{"product_id":"bayesian-statistics-9781118332573","title":"Bayesian Statistics","description":"\u003cb\u003eBook Synopsis\u003c\/b\u003e\u003cbr\u003eBayesian Statistics is the school of thought that combines prior beliefs with the likelihood of a hypothesis to arrive at posterior beliefs. The first edition of Peter Lee   s book appeared in 1989, but the subject has moved ever onwards, with increasing emphasis on Monte Carlo based techniques.\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTrade Review\u003c\/b\u003e\u003cbr\u003e\u003cp\u003e“As a lifelong non-statistician and sporadic “user” of statistics, I have not come across another advanced statistics book (as I would characterize this one) that offers so much to the non-expert and, I’ll bet, to the expert as well. The book has my highest recommendation.”  (\u003ci\u003eComputing Reviews\u003c\/i\u003e, 7 January 2013)\u003c\/p\u003e\u003cbr\u003e\u003cbr\u003e\u003cb\u003eTable of Contents\u003c\/b\u003e\u003cbr\u003e\u003cp\u003ePreface xix\u003c\/p\u003e \u003cp\u003ePreface to the First Edition xxi\u003c\/p\u003e \u003cp\u003e\u003cb\u003e1 Preliminaries 1\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e1.1 Probability and Bayes’ Theorem 1\u003c\/p\u003e \u003cp\u003e1.1.1 Notation 1\u003c\/p\u003e \u003cp\u003e1.1.2 Axioms for probability 2\u003c\/p\u003e \u003cp\u003e1.1.3 ‘Unconditional’ probability 5\u003c\/p\u003e \u003cp\u003e1.1.4 Odds 6\u003c\/p\u003e \u003cp\u003e1.1.5 Independence 7\u003c\/p\u003e \u003cp\u003e1.1.6 Some simple consequences of the axioms; Bayes’ Theorem 7\u003c\/p\u003e \u003cp\u003e1.2 Examples on Bayes’ Theorem 9\u003c\/p\u003e \u003cp\u003e1.2.1 The Biology of Twins 9\u003c\/p\u003e \u003cp\u003e1.2.2 A political example 10\u003c\/p\u003e \u003cp\u003e1.2.3 A warning 10\u003c\/p\u003e \u003cp\u003e1.3 Random variables 12\u003c\/p\u003e \u003cp\u003e1.3.1 Discrete random variables 12\u003c\/p\u003e \u003cp\u003e1.3.2 The binomial distribution 13\u003c\/p\u003e \u003cp\u003e1.3.3 Continuous random variables 14\u003c\/p\u003e \u003cp\u003e1.3.4 The normal distribution 16\u003c\/p\u003e \u003cp\u003e1.3.5 Mixed random variables 17\u003c\/p\u003e \u003cp\u003e1.4 Several random variables 17\u003c\/p\u003e \u003cp\u003e1.4.1 Two discrete random variables 17\u003c\/p\u003e \u003cp\u003e1.4.2 Two continuous random variables 18\u003c\/p\u003e \u003cp\u003e1.4.3 Bayes’ Theorem for random variables 20\u003c\/p\u003e \u003cp\u003e1.4.4 Example 21\u003c\/p\u003e \u003cp\u003e1.4.5 One discrete variable and one continuous variable 21\u003c\/p\u003e \u003cp\u003e1.4.6 Independent random variables 22\u003c\/p\u003e \u003cp\u003e1.5 Means and variances 23\u003c\/p\u003e \u003cp\u003e1.5.1 Expectations 23\u003c\/p\u003e \u003cp\u003e1.5.2 The expectation of a sum and of a product 24\u003c\/p\u003e \u003cp\u003e1.5.3 Variance, precision and standard deviation 25\u003c\/p\u003e \u003cp\u003e1.5.4 Examples 25\u003c\/p\u003e \u003cp\u003e1.5.5 Variance of a sum; covariance and correlation 27\u003c\/p\u003e \u003cp\u003e1.5.6 Approximations to the mean and variance of a function of a random variable 28\u003c\/p\u003e \u003cp\u003e1.5.7 Conditional expectations and variances 29\u003c\/p\u003e \u003cp\u003e1.5.8 Medians and modes 31\u003c\/p\u003e \u003cp\u003e1.6 Exercises on Chapter 1 31\u003c\/p\u003e \u003cp\u003e\u003cb\u003e2 Bayesian inference for the normal distribution 36\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e2.1 Nature of Bayesian inference 36\u003c\/p\u003e \u003cp\u003e2.1.1 Preliminary remarks 36\u003c\/p\u003e \u003cp\u003e2.1.2 Post is prior times likelihood 36\u003c\/p\u003e \u003cp\u003e2.1.3 Likelihood can be multiplied by any constant 38\u003c\/p\u003e \u003cp\u003e2.1.4 Sequential use of Bayes’ Theorem 38\u003c\/p\u003e \u003cp\u003e2.1.5 The predictive distribution 