Likelihood, Bayesian and MCMC methods in quantitative genetics

Başlık:

Yazar:

Sorensen, Daniel

ISBN:

9780387954400

Ek Yazar:

Sorensen, Daniel

Yayın Bilgisi:

New York : Springer Verlag , 2002.

Fiziksel Tanım:

740 s. ; 24 sm.

Özet:

Contents Preface I Review of Probability and Distribution Theory 1 1 Probability and Random Variables 3 1.1 Introduction ........................... 3 1.2 Univariate Discrete Distributions ............... 4 1.2.1 The Bernoulli and Binomial Distributions ...... 7 1.2.2 The Poisson Distribution ............... 10 1.2.3 Binomial Distribution: Normal Approximation ... 12 1.3 Univariate Continuous Distributions ............. 13 1.3.1 The Uniform, Beta, Gamma, Normal, and Student-t Distributions .............. 18 1.4 Multivariate Probability Distributions ............ 29 1.4.1 The Multinomial Distribution ............. 37 1.4.2 The Dirichlet Distribution ............... 40 1.4.3 The d-Dimensional Uniform Distribution ...... 40 1.4.4 The Multivariate Normal Distribution ........ 41 1.4.5 The Chi-square Distribution .............. 53 1.4.6 The Wishart and Inverse Wishart Distributions ... 55 1.4.7 The Multivariate-t Distribution ............ 60 1.5 Distributions with Constrained Sample Space ........ 62 1.6 Iterated Expectations ..................... 67 2 Functions of Random Variables 77 2.1 Introduction ........................... 77 2.2 Functions of a Single Random Variable ............ 78 2.2.1 Discrete Random Variables .............. 78 2.2.2 Continuous Random Variables ............ 79 2.2.3 Approximating the Mean and Variance ....... 89 2.2.4 Delta Method ...................... 93 2.3 Functions of Several Random Variables ............ 95 2.3.1 Linear Transformations ................ Ill 2.3.2 Approximating the Mean and Covariance Matrix . . 114 II Methods of Inference 117 3 An Introduction to Likelihood Inference 119 3.1 Introduction ........................... 119 3.2 The Likelihood Function .................... 120 3.3 The Maximum Likelihood Estimator ............. 122 3.4 Likelihood Inference in a Gaussian Model .......... 125 3.5 Fisher’s Information Measure ................. 128 3.5.1 Single Parameter Case ................. 128 3.5.2 Alternative Representation of Information ...... 131 3.5.3 Mean and Variance of the Score Function ...... 134 3.5.4 Multiparameter Case .................. 135 3.5.5 Cramer-Rao Lower Bound .............. 138 3.6 Sufficiency ............................ 142 3.7 Asymptotic Properties: Single Parameter Models ...... 143 3.7.1 Probability of the Data Given the Parameter .... 144 3.7.2 Consistency ....................... 146 3.7.3 Asymptotic Normality and Efficiency ......... 147 3.8 Asymptotic Properties: Multiparameter Models ....... 152 3.9 Functional Invariance ..................... 153 3.9.1 Illustration of Functional Invariance ......... 153 3.9.2 Invariance in a Single Parameter Model ....... 157 3.9.3 Invariance in a Multiparameter Model ........ 159 4 Further Topics in Likelihood Inference 161 4.1 Introduction ........................... 161 4.2 Computation of Maximum Likelihood Estimates ...... 162 4.3 Evaluation of Hypotheses ................... 166 4.3.1 Likelihood Ratio Tests ................. 166 4.3.2 Confidence Regions ................... 177 4.3.3 Wald’s Test ....................... 179 4.3.4 Score Test ........................ 179 4.4 Nuisance Parameters ...................... 181 Contents xiii 4.4.1 Loss of Efficiency Due to Nuisance Parameters . . . 182 4.4.2 Marginal Likelihoods .................. 182 4.4.3 Profile Likelihoods ................... 186 4.5 Analysis of a Multinomial Distribution ............ 190 4.5.1 Amount of Information per Observation ....... 199 4.6 Analysis of Linear Logistic Model .............. 202 4.6.1 The Logistic Distribution ............... 204 4.6.2 Likelihood Function under Bernoulli Sampling . . . 204 4.6.3 Mixed Effects Linear Logistic Model ......... 