دانلود کتاب Principles of Uncertainty (Chapman & Hall/CRC Texts in Statistical Science)

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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
8.5 Uncertain mean and precision
8.5.1 Summary
8.5.2 Exercise
8.6 The normal linear model, uncertain precision
8.6.1 Summary
8.6.2 Exercise
8.7 The Wishart distribution
8.7.1 The trace of a square matrix
8.7.2 The Wishart distribution
8.7.3 Jacobian of a linear transformation of a symmetric matrix
8.7.4 Determinant of the triangular decomposition
8.7.5 Integrating the Wishart density
8.7.6 Multivariate normal distribution with uncertain precision and certain mean
8.7.7 Summary
8.7.8 Exercise
8.8 Both mean and precision matrix uncertain
8.8.1 Summary
8.8.2 Exercise
8.9 The Beta and Dirichlet distributions
8.9.1 Refining and coarsening
8.9.2 The marginal distribution of X
8.9.3 A relationship to the Gamma distribution
8.9.4 Stick breaking
8.9.5 Summary
8.9.6 Exercises
8.10 The exponential families
8.10.1 Summary
8.10.2 Exercises
8.10.3 Utility
8.11 Large sample theory for Bayesians
8.11.1 A supplement on convex functions and Jensen\'s Inequality
8.11.2 Resuming the main argument
8.11.3 Exercises
8.11.4 References
8.12 Some general perspective
9 Hierarchical Structuring of a Model
9.1 Introduction
9.1.1 Summary
9.1.2 Exercises
9.1.3 More history and related literature
9.1.4 Minimax as a decision criterion
9.2 Missing data
9.2.1 Examples
9.2.2 Bayesian analysis of missing data
9.2.3 Summary
9.2.4 Remarks and further reading
9.2.5 Exercises
9.3 Meta-analysis
9.3.1 Summary
9.4 Model uncertainty/model choice
9.4.1 Summary
9.4.2 Further reading
9.5 Graphical hierarchical models
9.5.1 Summary
9.5.2 Exercises
9.5.3 Additional references
9.6 Causation
9.7 Non-parametric Bayes
9.7.1 Gaussian Process
9.7.2 Exercise
9.7.3 Further reading
9.7.4 Dirichlet Process
9.7.5 The Generalized Polya Urn Scheme and the Chinese Restaurant Process
9.7.6 Stick breaking representation
9.7.7 Acknowledgment
9.7.8 Exercise
10 Markov Chain Monte Carlo
10.1 Introduction
10.2 Simulation
10.2.1 Summary
10.2.2 Exercises
10.2.3 References
10.3 The Metropolis-Hastings algorithm
10.3.1 Literature
10.3.2 Summary
10.3.3 Exercises
10.4 Extensions and special cases
10.4.1 Summary
10.4.2 Exercises
10.5 Practical considerations
10.5.1 Summary
10.5.2 Exercises
10.6 Variable dimensions: Reversible jumps
10.6.1 Summary
10.6.2 Exercises
10.7 Hamiltonian Monte Carlo
10.7.1 Further reading
10.8 Variational Inference
10.8.1 Summary
10.8.2 Exercises
10.8.3 Further reading:
11 Multiparty Problems
11.1 More than one decision maker
11.2 A simple three-stage game
11.2.1 Summary
11.2.2 References and notes
11.2.3 Exercises
11.3 Private information
11.3.1 Other views
11.3.2 References and notes
11.3.3 Summary
11.3.4 Exercises
11.4 Design for another\'s analysis
11.4.1 Notes and references
11.4.2 Summary
11.4.3 Exercises
11.4.4 Research problem
11.4.5 Career problem
11.5 Optimal Bayesian randomization
11.5.1 Notes and references
11.5.2 Summary
11.5.3 Exercises
11.6 Simultaneous moves
11.6.1 Minimax theory for two person constant-sum games
11.6.2 Comments from a Bayesian perspective
11.6.3 An example: Bank runs
11.6.4 Example: Prisoner\'s Dilemma
11.6.5 Notes and references
11.6.6 Iterated Prisoner\'s Dilemma
11.6.7 Centipede Game
11.6.8 Guessing a multiple of the average
11.6.9 References
11.6.10 Summary
11.6.11 Exercises
11.7 The Allais and Ellsberg Paradoxes
11.7.1 The Allais Paradox
11.7.2 The Ellsberg Paradox
11.7.3 What do these resolutions of the paradoxes imply for elicitation?
11.7.4 Notes and references
11.7.5 Summary
11.7.6 Exercises
11.8 Forming a Bayesian group
11.8.1 Summary
11.8.2 Notes and references
11.8.3 Exercises
Appendix A: The minimax theorem
11.A.1 Notes and references
12 Exploration of Old Ideas: A Critique of Classical Statistics
12.1 Introduction
12.1.1 Summary
12.1.2 Exercises
12.2 Testing
12.2.1 Further reading
12.2.2 Summary
12.2.3 Exercises
12.3 Confidence intervals and sets
12.3.1 Summary
12.4 Estimation
12.4.1 Further reading
12.4.2 Summary
12.4.3 Exercise
12.5 Choosing among models
12.6 Goodness of fit
12.7 Sampling theory statistics
12.8 \"Objective\" Bayesian methods
12.9 The Frequentist Guarantee
12.10 Final word about foundations
13 Epilogue: Applications
13.1 Computation
13.2 A final thought
Bibliography
Subject Index
Person Index