【下载】Likelihood, Bayesian,and MCMC Methods in Quantitative Genetics.2002

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Likelihood, Bayesian and MCMC Methods in Quantitative Genetics (Hardcover)
Daniel Sorensen (Author), Daniel Gianola (Author)

Editorial Reviews
Review
From the reviews:
BIOINFORMATICS
"I found the coverage of material to be excellent: well chosen and well written, and I didn’t spot a single typographical error…It can serve as a resource book for masters-level taught courses, but will be most useful for PhD students and other researchers who need to fill in the gaps in their knowledge, grasp the intuition behind statistical techniques, models, and algorithms, and find pointers to more extensive treatments. Overall, I find that the authors have succeeded admirably in their goals. I highly recommend this excellent book to any researcher seeking a graduate-level introduction to the modern statistical methods applied in quantitative genetics."
"Just one personal sentence as an Introduction: I like the book so much that I have decided to include several parts of it in my own lectures. … it may be understood more easily by students and researchers that lack a strong background in statistics and mathematics. … most examples are nicely explained. … Summing up, I am convinced that this excellent book should be a standard book for researchers and students with a background in genetics who are interested in Bayesian and MCMC methods." (Andreas Ziegler, Metrika, February, 2004)
"Both authors … have made significant contributions to development of statistical methods in quantitative genetics and in particular have been at the forefront of the adoption of MCMC methods for Bayesian analysis, which can be applied to an enormous range of problems … . their coverage of likelihood methods is both extensive and fair. … this is a valuable book, in that it presents so much background essential for the subsequent application and merits a much broader market that it is likely to get." (William G. Hill, Genetical Research, Vol. 81, 2003)
"The coverage of Bayesian theory is extensive, and includes a discussion of information and entropy, and of the notion ‘uninformative’ priors, as well as model assessment and model averaging. … I found the coverage of material to be excellent: well chosen and well written, and I didn’t spot a single typographical error. … the authors have succeeded admirably in their goals. I highly recommend this excellent book to any researcher seeking a graduate-level introduction to the modern statistical methods applied in quantitative genetics." (David Balding, Bioinformatics, July, 2003)
"The book is aimed at students and researchers in agriculture, biology and medicine. … Statisticians will appreciate the attempt to relate biological to statistical parameters. In conclusion the book shows that the authors have a lot of experience with applications of statistics to quantitative genetics. Much more details are given in this book than usual, so it can be considered and recommended for classroom use." (Prof. Dr. W. Urfer, Statistical Papers, Vol. 46 (4), 2005)
" [T]he book is worth owning for anyone interested in applying likelihood or Bayesian models, especially realistic models that may require MCMC for implementation." (Journal of the American Statistical Associaton)
Product Description

Over the last ten years the introduction of computer intensive statistical methods has opened new horizons concerning the probability models that can be fitted to genetic data, the scale of the problems that can be tackled and the nature of the questions that can be posed. In particular, the application of Bayesian and likelihood methods to statistical genetics has been facilitated enormously by these methods. Techniques generally referred to as Markov chain Monte Carlo (MCMC) have played a major role in this process, stimulating synergies among scientists in different fields, such as mathematicians, probabilists, statisticians, computer scientists and statistical geneticists. Specifically, the MCMC "revolution" has made a deep impact in quantitative genetics. This can be seen, for example, in the vast number of papers dealing with complex hierarchical models and models for detection of genes affecting quantitative or meristic traits in plants, animals and humans that have been published recently. This book, suitable for numerate biologists and for applied statisticians, provides the foundations of likelihood, Bayesian and MCMC methods in the context of genetic analysis of quantitative traits. Most students in biology and agriculture lack the formal background needed to learn these modern biometrical techniques. Although a number of excellent texts in these areas have become available in recent years, the basic ideas and tools are typically described in a technically demanding style, and have been written by and addressed to professional statisticians. For this reason, considerable more detail is offered than what may be warranted for a more mathematically apt audience. The book is divided into four parts. Part I gives a review of probability and distribution theory. Parts II and III present methods of inference and MCMC methods. Part IV discusses several models that can be applied in quantitative genetics, primarily from a Bayesian perspective. An effort has been made to relate biological to statistical parameters throughout, and examples are used profusely to motivate the developments. Daniel Sorensen is Research Leader in Biometrical Genetics, at the Department of Animal Breeding and Genetics in the Danish Institute of Agricultural Sciences. Daniel Gianola is Professor in the Animal Sciences, Biostatistics and Medical Informatics, and Dairy Science Departments of the University of Wisconsin-Madison. Gianola and Sorensen pioneered the introduction of Bayesian and MCMC methods in animal breeding. The authors have published and lectured extensively in applications of statistics to quantitative genetics.


