Bayesian Statistics - Peter M. Lee

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Résumé :

Preface xix

Preface to the First Edition xxi

1 Preliminaries 1

1.1 Probability and Bayes' Theorem 1

1.1.1 Notation 1

1.1.2 Axioms for probability 2

1.1.3 'Unconditional' probability 5

1.1.4 Odds 6

1.1.5 Independence 7

1.1.6 Some simple consequences of the axioms...

Biographie:
covariance and correlation 27

1.5.6 Approximations to the mean and variance of a function of a random variable 28

1.5.7 Conditional expectations and variances 29

1.5.8 Medians and modes 31

1.6 Exercises on Chapter 1 31

2 Bayesian inference for the normal distribution 36

2.1 Nature of Bayesian inference 36

2.1.1 Preliminary remarks 36

2.1.2 Post is prior times likelihood 36

2.1.3 Likelihood can be multiplied by any constant 38

2.1.4 Sequential use of Bayes' Theorem 38

2.1.5 The predictive distribution 39

2.1.6 A warning 39

2.2 Normal prior and likelihood 40

2.2.1 Posterior from a normal prior and likelihood 40

2.2.2 Example 42

2.2.3 Predictive distribution 43

2.2.4 The nature of the assumptions made 44

2.3 Several normal observations with a normal prior 44

2.3.1 Posterior distribution 44

2.3.2 Example 46

2.3.3 Predictive distribution 47

2.3.4 Robustness 47

2.4 Dominant likelihoods 48

2.4.1 Improper priors 48

2.4.2 Approximation of proper priors by improper priors 49

2.5 Locally uniform priors 50

2.5.1 Bayes' postulate 50

2.5.2 Data translated likelihoods 52

2.5.3 Transformation of unknown parameters 52

2.6 Highest density regions 54

2.6.1 Need for summaries of posterior information 54

2.6.2 Relation to classical statistics 55

2.7 Normal variance 55

2.7.1 A suitable prior for the normal variance 55

2.7.2 Reference prior for the normal variance 58

2.8 HDRs for the normal variance 59

2.8.1 What distribution should we be considering? 59

2.8.2 Example 59

2.9 The role of sufficiency 60

2.9.1 Definition of sufficiency 60

2.9.2 Neyman's factorization theorem 61

2.9.3 Sufficiency principle 63

2.9.4 Examples 63

2.9.5 Order statistics and minimal sufficient statistics 65

2.9.6 Examples on minimal sufficiency 66

2.10 Conjugate prior distributions 67

2.10.1 Definition and difficulties 67

2.10.2 Examples 68

2.10.3 Mixtures of conjugate densities 69

2.10.4 Is your prior really conjugate? 71

2.11 The exponential family 71

2.11.1 Definition 71

2.11.2 Examples 72

2.11.3 Conjugate densities 72

2.11.4 Two-parameter exponential family 73

2.12 Normal mean and variance both unknown 73

2.12.1 Formulation of the problem 73

2.12.2 Marginal distribution of the mean 75

2.12.3 Example of the posterior density for the mean 76

2.12.4 Marginal distribution of the variance 77

2.12.5 Example of the posterior density of the variance 77

2.12.6 Conditional density of the mean for given variance 77

2.13 Conjugate joint prior for the normal distribution 78

2.13.1 The form of the conjugate prior 78

2.13.2 Derivation of the posterior 80

2.13.3 Example 81

2.13.4 Concluding remarks 82

2.14 Exercises on Chapter 2 82

3 Some other common distributions 85

3.1 The binomial distribution 85

3.1.1 Conjugate prior 85

3.1.2 Odds and log-odds 88

3.1.3 Highest density regions 90

3.1.4 Example 91

3.1.5 Predictive distribution 92

3.2 Reference prior for the binomial likelihood 92

3.2.1 Bayes' postulate 92

3.2.2 Haldane's prior 93

3.2.3 The arc-sine distribution 94

3.2.4 Conclusion 95

3.3 Jeffreys' rule 96

3.3.1 Fisher's information 96

3.3.2 The information from several observations 97

3.3.3 Jeffreys' prior 98

3.3.4 Examples 98

3.3.5 Warning 100

3.3.6 Several unknown parameter...

Sommaire:
Bayes' Theorem 7

1.2 Examples on Bayes' Theorem 9

1.2.1 The Biology of Twins 9

1.2.2 A political example 10

1.2.3 A warning 10

1.3 Random variables 12

1.3.1 Discrete random variables 12

1.3.2 The binomial distribution 13

1.3.3 Continuous random variables 14

1.3.4 The normal distribution 16

1.3.5 Mixed random variables 17

1.4 Several random variables 17

1.4.1 Two discrete random variables 17

1.4.2 Two continuous random variables 18

1.4.3 Bayes' Theorem for random variables 20

1.4.4 Example 21

1.4.5 One discrete variable and one continuous variable 21

1.4.6 Independent random variables 22

1.5 Means and variances 23

1.5.1 Expectations 23

1.5.2 The expectation of a sum and of a product 24

1.5.3 Variance, precision and standard deviation 25

1.5.4 Examples 25

1.5.5 Variance of a sum...