39\u003c\/p\u003e \u003cp\u003e2.1.6 A warning 39\u003c\/p\u003e \u003cp\u003e2.2 Normal prior and likelihood 40\u003c\/p\u003e \u003cp\u003e2.2.1 Posterior from a normal prior and likelihood 40\u003c\/p\u003e \u003cp\u003e2.2.2 Example 42\u003c\/p\u003e \u003cp\u003e2.2.3 Predictive distribution 43\u003c\/p\u003e \u003cp\u003e2.2.4 The nature of the assumptions made 44\u003c\/p\u003e \u003cp\u003e2.3 Several normal observations with a normal prior 44\u003c\/p\u003e \u003cp\u003e2.3.1 Posterior distribution 44\u003c\/p\u003e \u003cp\u003e2.3.2 Example 46\u003c\/p\u003e \u003cp\u003e2.3.3 Predictive distribution 47\u003c\/p\u003e \u003cp\u003e2.3.4 Robustness 47\u003c\/p\u003e \u003cp\u003e2.4 Dominant likelihoods 48\u003c\/p\u003e \u003cp\u003e2.4.1 Improper priors 48\u003c\/p\u003e \u003cp\u003e2.4.2 Approximation of proper priors by improper priors 49\u003c\/p\u003e \u003cp\u003e2.5 Locally uniform priors 50\u003c\/p\u003e \u003cp\u003e2.5.1 Bayes’ postulate 50\u003c\/p\u003e \u003cp\u003e2.5.2 Data translated likelihoods 52\u003c\/p\u003e \u003cp\u003e2.5.3 Transformation of unknown parameters 52\u003c\/p\u003e \u003cp\u003e2.6 Highest density regions 54\u003c\/p\u003e \u003cp\u003e2.6.1 Need for summaries of posterior information 54\u003c\/p\u003e \u003cp\u003e2.6.2 Relation to classical statistics 55\u003c\/p\u003e \u003cp\u003e2.7 Normal variance 55\u003c\/p\u003e \u003cp\u003e2.7.1 A suitable prior for the normal variance 55\u003c\/p\u003e \u003cp\u003e2.7.2 Reference prior for the normal variance 58\u003c\/p\u003e \u003cp\u003e2.8 HDRs for the normal variance 59\u003c\/p\u003e \u003cp\u003e2.8.1 What distribution should we be considering? 59\u003c\/p\u003e \u003cp\u003e2.8.2 Example 59\u003c\/p\u003e \u003cp\u003e2.9 The role of sufficiency 60\u003c\/p\u003e \u003cp\u003e2.9.1 Definition of sufficiency 60\u003c\/p\u003e \u003cp\u003e2.9.2 Neyman’s factorization theorem 61\u003c\/p\u003e \u003cp\u003e2.9.3 Sufficiency principle 63\u003c\/p\u003e \u003cp\u003e2.9.4 Examples 63\u003c\/p\u003e \u003cp\u003e2.9.5 Order statistics and minimal sufficient statistics 65\u003c\/p\u003e \u003cp\u003e2.9.6 Examples on minimal sufficiency 66\u003c\/p\u003e \u003cp\u003e2.10 Conjugate prior distributions 67\u003c\/p\u003e \u003cp\u003e2.10.1 Definition and difficulties 67\u003c\/p\u003e \u003cp\u003e2.10.2 Examples 68\u003c\/p\u003e \u003cp\u003e2.10.3 Mixtures of conjugate densities 69\u003c\/p\u003e \u003cp\u003e2.10.4 Is your prior really conjugate? 71\u003c\/p\u003e \u003cp\u003e2.11 The exponential family 71\u003c\/p\u003e \u003cp\u003e2.11.1 Definition 71\u003c\/p\u003e \u003cp\u003e2.11.2 Examples 72\u003c\/p\u003e \u003cp\u003e2.11.3 Conjugate densities 72\u003c\/p\u003e \u003cp\u003e2.11.4 Two-parameter exponential family 73\u003c\/p\u003e \u003cp\u003e2.12 Normal mean and variance both unknown 73\u003c\/p\u003e \u003cp\u003e2.12.1 Formulation of the problem 73\u003c\/p\u003e \u003cp\u003e2.12.2 Marginal distribution of the mean 75\u003c\/p\u003e \u003cp\u003e2.12.3 Example of the posterior density for the mean 76\u003c\/p\u003e \u003cp\u003e2.12.4 Marginal distribution of the variance 77\u003c\/p\u003e \u003cp\u003e2.12.5 Example of the posterior density of the variance 77\u003c\/p\u003e \u003cp\u003e2.12.6 Conditional density of the mean for given variance 77\u003c\/p\u003e \u003cp\u003e2.13 Conjugate joint prior for the normal distribution 78\u003c\/p\u003e \u003cp\u003e2.13.1 The form of the conjugate prior 78\u003c\/p\u003e \u003cp\u003e2.13.2 Derivation of the posterior 80\u003c\/p\u003e \u003cp\u003e2.13.3 Example 81\u003c\/p\u003e \u003cp\u003e2.13.4 Concluding remarks 82\u003c\/p\u003e \u003cp\u003e2.14 Exercises on Chapter 2 82\u003c\/p\u003e \u003cp\u003e\u003cb\u003e3 Some other common distributions 85\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e3.1 