Principles of uncertainty / Joseph B. Kadane.

Format:

Book

Author/Creator:

Kadane, Joseph B., author.

Series:

Texts in statistical science.

Texts in statistical science

Language:

English

Subjects (All):

Bayesian statistical decision theory.

Mathematical statistics.

Physical Description:

1 online resource (xxvi, 498 pages) : illustrations.

Edition:

Second edition.

Place of Publication:

Boca Raton, FL ; London ; New York : CRC Press, Taylor & Francis Group, 2021.

Summary:

"Like the De Groot winning first edition, the second edition of Principles of Uncertainty is an accessible, comprehensive guide to the theory of Bayesian Statistics written in an appealing, inviting style, and packed with interesting examples. It presents an accessible, comprehensive guide to the subjective Bayesian approach which has played a pivotal role in game theory, economics, and the recent boom in Markov Chain Monte Carlo methods. This new edition has been updated throughout and features a new chapter on Nonparametric Bayesian Methods"-- Provided by publisher.

Contents:

Cover

Half Title

Title Page

Copyright Page

Dedication

Table of Contents

List of Figures

List of Tables

Foreword

Preface

1 Probability

1.1 Avoiding being a sure loser

1.1.1 Interpretation

1.1.2 Notes and other views

1.1.3 Summary

1.1.4 Exercises

1.2 Disjoint events

1.2.1 Summary

1.2.2 A supplement on induction

1.2.3 A supplement on indexed mathematical expressions

1.2.4 Intersections of events

1.2.5 Summary

1.2.6 Exercises

1.3 Events not necessarily disjoint

1.3.1 A supplement on proofs of set inclusion

1.3.2 Boole's Inequality

1.3.3 Summary

1.3.4 Exercises

1.4 Random variables, also known as uncertain quantities

1.4.1 Summary

1.4.2 Exercises

1.5 Finite number of values

1.5.1 Summary

1.5.2 Exercises

1.6 Other properties of expectation

1.6.1 Summary

1.6.2 Exercises

1.7 Coherence implies not a sure loser

1.7.1 Summary

1.7.2 Exercises

1.8 Expectations and limits

1.8.1 A supplement on limits

1.8.2 Resuming the discussion of expectations and limits

1.8.3 Reference

1.8.4 Exercises

2 Conditional Probability and Bayes Theorem

2.1 Conditional probability

2.1.1 Summary

2.1.2 Exercises

2.2 The birthday problem

2.2.1 Exercises

2.2.2 A supplement on computing

2.2.3 References

2.2.4 Exercises

2.3 Simpson's Paradox

2.3.1 Notes

2.3.2 Exercises

2.4 Bayes Theorem

2.4.1 Notes and other views

2.4.2 Exercises

2.5 Independence of events

2.5.1 Summary

2.5.2 Exercises

2.6 The Monty Hall problem

2.6.1 Exercises

2.7 Gambler's Ruin problem

2.7.1 Changing stakes

2.7.2 Summary

2.7.3 References

2.7.4 Exercises

2.8 Iterated expectations and independence

2.8.1 Summary

2.8.2 Exercises

2.9 The binomial and multinomial distributions

2.9.1 Refining and coarsening.

2.9.2 Why these distributions have these names

2.9.3 Summary

2.9.4 Exercises

2.10 Sampling without replacement

2.10.1 Polya's Urn Scheme

2.10.2 Summary

2.10.3 References

2.10.4 Exercises

2.11 Variance and covariance

2.11.1 An application of the Cauchy-Schwarz Inequality

2.11.2 Remark

2.11.3 Summary

2.11.4 Exercises

2.12 A short introduction to multivariate thinking

2.12.1 A supplement on vectors and matrices

2.12.2 Least squares

2.12.3 A limitation of correlation in expressing negative association between non-independent random variables

