## An Introduction to Bayesian Analysis: Theory and MethodsThough there are many recent additions to graduate-level introductory books on Bayesian analysis, none has quite our blend of theory, methods, and ap plications. We believe a beginning graduate student taking a Bayesian course or just trying to find out what it means to be a Bayesian ought to have some familiarity with all three aspects. More specialization can come later. Each of us has taught a course like this at Indian Statistical Institute or Purdue. In fact, at least partly, the book grew out of those courses. We would also like to refer to the review (Ghosh and Samanta (2002b)) that first made us think of writing a book. The book contains somewhat more material than can be covered in a single semester. We have done this intentionally, so that an instructor has some choice as to what to cover as well as which of the three aspects to emphasize. Such a choice is essential for the instructor. The topics include several results or methods that have not appeared in a graduate text before. In fact, the book can be used also as a second course in Bayesian analysis if the instructor supplies more details. Chapter 1 provides a quick review of classical statistical inference. Some knowledge of this is assumed when we compare different paradigms. Following this, an introduction to Bayesian inference is given in Chapter 2 emphasizing the need for the Bayesian approach to statistics. |

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### Contents

LXXXII | 149 |

LXXXIV | 155 |

LXXXV | 156 |

LXXXVI | 159 |

LXXXVII | 161 |

LXXXVIII | 163 |

LXXXIX | 164 |

XC | 165 |

XI | 20 |

XII | 21 |

XIII | 23 |

XIV | 24 |

XV | 29 |

XVII | 30 |

XVIII | 35 |

XIX | 37 |

XX | 38 |

XXI | 40 |

XXII | 41 |

XXIV | 42 |

XXV | 48 |

XXVI | 49 |

XXVII | 50 |

XXVIII | 51 |

XXX | 53 |

XXXI | 54 |

XXXII | 55 |

XXXIV | 57 |

XXXVI | 58 |

XXXVII | 64 |

XXXIX | 67 |

XL | 68 |

XLI | 70 |

XLII | 71 |

XLIII | 72 |

XLIV | 74 |

XLV | 75 |

XLVII | 76 |

XLIX | 81 |

L | 83 |

LI | 84 |

LIII | 91 |

LIV | 92 |

LV | 93 |

LVI | 94 |

LVII | 99 |

LVIII | 100 |

LX | 101 |

LXI | 107 |

LXII | 109 |

LXIII | 113 |

LXV | 115 |

LXVI | 119 |

LXVII | 120 |

LXVIII | 122 |

LXIX | 123 |

LXX | 125 |

LXXI | 126 |

LXXII | 129 |

LXXIII | 132 |

LXXIV | 135 |

LXXV | 136 |

LXXVI | 138 |

LXXVII | 139 |

LXXVIII | 140 |

LXXIX | 145 |

LXXX | 146 |

LXXXI | 147 |

XCI | 168 |

XCII | 172 |

XCIII | 176 |

XCV | 178 |

XCVI | 179 |

XCVIII | 185 |

XCIX | 188 |

C | 190 |

CI | 191 |

CII | 194 |

CIII | 199 |

CIV | 205 |

CV | 207 |

CVI | 208 |

CVII | 211 |

CVIII | 215 |

CIX | 216 |

CX | 218 |

CXI | 220 |

CXII | 223 |

CXIII | 225 |

CXIV | 231 |

CXV | 233 |

CXVI | 238 |

CXVII | 241 |

CXVIII | 245 |

CXIX | 246 |

CXX | 251 |

CXXI | 252 |

CXXII | 255 |

CXXIII | 256 |

CXXIV | 259 |

CXXV | 260 |

CXXVI | 262 |

CXXVII | 263 |

CXXVIII | 264 |

CXXIX | 268 |

CXXX | 269 |

CXXXI | 271 |

CXXXII | 272 |

CXXXIII | 273 |

CXXXIV | 276 |

CXXXV | 284 |

CXXXVI | 285 |

CXXXVII | 289 |

CXXXVIII | 292 |

CXXXIX | 293 |

CXL | 296 |

CXLI | 299 |

CXLII | 302 |

CXLIII | 303 |

CXLV | 306 |

CXLVI | 307 |

CXLVII | 311 |

CXLVIII | 313 |

CXLIX | 315 |

316 | |

339 | |

344 | |

### Common terms and phrases

assume asymptotic Bayes estimate Bayes factor Bayesian analysis Bayesian approach Berger binomial Chapter class of priors classical statistics compute conditional conjugate prior Consider corresponding credible interval decision defined Delampady denote discussed elicitation error Example exponential family finite frequentist Ghosh given hence hyperparameters improper prior independent inference integral invariant Jeffreys prior Laplace approximation Let X1 likelihood function linear loss function lower bounds M-H algorithm Markov chain matrix maximize MCMC measure method minimax model selection multivariate noninformative priors normal Note null hypothesis objective priors observed obtained P-value parameter space posterior density posterior distribution posterior mean posterior mode posterior probability predictive P-value prior density prior distribution probability distribution random variables reference prior regression robustness sample Section specified sufficient statistic Suppose symmetric Theorem unbiased estimate uniform prior unimodal unknown values variance vector versus H1 vide wavelet Xn be i.i.d.