The binomial distribution 85\u003c\/p\u003e \u003cp\u003e3.1.1 Conjugate prior 85\u003c\/p\u003e \u003cp\u003e3.1.2 Odds and log-odds 88\u003c\/p\u003e \u003cp\u003e3.1.3 Highest density regions 90\u003c\/p\u003e \u003cp\u003e3.1.4 Example 91\u003c\/p\u003e \u003cp\u003e3.1.5 Predictive distribution 92\u003c\/p\u003e \u003cp\u003e3.2 Reference prior for the binomial likelihood 92\u003c\/p\u003e \u003cp\u003e3.2.1 Bayes’ postulate 92\u003c\/p\u003e \u003cp\u003e3.2.2 Haldane’s prior 93\u003c\/p\u003e \u003cp\u003e3.2.3 The arc-sine distribution 94\u003c\/p\u003e \u003cp\u003e3.2.4 Conclusion 95\u003c\/p\u003e \u003cp\u003e3.3 Jeffreys’ rule 96\u003c\/p\u003e \u003cp\u003e3.3.1 Fisher’s information 96\u003c\/p\u003e \u003cp\u003e3.3.2 The information from several observations 97\u003c\/p\u003e \u003cp\u003e3.3.3 Jeffreys’ prior 98\u003c\/p\u003e \u003cp\u003e3.3.4 Examples 98\u003c\/p\u003e \u003cp\u003e3.3.5 Warning 100\u003c\/p\u003e \u003cp\u003e3.3.6 Several unknown parameters 100\u003c\/p\u003e \u003cp\u003e3.3.7 Example 101\u003c\/p\u003e \u003cp\u003e3.4 The Poisson distribution 102\u003c\/p\u003e \u003cp\u003e3.4.1 Conjugate prior 102\u003c\/p\u003e \u003cp\u003e3.4.2 Reference prior 103\u003c\/p\u003e \u003cp\u003e3.4.3 Example 104\u003c\/p\u003e \u003cp\u003e3.4.4 Predictive distribution 104\u003c\/p\u003e \u003cp\u003e3.5 The uniform distribution 106\u003c\/p\u003e \u003cp\u003e3.5.1 Preliminary definitions 106\u003c\/p\u003e \u003cp\u003e3.5.2 Uniform distribution with a fixed lower endpoint 107\u003c\/p\u003e \u003cp\u003e3.5.3 The general uniform distribution 108\u003c\/p\u003e \u003cp\u003e3.5.4 Examples 110\u003c\/p\u003e \u003cp\u003e3.6 Reference prior for the uniform distribution 110\u003c\/p\u003e \u003cp\u003e3.6.1 Lower limit of the interval fixed 110\u003c\/p\u003e \u003cp\u003e3.6.2 Example 111\u003c\/p\u003e \u003cp\u003e3.6.3 Both limits unknown 111\u003c\/p\u003e \u003cp\u003e3.7 The tramcar problem 113\u003c\/p\u003e \u003cp\u003e3.7.1 The discrete uniform distribution 113\u003c\/p\u003e \u003cp\u003e3.8 The first digit problem; invariant priors 114\u003c\/p\u003e \u003cp\u003e3.8.1 A prior in search of an explanation 114\u003c\/p\u003e \u003cp\u003e3.8.2 The problem 114\u003c\/p\u003e \u003cp\u003e3.8.3 A solution 115\u003c\/p\u003e \u003cp\u003e3.8.4 Haar priors 117\u003c\/p\u003e \u003cp\u003e3.9 The circular normal distribution 117\u003c\/p\u003e \u003cp\u003e3.9.1 Distributions on the circle 117\u003c\/p\u003e \u003cp\u003e3.9.2 Example 119\u003c\/p\u003e \u003cp\u003e3.9.3 Construction of an HDR by numerical integration 120\u003c\/p\u003e \u003cp\u003e3.9.4 Remarks 122\u003c\/p\u003e \u003cp\u003e3.10 Approximations based on the likelihood 122\u003c\/p\u003e \u003cp\u003e3.10.1 Maximum likelihood 122\u003c\/p\u003e \u003cp\u003e3.10.2 Iterative methods 123\u003c\/p\u003e \u003cp\u003e3.10.3 Approximation to the posterior density 123\u003c\/p\u003e \u003cp\u003e3.10.4 Examples 124\u003c\/p\u003e \u003cp\u003e3.10.5 Extension to more than one parameter 126\u003c\/p\u003e \u003cp\u003e3.10.6 Example 127\u003c\/p\u003e \u003cp\u003e3.11 Reference posterior distributions 128\u003c\/p\u003e \u003cp\u003e3.11.1 The information provided by an experiment 128\u003c\/p\u003e \u003cp\u003e3.11.2 Reference priors under asymptotic normality 130\u003c\/p\u003e \u003cp\u003e3.11.3 Uniform distribution of unit length 131\u003c\/p\u003e \u003cp\u003e3.11.4 Normal mean and variance 132\u003c\/p\u003e \u003cp\u003e3.11.5 Technical complications 134\u003c\/p\u003e \u003cp\u003e3.12 Exercises on Chapter 3 134\u003c\/p\u003e \u003cp\u003e\u003cb\u003e4 Hypothesis testing 138\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e4.1 Hypothesis testing 138\u003c\/p\u003e \u003cp\u003e4.1.1 Introduction 138\u003c\/p\u003e \u003cp\u003e4.1.2 Classical hypothesis testing 