Bayesian statistics [[electronic resource] ] : an introduction / / Peter M. Lee

• Autore: Lee Peter M
• Pubblicazione: Chichester, West Sussex ; ; Hoboken, N.J., 2012
• Edizione: 4th ed.
• Descrizione fisica: xxiii, 462 p
• Disciplina: 519.5/42
• Soggetto topico: Bayesian statistical decision theory; Mathematical statistics
• Nota di bibliografia: Includes bibliographical references and index.
• Nota di contenuto: Intro -- Bayesian Statistics -- Contents -- Preface -- Preface to the First Edition -- 1 Preliminaries -- 1.1 Probability and Bayes' Theorem -- 1.1.1 Notation -- 1.1.2 Axioms for probability -- 1.1.3 'Unconditional' probability -- 1.1.4 Odds -- 1.1.5 Independence -- 1.1.6 Some simple consequences of the axioms -- Bayes' Theorem -- 1.2 Examples on Bayes' Theorem -- 1.2.1 The Biology of Twins -- 1.2.2 A political example -- 1.2.3 A warning -- 1.3 Random variables -- 1.3.1 Discrete random variables -- 1.3.2 The binomial distribution -- 1.3.3 Continuous random variables -- 1.3.4 The normal distribution -- 1.3.5 Mixed random variables -- 1.4 Several random variables -- 1.4.1 Two discrete random variables -- 1.4.2 Two continuous random variables -- 1.4.3 Bayes' Theorem for random variables -- 1.4.4 Example -- 1.4.5 One discrete variable and one continuous variable -- 1.4.6 Independent random variables -- 1.5 Means and variances -- 1.5.1 Expectations -- 1.5.2 The expectation of a sum and of a product -- 1.5.3 Variance, precision and standard deviation -- 1.5.4 Examples -- 1.5.5 Variance of a sum -- covariance and correlation -- 1.5.6 Approximations to the mean and variance of a function of a random variable -- 1.5.7 Conditional expectations and variances -- 1.5.8 Medians and modes -- 1.6 Exercises on Chapter 1 -- 2 Bayesian inference for the normal distribution -- 2.1 Nature of Bayesian inference -- 2.1.1 Preliminary remarks -- 2.1.2 Post is prior times likelihood -- 2.1.3 Likelihood can be multiplied by any constant -- 2.1.4 Sequential use of Bayes' Theorem -- 2.1.5 The predictive distribution -- 2.1.6 A warning -- 2.2 Normal prior and likelihood -- 2.2.1 Posterior from a normal prior and likelihood -- 2.2.2 Example -- 2.2.3 Predictive distribution -- 2.2.4 The nature of the assumptions made -- 2.3 Several normal observations with a normal prior. ; 2.3.1 Posterior distribution -- 2.3.2 Example -- 2.3.3 Predictive distribution -- 2.3.4 Robustness -- 2.4 Dominant likelihoods -- 2.4.1 Improper priors -- 2.4.2 Approximation of proper priors by improper priors -- 2.5 Locally uniform priors -- 2.5.1 Bayes' postulate -- 2.5.2 Data translated likelihoods -- 2.5.3 Transformation of unknown parameters -- 2.6 Highest density regions -- 2.6.1 Need for summaries of posterior information -- 2.6.2 Relation to classical statistics -- 2.7 Normal variance -- 2.7.1 A suitable prior for the normal variance -- 2.7.2 Reference prior for the normal variance -- 2.8 HDRs for the normal variance -- 2.8.1 What distribution should we be considering? -- 2.8.2 Example -- 2.9 The role of sufficiency -- 2.9.1 Definition of sufficiency -- 2.9.2 Neyman's factorization theorem -- 2.9.3 Sufficiency principle -- 2.9.4 Examples -- 2.9.5 Order statistics and minimal sufficient statistics -- 2.9.6 Examples on minimal sufficiency -- 2.10 Conjugate prior distributions -- 2.10.1 Definition and difficulties -- 2.10.2 Examples -- 2.10.3 Mixtures of conjugate densities -- 2.10.4 Is your prior really conjugate? -- 2.11 The exponential family -- 2.11.1 Definition -- 2.11.2 Examples -- 2.11.3 Conjugate densities -- 2.11.4 Two-parameter exponential family -- 2.12 Normal mean and variance both unknown -- 2.12.1 Formulation of the problem -- 2.12.2 Marginal distribution of the mean -- 2.12.3 Example of the posterior density for the mean -- 2.12.4 Marginal distribution