2.12.4 Covariance matrices

2.12.5 Conditional variances and covariances

2.12.6 Summary

2.12.7 Exercises

2.13 Tchebychev's Inequality

2.13.1 Interpretations

2.13.2 Summary

2.13.3 Exercises

3 Discrete Random Variables

3.1 Countably many possible values

3.1.1 A supplement on in nity

3.1.2 Notes

3.1.3 Summary

3.1.4 Exercises

3.2 Finite additivity

3.2.1 Summary

3.2.2 References

3.2.3 Exercises

3.3 Countable additivity

3.3.1 Summary

3.3.2 References

3.3.3 Can we use countable additivity to handle countably many bets simultaneously?

3.3.4 Exercises

3.3.5 A supplement on calculus-based methods of demonstrating the convergence of series

3.4 Properties of countable additivity

3.4.1 Summary

3.5 Dynamic sure loss

3.5.1 Summary

3.5.2 Discussion

3.5.3 Other views

3.6 Probability generating functions

3.6.1 Summary

3.6.2 Exercises

3.7 Geometric random variables

3.7.1 Summary

3.7.2 Exercises

3.8 The negative binomial random variable

3.8.1 Summary

3.8.2 Exercises

3.9 The Poisson random variable

3.9.1 Summary

3.9.2 Exercises

3.10 Cumulative distribution function

3.10.1 Introduction

3.10.2 An interesting relationship between cdf's and expectations.

3.10.3 Summary

3.10.4 Exercises

3.11 Dominated and bounded convergence

3.11.1 Summary

3.11.2 Exercises

4 Continuous Random Variables

4.1 Introduction

4.1.1 The cumulative distribution function

4.1.2 Summary and reference

4.1.3 Exercises

4.2 Joint distributions

4.2.1 Summary

4.2.2 Exercises

4.3 Conditional distributions and independence

4.3.1 Summary

4.3.2 Exercises

4.4 Existence and properties of expectations

4.4.1 Summary

4.4.2 Exercises

4.5 Extensions

4.5.1 An interesting relationship between cdf's and expectations of continuous random variables

4.6 Chapter retrospective so far

4.7 Bounded and dominated convergence

4.7.1 A supplement about limits of sequences and Cauchy's criterion

4.7.2 Exercises

4.7.3 References

4.7.4 A supplement on Riemann integrals

4.7.5 Summary

4.7.6 Exercises

4.7.7 Bounded and dominated convergence for Riemann integrals

4.7.8 Summary

4.7.9 Exercises

4.7.10 References

4.7.11 A supplement on uniform convergence

4.7.12 Bounded and dominated convergence for Riemann expectations

4.7.13 Summary

4.7.14 Exercises

4.7.15 Discussion

4.8 The Riemann-Stieltjes integral

4.8.1 Definition of the Riemann-Stieltjes integral

4.8.2 The Riemann-Stieltjes integral in the nite discrete case

4.8.3 The Riemann-Stieltjes integral in the countable discrete case

4.8.4 The Riemann-Stieltjes integral when F has a derivative

4.8.5 Other cases of the Riemann-Stieltjes integral

4.8.6 Summary

4.8.7 Exercises

4.9 The McShane-Stieltjes integral

4.9.1 Extension of the McShane integral to unbounded sets

4.9.2 Properties of the McShane integral

4.9.3 McShane probabilities

4.9.4 Comments and relationship to other literature

4.9.5 Summary

4.9.6 Exercises

4.10 The road from here.

4.11 The strong law of large numbers

4.11.1 Random variables (otherwise known as uncertain quantities) more precisely

4.11.2 Modes of convergence of random variables

4.11.3 Four algebraic lemmas

4.11.4 The strong law of large numbers

4.11.5 Summary

4.11.6 Exercises

4.11.7 Reference

5 Transformations

5.1 Introduction

5.2 Discrete random variables