138\u003c\/p\u003e \u003cp\u003e4.1.3 Difficulties with the classical approach 139\u003c\/p\u003e \u003cp\u003e4.1.4 The Bayesian approach 140\u003c\/p\u003e \u003cp\u003e4.1.5 Example 142\u003c\/p\u003e \u003cp\u003e4.1.6 Comment 143\u003c\/p\u003e \u003cp\u003e4.2 One-sided hypothesis tests 143\u003c\/p\u003e \u003cp\u003e4.2.1 Definition 143\u003c\/p\u003e \u003cp\u003e4.2.2 P-values 144\u003c\/p\u003e \u003cp\u003e4.3 Lindley’s method 145\u003c\/p\u003e \u003cp\u003e4.3.1 A compromise with classical statistics 145\u003c\/p\u003e \u003cp\u003e4.3.2 Example 145\u003c\/p\u003e \u003cp\u003e4.3.3 Discussion 146\u003c\/p\u003e \u003cp\u003e4.4 Point (or sharp) null hypotheses with prior information 146\u003c\/p\u003e \u003cp\u003e4.4.1 When are point null hypotheses reasonable? 146\u003c\/p\u003e \u003cp\u003e4.4.2 A case of nearly constant likelihood 147\u003c\/p\u003e \u003cp\u003e4.4.3 The Bayesian method for point null hypotheses 148\u003c\/p\u003e \u003cp\u003e4.4.4 Sufficient statistics 149\u003c\/p\u003e \u003cp\u003e4.5 Point null hypotheses for the normal distribution 150\u003c\/p\u003e \u003cp\u003e4.5.1 Calculation of the Bayes’ factor 150\u003c\/p\u003e \u003cp\u003e4.5.2 Numerical examples 151\u003c\/p\u003e \u003cp\u003e4.5.3 Lindley’s paradox 152\u003c\/p\u003e \u003cp\u003e4.5.4 A bound which does not depend on the prior distribution 154\u003c\/p\u003e \u003cp\u003e4.5.5 The case of an unknown variance 155\u003c\/p\u003e \u003cp\u003e4.6 The Doogian philosophy 157\u003c\/p\u003e \u003cp\u003e4.6.1 Description of the method 157\u003c\/p\u003e \u003cp\u003e4.6.2 Numerical example 157\u003c\/p\u003e \u003cp\u003e4.7 Exercises on Chapter 4 158\u003c\/p\u003e \u003cp\u003e\u003cb\u003e5 Two-sample problems 162\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e5.1 Two-sample problems – both variances unknown 162\u003c\/p\u003e \u003cp\u003e5.1.1 The problem of two normal samples 162\u003c\/p\u003e \u003cp\u003e5.1.2 Paired comparisons 162\u003c\/p\u003e \u003cp\u003e5.1.3 Example of a paired comparison problem 163\u003c\/p\u003e \u003cp\u003e5.1.4 The case where both variances are known 163\u003c\/p\u003e \u003cp\u003e5.1.5 Example 164\u003c\/p\u003e \u003cp\u003e5.1.6 Non-trivial prior information 165\u003c\/p\u003e \u003cp\u003e5.2 Variances unknown but equal 165\u003c\/p\u003e \u003cp\u003e5.2.1 Solution using reference priors 165\u003c\/p\u003e \u003cp\u003e5.2.2 Example 167\u003c\/p\u003e \u003cp\u003e5.2.3 Non-trivial prior information 167\u003c\/p\u003e \u003cp\u003e5.3 Variances unknown and unequal (Behrens–Fisher problem) 168\u003c\/p\u003e \u003cp\u003e5.3.1 Formulation of the problem 168\u003c\/p\u003e \u003cp\u003e5.3.2 Patil’s approximation 169\u003c\/p\u003e \u003cp\u003e5.3.3 Example 170\u003c\/p\u003e \u003cp\u003e5.3.4 Substantial prior information 170\u003c\/p\u003e \u003cp\u003e5.4 The Behrens–Fisher controversy 171\u003c\/p\u003e \u003cp\u003e5.4.1 The Behrens–Fisher problem from a classical standpoint 171\u003c\/p\u003e \u003cp\u003e5.4.2 Example 172\u003c\/p\u003e \u003cp\u003e5.4.3 The controversy 173\u003c\/p\u003e \u003cp\u003e5.5 Inferences concerning a variance ratio 173\u003c\/p\u003e \u003cp\u003e5.5.1 Statement of the problem 173\u003c\/p\u003e \u003cp\u003e5.5.2 Derivation of the F distribution 174\u003c\/p\u003e \u003cp\u003e5.5.3 Example 175\u003c\/p\u003e \u003cp\u003e5.6 Comparison of two proportions; the 2 × 2 table 176\u003c\/p\u003e \u003cp\u003e5.6.1 Methods based on the log-odds ratio 176\u003c\/p\u003e \u003cp\u003e5.6.2 Example 177\u003c\/p\u003e \u003cp\u003e5.6.3 The inverse root-sine transformation 178\u003c\/p\u003e \u003cp\u003e5.6.4 Other methods 178\u003c\/p\u003e \u003cp\u003e5.7 Exercises on Chapter 5 179\u003c\/p\u003e \u003cp\u003e\u003cb\u003e6 Correlation, regression and the analysis of variance 