of the variance -- 2.12.5 Example of the posterior density of the variance -- 2.12.6 Conditional density of the mean for given variance -- 2.13 Conjugate joint prior for the normal distribution -- 2.13.1 The form of the conjugate prior -- 2.13.2 Derivation of the posterior -- 2.13.3 Example -- 2.13.4 Concluding remarks -- 2.14 Exercises on Chapter 2. ; 3 Some other common distributions -- 3.1 The binomial distribution -- 3.1.1 Conjugate prior -- 3.1.2 Odds and log-odds -- 3.1.3 Highest density regions -- 3.1.4 Example -- 3.1.5 Predictive distribution -- 3.2 Reference prior for the binomial likelihood -- 3.2.1 Bayes' postulate -- 3.2.2 Haldane's prior -- 3.2.3 The arc-sine distribution -- 3.2.4 Conclusion -- 3.3 Jeffreys' rule -- 3.3.1 Fisher's information -- 3.3.2 The information from several observations -- 3.3.3 Jeffreys' prior -- 3.3.4 Examples -- 3.3.5 Warning -- 3.3.6 Several unknown parameters -- 3.3.7 Example -- 3.4 The Poisson distribution -- 3.4.1 Conjugate prior -- 3.4.2 Reference prior -- 3.4.3 Example -- 3.4.4 Predictive distribution -- 3.5 The uniform distribution -- 3.5.1 Preliminary definitions -- 3.5.2 Uniform distribution with a fixed lower endpoint -- 3.5.3 The general uniform distribution -- 3.5.4 Examples -- 3.6 Reference prior for the uniform distribution -- 3.6.1 Lower limit of the interval fixed -- 3.6.2 Example -- 3.6.3 Both limits unknown -- 3.7 The tramcar problem -- 3.7.1 The discrete uniform distribution -- 3.8 The first digit problem -- invariant priors -- 3.8.1 A prior in search of an explanation -- 3.8.2 The problem -- 3.8.3 A solution -- 3.8.4 Haar priors -- 3.9 The circular normal distribution -- 3.9.1 Distributions on the circle -- 3.9.2 Example -- 3.9.3 Construction of an HDR by numerical integration -- 3.9.4 Remarks -- 3.10 Approximations based on the likelihood -- 3.10.1 Maximum likelihood -- 3.10.2 Iterative methods -- 3.10.3 Approximation to the posterior density -- 3.10.4 Examples -- 3.10.5 Extension to more than one parameter -- 3.10.6 Example -- 3.11 Reference posterior distributions -- 3.11.1 The information provided by an experiment -- 3.11.2 Reference priors under asymptotic normality -- 3.11.3 Uniform distribution of unit length. ; 3.11.4 Normal mean and variance -- 3.11.5 Technical complications -- 3.12 Exercises on Chapter 3 -- 4 Hypothesis testing -- 4.1 Hypothesis testing -- 4.1.1 Introduction -- 4.1.2 Classical hypothesis testing -- 4.1.3 Difficulties with the classical approach -- 4.1.4 The Bayesian approach -- 4.1.5 Example -- 4.1.6 Comment -- 4.2 One-sided hypothesis tests -- 4.2.1 Definition -- 4.2.2 P-values -- 4.3 Lindley's method -- 4.3.1 A compromise with classical statistics -- 4.3.2 Example -- 4.3.3 Discussion -- 4.4 Point (or sharp) null hypotheses with prior information -- 4.4.1 When are point null hypotheses reasonable? -- 4.4.2 A case of nearly constant likelihood -- 4.4.3 The Bayesian method for point null hypotheses -- 4.4.4 Sufficient statistics -- 4.5 Point null hypotheses for the normal distribution -- 4.5.1 Calculation of the Bayes' factor -- 4.5.2 Numerical examples -- 4.5.3 Lindley's paradox -- 4.5.4 A bound which does not depend on the prior distribution -- 4.5.5 The case of an unknown variance -- 4.6 The Doogian philosophy -- 4.6.1 Description of the method -- 4.6.2 Numerical example -- 4.7 Exercises on Chapter 4 -- 5 Two-sample problems -- 5.1 Two-sample problems - both variances unknown -- 5.1.1 The problem of two normal samples -- 5.1.2 Paired comparisons -- 5.1.3 Example of a paired comparison problem -- 5.1.4 The case where both variances are known -- 5.1.5 Example -- 5.1.6 Non-trivial prior information -- 5.2 Variances unknown but equal -- 5.2.1 Solution using reference priors -- 5.2.2 Example -- 5.2.3 Non-trivial prior information -- 5.3 Variances unknown and unequal (Behrens-Fisher problem) -- 5.3.1 Formulation of the problem -- 5.3.2 Patil's approximation -- 5.3.3 Example -- 5.3.4 Substantial prior information -- 5.4 The Behrens-Fisher controversy -- 5.4.1 The Behrens-Fisher problem from a classical standpoint -- 5.4.2 Example. ; 5.4.3 The controversy -- 5.5 Inferences concerning a variance ratio -- 5.5.1 Statement of the problem -- 5.5.2 Derivation of the F distribution -- 5.5.3 Example -- 5.6 Comparison of two proportions -- the 2 × 2 table -- 5.6.1 Methods based on the log-odds ratio -- 5.6.2 Example -- 5.6.3 The inverse root-sine transformation -- 5.6.4 Other methods -- 5.7 Exercises on Chapter 5 -- 6 Correlation, regression and the analysis of variance -- 6.1 Theory of the correlation coefficient -- 6.1.1 Definitions -- 6.1.2 Approximate posterior distribution of the correlation coefficient -- 6.1.3 The hyperbolic tangent substitution -- 6.1.4 Reference prior -- 6.1.5 Incorporation of prior information -- 6.2 Examples on the use of the correlation coefficient -- 6.2.1 Use of the hyperbolic tangent transformation -- 6.2.2 Combination of several correlation coefficients -- 6.2.3 The squared correlation coefficient -- 6.3 Regression and the bivariate normal model -- 6.3.1 The model -- 6.3.2 Bivariate linear regression -- 6.3.3 Example -- 6.3.4 Case of known variance -- 6.3.5 The mean value at a given value of the explanatory variable -- 6.3.6 Prediction of observations at a given value of the explanatory variable -- 6.3.7 Continuation of the example -- 6.3.8 Multiple regression -- 6.3.9 Polynomial regression -- 6.4 Conjugate prior for the bivariate regression model -- 6.4.1 The problem of updating a regression line -- 6.4.2 Formulae for recursive construction of a regression line -- 6.4.3 Finding an appropriate prior -- 6.5 Comparison of several means - the one way model -- 6.5.1 Description of the one way layout -- 6.5.2 Integration over the nuisance parameters -- 6.5.3 Derivation of the F distribution -- 6.5.4 Relationship to the analysis of variance -- 6.5.5 Example -- 6.5.6 Relationship to a simple linear regression model -- 6.5.7 Investigation of contrasts. ; 6.6 The two way layout.
• Sommario/riassunto: Bayesian 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. This new fourth edition looks at recent techniques such as variational methods, Bayesian importance sampling, approximate Bayesian computation and Reversible Jump Markov Chain Monte Carlo (RJMCMC), providing a concise account of the way in which the Bayesian approach to statistics develops as well as how it contrasts with the conventional approach. The theory is built up step by step, and important notions such as sufficiency are brought out of a discussion of the salient features of specific examples. This edition: Includes expanded coverage of Gibbs sampling, including more numerical examples and treatments of OpenBUGS, R2WinBUGS and R2OpenBUGS. Presents significant new material on recent techniques such as Bayesian importance sampling, variational Bayes, Approximate Bayesian Computation (ABC) and Reversible Jump Markov Chain Monte Carlo (RJMCMC). Provides extensive examples throughout the book to complement the theory presented. Accompanied by a supporting website featuring new material and solutions. More and more students are realizing that they need to learn Bayesian statistics to meet their academic and professional goals. This book is best suited for use as a main text in courses on Bayesian statistics for third and fourth year undergraduates and postgraduate students.
• Titolo autorizzato: Bayesian Statistics
• ISBN: 1-280-77576-9; 9786613686152; 1-118-35975-5
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910820114203321
• Lo trovi qui: Univ. Federico II