5.2.1 Summary

5.2.2 Exercises

5.3 Univariate continuous distributions

5.3.1 Summary

5.3.2 Exercises

5.3.3 A note to the reader

5.4 Linear spaces

5.4.1 A mathematical note

5.4.2 Inner products

5.4.3 Summary

5.4.4 Exercises

5.5 Permutations

5.5.1 Summary

5.5.2 Exercises

5.6 Number systems

DeMoivre's Formula

5.6.1 A supplement with more facts about Taylor series

5.6.2 DeMoivre's Formula

5.6.3 Complex numbers in polar co-ordinates

5.6.4 The fundamental theorem of algebra

5.6.5 Summary

5.6.6 Exercises

5.6.7 Notes

5.7 Determinants

5.7.1 Summary

5.7.2 Exercises

5.7.3 Real matrices

5.7.4 References

5.8 Eigenvalues, eigenvectors and decompositions

5.8.1 Projection matrices

5.8.2 Generalizations

5.8.3 Summary

5.8.4 Exercises

5.9 Non-linear transformations

5.9.1 Summary

5.9.2 Exercise

5.10 The Borel-Kolmogorov Paradox

5.10.1 Summary

5.10.2 Exercises

6 Normal Distribution

6.1 Introduction

6.2 Moment generating functions

6.2.1 Summary

6.2.2 Exercises

6.3 The normal distribution

6.3.1 Remark

6.3.2 Exercises

6.4 Multivariate normal distributions

6.4.1 Exercises

6.5 The Central Limit Theorem

6.5.1 A supplement on a relation between convergence in probability and weak convergence

6.5.2 A supplement on uniform continuity and points of accumulation

6.5.3 Exercises

6.5.4 Resuming the proof of the central limit theorem.

6.5.5 Supplement on the sup-norm

6.5.6 Resuming the development of the central limit theorem

6.5.7 The delta method

6.5.8 A heuristic explanation of smudging

6.5.9 Summary

6.5.10 Exercises

6.6 The Weak Law of Large Numbers

6.6.1 Exercises

6.6.2 Related literature

6.6.3 Remark

6.7 Stein's Method

7 Making Decisions

7.1 Introduction

7.2 An example

7.2.1 Remarks on the use of these ideas

7.2.2 Summary

7.2.3 Exercises

7.3 In greater generality

7.3.1 A supplement on regret

7.3.2 Notes and other views

7.3.3 Summary

7.3.4 Exercises

7.4 The St. Petersburg Paradox

7.4.1 Summary

7.4.2 Notes and references

7.4.3 Exercises

7.5 Risk aversion

7.5.1 A supplement on finite differences and derivatives

7.5.2 Resuming the discussion of risk aversion

7.5.3 References

7.5.4 Summary

7.5.5 Exercises

7.6 Log (fortune) as utility

7.6.1 A supplement on optimization

7.6.2 Resuming the maximization of log fortune in various circumstances

7.6.3 Interpretation

7.6.4 Summary

7.6.5 Exercises

7.7 Decisions after seeing data

7.7.1 Summary

7.7.2 Exercise

7.8 The expected value of sample information

7.8.1 Summary

7.8.2 Exercise

7.9 An example

7.9.1 Summary

7.9.2 Exercises

7.9.3 Further reading

7.10 Randomized decisions

7.10.1 Summary

7.10.2 Exercise

7.11 Sequential decisions

7.11.1 Notes

7.11.2 Summary

7.11.3 Exercise

8 Conjugate Analysis

8.1 A simple normal-normal case

8.1.1 Summary

8.1.2 Exercises

8.2 A multivariate normal case, known precision

8.2.1 Shrinkage and Stein's Paradox

8.2.2 Summary

8.2.3 Exercises

8.3 The normal linear model with known precision

8.3.1 Summary

8.3.2 Further reading

8.3.3 Exercises

8.4 The gamma distribution

8.4.1 Summary

8.4.2 Exercises.

8.4.3 Reference.

Notes:

Includes bibliographical references (pages 463-480) and indexes.

Description based on print version record.

ISBN:

1-351-68335-7

1-315-16756-5

1-351-68336-5

9781315167565

OCLC:

1154112761