182\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e6.1 Theory of the correlation coefficient 182\u003c\/p\u003e \u003cp\u003e6.1.1 Definitions 182\u003c\/p\u003e \u003cp\u003e6.1.2 Approximate posterior distribution of the correlation coefficient 184\u003c\/p\u003e \u003cp\u003e6.1.3 The hyperbolic tangent substitution 186\u003c\/p\u003e \u003cp\u003e6.1.4 Reference prior 188\u003c\/p\u003e \u003cp\u003e6.1.5 Incorporation of prior information 189\u003c\/p\u003e \u003cp\u003e6.2 Examples on the use of the correlation coefficient 189\u003c\/p\u003e \u003cp\u003e6.2.1 Use of the hyperbolic tangent transformation 189\u003c\/p\u003e \u003cp\u003e6.2.2 Combination of several correlation coefficients 189\u003c\/p\u003e \u003cp\u003e6.2.3 The squared correlation coefficient 190\u003c\/p\u003e \u003cp\u003e6.3 Regression and the bivariate normal model 190\u003c\/p\u003e \u003cp\u003e6.3.1 The model 190\u003c\/p\u003e \u003cp\u003e6.3.2 Bivariate linear regression 191\u003c\/p\u003e \u003cp\u003e6.3.3 Example 193\u003c\/p\u003e \u003cp\u003e6.3.4 Case of known variance 194\u003c\/p\u003e \u003cp\u003e6.3.5 The mean value at a given value of the explanatory variable 194\u003c\/p\u003e \u003cp\u003e6.3.6 Prediction of observations at a given value of the explanatory variable 195\u003c\/p\u003e \u003cp\u003e6.3.7 Continuation of the example 195\u003c\/p\u003e \u003cp\u003e6.3.8 Multiple regression 196\u003c\/p\u003e \u003cp\u003e6.3.9 Polynomial regression 196\u003c\/p\u003e \u003cp\u003e6.4 Conjugate prior for the bivariate regression model 197\u003c\/p\u003e \u003cp\u003e6.4.1 The problem of updating a regression line 197\u003c\/p\u003e \u003cp\u003e6.4.2 Formulae for recursive construction of a regression line 197\u003c\/p\u003e \u003cp\u003e6.4.3 Finding an appropriate prior 199\u003c\/p\u003e \u003cp\u003e6.5 Comparison of several means – the one way model 200\u003c\/p\u003e \u003cp\u003e6.5.1 Description of the one way layout 200\u003c\/p\u003e \u003cp\u003e6.5.2 Integration over the nuisance parameters 201\u003c\/p\u003e \u003cp\u003e6.5.3 Derivation of the F distribution 203\u003c\/p\u003e \u003cp\u003e6.5.4 Relationship to the analysis of variance 203\u003c\/p\u003e \u003cp\u003e6.5.5 Example 204\u003c\/p\u003e \u003cp\u003e6.5.6 Relationship to a simple linear regression model 206\u003c\/p\u003e \u003cp\u003e6.5.7 Investigation of contrasts 207\u003c\/p\u003e \u003cp\u003e6.6 The two way layout 209\u003c\/p\u003e \u003cp\u003e6.6.1 Notation 209\u003c\/p\u003e \u003cp\u003e6.6.2 Marginal posterior distributions 210\u003c\/p\u003e \u003cp\u003e6.6.3 Analysis of variance 212\u003c\/p\u003e \u003cp\u003e6.7 The general linear model 212\u003c\/p\u003e \u003cp\u003e6.7.1 Formulation of the general linear model 212\u003c\/p\u003e \u003cp\u003e6.7.2 Derivation of the posterior 214\u003c\/p\u003e \u003cp\u003e6.7.3 Inference for a subset of the parameters 215\u003c\/p\u003e \u003cp\u003e6.7.4 Application to bivariate linear regression 216\u003c\/p\u003e \u003cp\u003e6.8 Exercises on Chapter 6 217\u003c\/p\u003e \u003cp\u003e\u003cb\u003e7 Other topics 221\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e7.1 The likelihood principle 221\u003c\/p\u003e \u003cp\u003e7.1.1 Introduction 221\u003c\/p\u003e \u003cp\u003e7.1.2 The conditionality principle 222\u003c\/p\u003e \u003cp\u003e7.1.3 The sufficiency principle 223\u003c\/p\u003e \u003cp\u003e7.1.4 The likelihood principle 223\u003c\/p\u003e \u003cp\u003e7.1.5 Discussion 225\u003c\/p\u003e \u003cp\u003e7.2 The stopping rule principle 226\u003c\/p\u003e \u003cp\u003e7.2.1 Definitions 226\u003c\/p\u003e \u003cp\u003e7.2.2 Examples 226\u003c\/p\u003e \u003cp\u003e7.2.3 The stopping rule principle 227\u003c\/p\u003e \u003cp\u003e7.2.4 Discussion 228\u003c\/p\u003e \u003cp\u003e7.3 Informative stopping rules 229\u003c\/p\u003e \u003cp\u003e7.3.1 