects, as well as worked out examples with R code provided in the online supplement.
? Includes contributions by experts in the field.

By David Angell

Irrationality and Transcendence in Number Theory

Book Description

Irrationality and Transcendence in Number Theory tells the story of irrational numbers from their discovery in the days of Pythagoras to the ideas behind the work of Baker and Mahler on transcendence in the 20th century. It focuses on themes of irrationality, algebraic and transcendental numbers, continued fractions, approximation of real numbers by rationals, and relations between automata and transcendence. This book serves as a guide and introduction to number theory for advanced undergraduates and early postgraduates. Readers are led through the developments in number theory from ancient to modern times. The book includes a wide range of exercises, from routine problems to surprising and thought-provoking extension material.

Features

Uses techniques from widely diverse areas of mathematics, including number theory, calculus, set theory, complex analysis, linear algebra, and the theory of computation.
Suitable as a primary textbook for advanced undergraduate courses in number theory, or as supplementary reading for interested postgraduates.
Each chapter concludes with an appendix setting out the basic facts needed from each topic, so that the book is accessible to readers without any specific specialist background.

Table of Contents

1. Introduction. 1.1. Irrational Surds. 1.2. Irrational Decimals. 1.3. Irrationality of the Exponential Constant. 1.4. Other Results, and Some Open Questions. Exercises. Appendix: Some Elementary Number Theory. 2. Hermite’s Method. 2.1. Irrationality of er. 2.2. Irrationality of π. 2.3. Irrational values of trigonometric functions. Exercises. Appendix: Some Results of Elementary Calculus. 3. Algebraic & Transcendental Numbers. 3.1. Definitions and Basic Properties. 3.2. Existence of Transcendental Numbers. 3.3. Approximation of Real Numbers by Rationals. 3.4. Irrationality of (3) : a sketch. Exercises. Appendix 1: Countable and Uncountable Sets. Appendix 2: The Mean Value Theorem. Appendix 3: The Prime Number Theorem. 4. Continued Fractions. Definition and Basic Properties. 4.2. Continued Fractions of Irrational Numbers. 4.3. Approximation Properties of Convergents. 4.4. Two important Approximation Problems. 4.5. A "Computational" Test for Rationality. 4.6. Further Approximation Properties of Convergents. 4.7. Computing the Continued Fraction of an Algebraic Irrational. 4.8. The Continued Fraction of e. Exercises. Appendix 1: A Property of Positive Fractions. Appendix 2: Simultaneous Equations with Integral Coefficients. Appendix 3: Cardinality of Sets of Sequences. Appendix 4: Basic Musical Terminology. 5. Hermite’s Method for Transcendence. 5.1. Transcendence of e. 5.2. Transcendence of π. 5.3. Some more Irrationality Proofs. 5.4. Transcendence of ea .5.5. Other Results. Exercises. Appendix 1: Roots and Coefficients of Polynomials. Appendix 2: Some Real and Complex Analysis. Appendix 3: Ordering Complex Numbers. 6. Automata and Transcendence. 6.1. Deterministic Finite Automata. 6.2 Mahler’s Transcendence Proof. 6.3 A More General Transcendence Result. 6.4. A Transcendence Proof for the Thue Sequence. 6.5. Automata and Functional Equations. 6.6. Conclusion. Exercises. Appendix 1: Alphabets, Languages and DFAs. Appendix 2: Some Results of Complex Analysis. Appendix 3: A Result on Linear Equations. 7. Lambert’s Irrationality Proofs. 7.1. Generalised Continued Fractions. 7.2. Further Continued Fractions. Exercises. Appendix: Some Results from Elementary Algebra and Calculus. Hints for Exercises. Bibliography. Index.

By Alicia A. Johnson, Miles Q. Ott, Mine Dogucu

Bayes Rules!
An Introduction to Applied Bayesian Modeling

Copyright Year 2022
ISBN 9781032191591
ISBN 9780367255398 (soft cover)
January 25, 2022 Forthcoming
547 Pages 134 Color & 102 B/W Illustrations

Book Description

An engaging, sophisticated, and fun introduction to the field of Bayesian Statistics, Bayes Rules! An Introduction to Bayesian Modeling with R brings the power of modern Bayesian thinking, modeling, and computing to a broad audience. In particular, it is an ideal resource for advanced undergraduate Statistics students and practitioners with comparable experience.

Bayes Rules! empowers readers to weave Bayesian approaches into their everyday practice. Discussions and applications are data driven. A natural progression from fundamental to multivariable, hierarchical models emphasizes a practical and generalizable model building process. The evaluation of these Bayesian models reflects the fact that a data analysis does not exist in a vacuum.

Table of Contents

List of Tables
List of Figures
Preface
About the Author
Chapter 1 The Big (Bayesian) Picture
Chapter 2 Bayes’ Rule
Chapter 3 The Beta-Binomial Bayesian Model
Chapter 4 Balance and Sequentiality in Bayesian Analyses
Chapter 5 Conjugate Families Chapter 6 Approximating the Posterior
Chapter 7 MCMC Under the Hood
Chapter 8 Posterior Inference and Prediction
Chapter 9 Simple Normal Regression
Chapter 10 Evaluating Regression Models
Chapter 11 Extending the Normal Regression Model
Chapter 12 Poisson and Negative Binomial Regression
Chapter 13 Logistic Regression
Chapter 14 Naive Bayes Classification
Chapter 15 Hierarchical Models are Exciting
Chapter 16 (Normal) Hierarchical Models Without Predictors
Chapter 17 (Normal) Hierarchical Models With Predictors
Chapter 18 Non-Normal Hierarchical Regression & Classification
Chapter 19 Adding More Layers Bibliography Index

By Osvaldo A. Martin, Ravin Kumar, Junpeng Lao

Bayesian Modeling and Computation in Python

Book Description

Table of Contents

By Vladimir Rakocevic

Fixed Point Results in W-Distance Spaces

Book Description

Features

Table of Contents

By Steven J. Rosenberg

An Invitation to Abstract Algebra

Book Description

Table of Contents

Edited By Mahlet G. Tadesse, Marina Vannucci

Handbook of Bayesian Variable Selection

Book Description

Features:

Table of Contents

By David Angell

Irrationality and Transcendence in Number Theory

By Alicia A. Johnson, Miles Q. Ott, Mine Dogucu

Bayes Rules! An Introduction to Applied Bayesian Modeling

Bayes Rules!
An Introduction to Applied Bayesian Modeling