208 5 An Introduction to Bayesian Inference 211 5.1 Introduction ........................... 211 5.2 Bayes Theorem: Discrete Case ................. 214 5.3 Bayes Theorem: Continuous Case ............... 223 5.4 Posterior Distributions ..................... 235 5.5 Bayesian Updating ....................... 249 5.6 Features of Posterior Distributions .............. 257 5.6.1 Posterior Probabilities ................. 258 5.6.2 Posterior Quantiles ................... 262 5.6.3 Posterior Modes .................... 264 5.6.4 Posterior Mean Vector and Covariance Matrix . . . 280 6 Bayesian Analysis of Linear Models 287 6.1 Introduction........................... 287 6.2 The Linear Regression Model ................. 287 6.2.1 Inference under Uniform Improper Priors ...... 288 6.2.2 Inference under Conjugate Priors ........... 297 6.2.3 Orthogonal Parameterization of the Model ..... 307 6.3 The Mixed Linear Model.................... 313 6.3.1 Bayesian View of the Mixed Effects Model ...... 313 6.3.2 Joint and Conditional Posterior Distributions .... 317 6.3.3 Marginal Distribution of Variance Components . . . 322 6.3.4 Marginal Distribution of Location Parameters .... 323 7 The Prior Distribution and Bayesian Analysis 327 7.1 Introduction ........................... 327 7.2 An Illustration of the Effect of Priors on Inferences ..... 328 7.3 A Rapid Tour of Bayesian Asymptotics ........... 330 7.3.1 Discrete Parameter ................... 330 7.3.2 Continuous Parameter ................. 331 7.4 Statistical Information and Entropy ............. 334 7.4.1 Information ....................... 334 7.4.2 Entropy of a Discrete Distribution .......... 337 7.4.3 Entropy of a Joint and Conditional Distribution . . 340 7.4.4 Entropy of a Continuous Distribution ........ 341 7.4.5 Information about a Parameter ............ 346 7.4.6 Fisher’s Information Revisited ............ 351 7.4.7 Prior and Posterior Discrepancy ........... 353 7.5 Priors Conveying Little Information ............. 356 7.5.1 The Uniform Prior ................... 356 7.5.2 Other Vague Priors ................... 358 7.5.3 Maximum Entropy Prior Distributions ........ 367 7.5.4 Reference Prior Distributions ............. 379 8 Bayesian Assessment of Hypotheses and Models 399 8.1 Introduction ........................... 399 8.2 Bayes Factors .......................... 400 8.2.1 Definition ........................ 400 8.2.2 Interpretation ...................... 402 8.2.3 The Bayes Factor and Hypothesis Testing ...... 403 8.2.4 Influence of the Prior Distribution .......... 412 8.2.5 Nested Models ..................... 414 8.2.6 Approximations to the Bayes Factor ......... 418 8.2.7 Partial and Intrinsic Bayes Factors .......... 422 8.3 Estimating the Marginal Likelihood ............. 424 8.4 Goodness of Fit and Model Complexity ........... 429 8.5 Goodness of Fit and Predictive Ability of a Model ..... 433 8.5.1 Analysis of Residuals .................. 434 8.5.2 Predictive Ability and Predictive Cross-Validation . 436 8.6 Bayesian Model Averaging ................... 439 8.6.1 General ......................... 439 8.6.2 Definitions ....................... 440 8.6.3 Predictive Ability of BMA ............... 441 9 Approximate Inference Via the EM Algorithm 443 9.1 Introduction ...........................

443 9.2 Complete and Incomplete Data ................ 444 9.3 The EM Algorithm ....................... 445 9.3.1 Form of the Algorithm ................. 445 9.3.2 Derivation ........................ 445 9.4 Monotonie Increase of lnp(0

Konu Terimi:

Genetics-Statistical methods

Nonte Carlo method

Markov processes

Bayesian statistical decision theory

Genetik - İstatistiksel metod

Mevcut:*

Library	Materyal Türü	Barkod	Yer Numarası	Durumu/İade Tarihi
Arıyor... Ferit Melen Library	Book	049555	575.60727 SORl 2002 k.1	Arıyor... Unknown

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