An example on capture and recapture of fish 229\u003c\/p\u003e \u003cp\u003e7.3.2 Choice of prior and derivation of posterior 230\u003c\/p\u003e \u003cp\u003e7.3.3 The maximum likelihood estimator 231\u003c\/p\u003e \u003cp\u003e7.3.4 Numerical example 231\u003c\/p\u003e \u003cp\u003e7.4 The likelihood principle and reference priors 232\u003c\/p\u003e \u003cp\u003e7.4.1 The case of Bernoulli trials and its general implications 232\u003c\/p\u003e \u003cp\u003e7.4.2 Conclusion 233\u003c\/p\u003e \u003cp\u003e7.5 Bayesian decision theory 234\u003c\/p\u003e \u003cp\u003e7.5.1 The elements of game theory 234\u003c\/p\u003e \u003cp\u003e7.5.2 Point estimators resulting from quadratic loss 236\u003c\/p\u003e \u003cp\u003e7.5.3 Particular cases of quadratic loss 237\u003c\/p\u003e \u003cp\u003e7.5.4 Weighted quadratic loss 238\u003c\/p\u003e \u003cp\u003e7.5.5 Absolute error loss 238\u003c\/p\u003e \u003cp\u003e7.5.6 Zero-one loss 239\u003c\/p\u003e \u003cp\u003e7.5.7 General discussion of point estimation 240\u003c\/p\u003e \u003cp\u003e7.6 Bayes linear methods 240\u003c\/p\u003e \u003cp\u003e7.6.1 Methodology 240\u003c\/p\u003e \u003cp\u003e7.6.2 Some simple examples 241\u003c\/p\u003e \u003cp\u003e7.6.3 Extensions 243\u003c\/p\u003e \u003cp\u003e7.7 Decision theory and hypothesis testing 243\u003c\/p\u003e \u003cp\u003e7.7.1 Relationship between decision theory and classical hypothesis testing 243\u003c\/p\u003e \u003cp\u003e7.7.2 Composite hypotheses 245\u003c\/p\u003e \u003cp\u003e7.8 Empirical Bayes methods 245\u003c\/p\u003e \u003cp\u003e7.8.1 Von Mises’ example 245\u003c\/p\u003e \u003cp\u003e7.8.2 The Poisson case 246\u003c\/p\u003e \u003cp\u003e7.9 Exercises on Chapter 7 247\u003c\/p\u003e \u003cp\u003e\u003cb\u003e8 Hierarchical models 253\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e8.1 The idea of a hierarchical model 253\u003c\/p\u003e \u003cp\u003e8.1.1 Definition 253\u003c\/p\u003e \u003cp\u003e8.1.2 Examples 254\u003c\/p\u003e \u003cp\u003e8.1.3 Objectives of a hierarchical analysis 257\u003c\/p\u003e \u003cp\u003e8.1.4 More on empirical Bayes methods 257\u003c\/p\u003e \u003cp\u003e8.2 The hierarchical normal model 258\u003c\/p\u003e \u003cp\u003e8.2.1 The model 258\u003c\/p\u003e \u003cp\u003e8.2.2 The Bayesian analysis for known overall mean 259\u003c\/p\u003e \u003cp\u003e8.2.3 The empirical Bayes approach 261\u003c\/p\u003e \u003cp\u003e8.3 The baseball example 262\u003c\/p\u003e \u003cp\u003e8.4 The Stein estimator 264\u003c\/p\u003e \u003cp\u003e8.4.1 Evaluation of the risk of the James–Stein estimator 267\u003c\/p\u003e \u003cp\u003e8.5 Bayesian analysis for an unknown overall mean 268\u003c\/p\u003e \u003cp\u003e8.5.1 Derivation of the posterior 270\u003c\/p\u003e \u003cp\u003e8.6 The general linear model revisited 272\u003c\/p\u003e \u003cp\u003e8.6.1 An informative prior for the general linear model 272\u003c\/p\u003e \u003cp\u003e8.6.2 Ridge regression 274\u003c\/p\u003e \u003cp\u003e8.6.3 A further stage to the general linear model 275\u003c\/p\u003e \u003cp\u003e8.6.4 The one way model 276\u003c\/p\u003e \u003cp\u003e8.6.5 Posterior variances of the estimators 277\u003c\/p\u003e \u003cp\u003e8.7 Exercises on Chapter 8 277\u003c\/p\u003e \u003cp\u003e\u003cb\u003e9 The Gibbs sampler and other numerical methods 281\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e9.1 Introduction to numerical methods 281\u003c\/p\u003e \u003cp\u003e9.1.1 Monte Carlo methods 281\u003c\/p\u003e \u003cp\u003e9.1.2 Markov chains 282\u003c\/p\u003e \u003cp\u003e9.2 The EM algorithm 283\u003c\/p\u003e \u003cp\u003e9.2.1 The idea of the EM algorithm 283\u003c\/p\u003e \u003cp\u003e9.2.2 Why the EM algorithm works 285\u003c\/p\u003e \u003cp\u003e9.2.3 Semi-conjugate prior with a normal likelihood 287\u003c\/p\u003e \u003cp\u003e9.2.4 The EM algorithm for the hierarchical normal model 288\u003c\/p\u003e \u003cp\u003e9.2.5 A particular case of the hierarchical normal model 290\u003c\/p\u003e \u003cp\u003e9.3 Data augmentation by Monte Carlo 291\u003c\/p\u003e \u003cp\u003e9.3.1 The genetic linkage example revisited 291\u003c\/p\u003e \u003cp\u003e9.3.2 Use of R 291\u003c\/p\u003e \u003cp\u003e9.3.3 The genetic linkage example in R 292\u003c\/p\u003e \u003cp\u003e9.3.4 Other possible uses for data augmentation 293\u003c\/p\u003e \u003cp\u003e9.4 The Gibbs sampler 294\u003c\/p\u003e \u003cp\u003e9.4.1 Chained data augmentation 294\u003c\/p\u003e \u003cp\u003e9.4.2 An example with observed data 296\u003c\/p\u003e \u003cp\u003e9.4.3 More on the semi-conjugate prior with a normal likelihood 299\u003c\/p\u003e \u003cp\u003e9.4.4 The Gibbs sampler as an extension of chained data augmentation 301\u003c\/p\u003e \u003cp\u003e9.4.5 An application to change-point analysis 302\u003c\/p\u003e \u003cp\u003e9.4.6 Other uses of the Gibbs sampler 306\u003c\/p\u003e \u003cp\u003e9.4.7 More about convergence 309\u003c\/p\u003e \u003cp\u003e9.5 Rejection sampling 311\u003c\/p\u003e \u003cp\u003e9.5.1 Description 311\u003c\/p\u003e \u003cp\u003e9.5.2 Example 311\u003c\/p\u003e \u003cp\u003e9.5.3 Rejection sampling for log-concave distributions 311\u003c\/p\u003e \u003cp\u003e9.5.4 A practical example 313\u003c\/p\u003e \u003cp\u003e9.6 The Metropolis–Hastings algorithm 317\u003c\/p\u003e \u003cp\u003e9.6.1 Finding an invariant distribution 317\u003c\/p\u003e \u003cp\u003e9.6.2 The Metropolis–Hastings algorithm 318\u003c\/p\u003e \u003cp\u003e9.6.3 Choice of a candidate density 320\u003c\/p\u003e \u003cp\u003e9.6.4 Example 321\u003c\/p\u003e \u003cp\u003e9.6.5 More realistic examples 322\u003c\/p\u003e \u003cp\u003e9.6.6 Gibbs as a special case of Metropolis–Hastings 322\u003c\/p\u003e \u003cp\u003e9.6.7 Metropolis within Gibbs 323\u003c\/p\u003e \u003cp\u003e9.7 Introduction to WinBUGS and OpenBUGS 323\u003c\/p\u003e \u003cp\u003e9.7.1 Information about WinBUGS and OpenBUGS 323\u003c\/p\u003e \u003cp\u003e9.7.2 Distributions in WinBUGS and OpenBUGS 324\u003c\/p\u003e \u003cp\u003e9.7.3 A simple example using WinBUGS 324\u003c\/p\u003e \u003cp\u003e9.7.4 The pump failure example revisited 327\u003c\/p\u003e \u003cp\u003e9.7.5 DoodleBUGS 327\u003c\/p\u003e \u003cp\u003e9.7.6 coda 329\u003c\/p\u003e \u003cp\u003e9.7.7 R2WinBUGS and R2OpenBUGS 329\u003c\/p\u003e \u003cp\u003e9.8 Generalized linear models 332\u003c\/p\u003e \u003cp\u003e9.8.1 Logistic regression 332\u003c\/p\u003e \u003cp\u003e9.8.2 A general framework 334\u003c\/p\u003e \u003cp\u003e9.9 Exercises on Chapter 9 335\u003c\/p\u003e \u003cp\u003e\u003cb\u003e10 Some approximate methods 340\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003e10.1 Bayesian importance sampling 340\u003c\/p\u003e \u003cp\u003e10.1.1 Importance sampling to find HDRs 343\u003c\/p\u003e \u003cp\u003e10.1.2 Sampling importance re-sampling 344\u003c\/p\u003e \u003cp\u003e10.1.3 Multidimensional applications 344\u003c\/p\u003e \u003cp\u003e10.2 Variational Bayesian methods: simple case 345\u003c\/p\u003e \u003cp\u003e10.2.1 Independent parameters 347\u003c\/p\u003e \u003cp\u003e10.2.2 Application to the normal distribution 349\u003c\/p\u003e \u003cp\u003e10.2.3 Updating the mean 350\u003c\/p\u003e \u003cp\u003e10.2.4 Updating the variance 351\u003c\/p\u003e \u003cp\u003e10.2.5 Iteration 352\u003c\/p\u003e \u003cp\u003e10.2.6 Numerical example 352\u003c\/p\u003e \u003cp\u003e10.3 Variational Bayesian methods: general case 353\u003c\/p\u003e \u003cp\u003e10.3.1 A mixture of multivariate normals 353\u003c\/p\u003e \u003cp\u003e10.4 ABC: Approximate Bayesian Computation 356\u003c\/p\u003e \u003cp\u003e10.4.1 The ABC rejection algorithm 356\u003c\/p\u003e \u003cp\u003e10.4.2 The genetic linkage example 358\u003c\/p\u003e \u003cp\u003e10.4.3 The ABC Markov Chain Monte Carlo algorithm 360\u003c\/p\u003e \u003cp\u003e10.4.4 The ABC Sequential Monte Carlo algorithm 362\u003c\/p\u003e \u003cp\u003e10.4.5 The ABC local linear regression algorithm 365\u003c\/p\u003e \u003cp\u003e10.4.6 Other variants of ABC 366\u003c\/p\u003e \u003cp\u003e10.5 Reversible jump Markov chain Monte Carlo 367\u003c\/p\u003e \u003cp\u003e10.5.1 RJMCMC algorithm 367\u003c\/p\u003e \u003cp\u003e10.6 Exercises on Chapter 10 369\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix A Common statistical distributions 373\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eA.1 Normal distribution 374\u003c\/p\u003e \u003cp\u003eA.2 Chi-squared distribution 375\u003c\/p\u003e \u003cp\u003eA.3 Normal approximation to chi-squared 376\u003c\/p\u003e \u003cp\u003eA.4 Gamma distribution 376\u003c\/p\u003e \u003cp\u003eA.5 Inverse chi-squared distribution 377\u003c\/p\u003e \u003cp\u003eA.6 Inverse chi distribution 378\u003c\/p\u003e \u003cp\u003eA.7 Log chi-squared distribution 379\u003c\/p\u003e \u003cp\u003eA.8 Student’s t distribution 380\u003c\/p\u003e \u003cp\u003eA.9 Normal\/chi-squared distribution 381\u003c\/p\u003e \u003cp\u003eA.10 Beta distribution 382\u003c\/p\u003e \u003cp\u003eA.11 Binomial distribution 383\u003c\/p\u003e \u003cp\u003eA.12 Poisson distribution 384\u003c\/p\u003e \u003cp\u003eA.13 Negative binomial distribution 385\u003c\/p\u003e \u003cp\u003eA.14 Hypergeometric distribution 386\u003c\/p\u003e \u003cp\u003eA.15 Uniform distribution 387\u003c\/p\u003e \u003cp\u003eA.16 Pareto distribution 388\u003c\/p\u003e \u003cp\u003eA.17 Circular normal distribution 389\u003c\/p\u003e \u003cp\u003eA.18 Behrens’ distribution 391\u003c\/p\u003e \u003cp\u003eA.19 Snedecor’s F distribution 393\u003c\/p\u003e \u003cp\u003eA.20 Fisher’s z distribution 393\u003c\/p\u003e \u003cp\u003eA.21 Cauchy distribution 394\u003c\/p\u003e \u003cp\u003eA.22 The probability that one beta variable is greater than another 395\u003c\/p\u003e \u003cp\u003eA.23 Bivariate normal distribution 395\u003c\/p\u003e \u003cp\u003eA.24 Multivariate normal distribution 396\u003c\/p\u003e \u003cp\u003eA.25 Distribution of the correlation coefficient 397\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix B Tables 399\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eB.1 Percentage points of the Behrens–Fisher distribution 399\u003c\/p\u003e \u003cp\u003eB.2 Highest density regions for the chi-squared distribution 402\u003c\/p\u003e \u003cp\u003eB.3 HDRs for the inverse chi-squared distribution 404\u003c\/p\u003e \u003cp\u003eB.4 Chi-squared corresponding to HDRs for log chi-squared 406\u003c\/p\u003e \u003cp\u003eB.5 Values of F corresponding to HDRs for log F 408\u003c\/p\u003e \u003cp\u003eAppendix C R programs 430\u003c\/p\u003e \u003cp\u003e\u003cb\u003eAppendix D Further reading 436\u003c\/b\u003e\u003c\/p\u003e \u003cp\u003eD.1 Robustness 436\u003c\/p\u003e \u003cp\u003eD.2 Nonparametric methods 436\u003c\/p\u003e \u003cp\u003eD.3 Multivariate estimation 436\u003c\/p\u003e \u003cp\u003eD.4 Time series and forecasting 437\u003c\/p\u003e \u003cp\u003eD.5 Sequential methods 437\u003c\/p\u003e \u003cp\u003eD.6 Numerical methods 437\u003c\/p\u003e \u003cp\u003eD.7 Bayesian networks 437\u003c\/p\u003e \u003cp\u003eD.8 General reading 438\u003c\/p\u003e \u003cp\u003eReferences 439\u003c\/p\u003e \u003cp\u003e\u003cb\u003eIndex 455\u003c\/b\u003e\u003c\/p\u003e","brand":"John Wiley \u0026 Sons Inc","offers":[{"title":"Default Title","offer_id":49406851678551,"sku":"9781118332573","price":42.7,"currency_code":"GBP","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0817\/1739\/5799\/files\/9781118332573.jpg?v=1730497342","url":"https:\/\/bookcurl.com\/products\/bayesian-statistics-9781118332573","provider":"Book Curl","version":"1.0","type":"link"}