Product Details

• Hardcover: 760 pages
• Publisher: Springer (August 12, 2002)
• Language: English
• ISBN-10: 0387954406
• ISBN-13: 978-0387954400

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关键词：Quantitative QUANTITATIV Likelihood GENETICS Bayesian Methods Bayesian Likelihood mcmc GENETICS

Contents

Preface v

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 TheMultinomial Distribution 37

1.4.2 The Dirichlet Distribution 40

1.4.3 The d-Dimensional UniformDistribution 40

1.4.4 TheMultivariate Normal Distribution 41

1.4.5 The Chi-square Distribution 53

1.4.6 The Wishart and Inverse Wishart Distributions 55

1.4.7 TheMultivariate-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 RandomVariable 78

2.2.1 Discrete RandomVariables 78

2.2.2 Continuous RandomVariables 79

2.2.3 Approximating theMean and Variance 89

2.2.4 DeltaMethod 93

2.3 Functions of Several RandomVariables 95

2.3.1 Linear Transformations 111

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 GaussianModel 125

3.5 Fisher’s InformationMeasure 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 Cram´er–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 ParameterModel 157

3.9.3 Invariance in aMultiparameterModel 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

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 aMultinomial Distribution 190

4.5.1 Amount of Information per Observation 199

4.6 Analysis of Linear Logistic Models 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 PosteriorModes 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 RegressionModel 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 TheMixed LinearModel 313

6.3.1 Bayesian View of theMixed EffectsModel 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 UniformPrior 356

7.5.2 Other Vague Priors 358

7.5.3 MaximumEntropy 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 NestedModels 414

8.2.6 Approximations to the Bayes Factor 418

8.2.7 Partial and Intrinsic Bayes Factors 422

8.3 Estimating theMarginal Likelihood 424

8.4 Goodness of Fit andModel 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 BayesianModel 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 EMAlgorithm 445

9.3.1 Formof the Algorithm 445

9.3.2 Derivation 445

9.4 Monotonic Increase of ln p (θ|y) 447

9.5 TheMissing Information Principle 448

9.5.1 Complete, Observed andMissing Information 448

9.5.2 Rate of Convergence of the EM Algorithm 449

9.6 EM Theory for Exponential Families 451

9.7 Standard Errors and Posterior Standard Deviations 452

9.7.1 TheMethod of Louis 453

9.7.2 Supplemented EM Algorithm (SEM) 454

9.7.3 TheMethod of Oakes 457

9.8 Examples 458

III Markov Chain Monte Carlo Methods 475

10 An Overview of Discrete Markov Chains 477

10.1 Introduction 477

10.2 Definitions 478

10.3 State of the System after n-Steps 479

10.4 Long-TermBehavior of theMarkov Chain 481

10.5 Stationary Distribution 481

10.6 Aperiodicity and Irreducibility 483

10.7 ReversibleMarkov Chains 487

10.8 Limiting Behavior 492

11 Markov Chain Monte Carlo 497

11.1 Introduction 497

11.2 Preliminaries 498

11.2.1 Notation 498

11.2.2 Transition Kernels 499

11.2.3 Varying Dimensionality 499

11.3 An Overview ofMarkov ChainMonte Carlo 500

11.4 The Metropolis–Hastings Algorithm 502

11.4.1 An Informal Derivation 502

11.4.2 AMore Formal Derivation 504

11.5 The Gibbs Sampler 509

11.5.1 Fully Conditional Posterior Distributions 510

11.5.2 The Gibbs Sampling Algorithm 510

11.6 Langevin–Hastings Algorithm 517

11.7 Reversible JumpMCMC 517

11.7.1 The Invariant Distribution518

11.7.2 Generating the Proposal 519

11.7.3 Specifying the Reversibility Condition 520

11.7.4 Derivation of the Acceptance Probability 522

11.7.5 Deterministic Proposals 523

11.7.6 Generating Proposals via the Identity Mapping 525

11.8 Data Augmentation 532

12 Implementation and Analysis of MCMC Samples 539

12.1 Introduction 539

12.2 A Single Long Chain or Several Short Chains? 540

12.3 Convergence Issues 541

12.3.1 Effect of Posterior Correlation on Convergence 541

12.3.2 Monitoring Convergence 547

12.4 Inferences fromtheMCMC Output 550

12.4.1 Estimators of Posterior Quantities 550

12.4.2 Monte Carlo Variance 553

12.5 Sensitivity Analysis 556

IV Applications in Quantitative Genetics 561

13 Gaussian and Thick-Tailed Linear Models 563

13.1 Introduction 563

13.2 The Univariate Linear Additive GeneticModel 564

13.2.1 A Gibbs Sampling Algorithm 566

13.3 Additive GeneticModel withMaternal Effects 570

13.3.1 Fully Conditional Posterior Distributions 575

13.4 TheMultivariate Linear Additive GeneticModel 576

13.4.1 Fully Conditional Posterior Distributions 580

13.5 A Blocked Gibbs Sampler for Gaussian Linear Models 584

13.6 LinearModels with Thick-Tailed Distributions 588

13.6.1 Motivation 588

13.6.2 A Student-tMixed EffectsModel 595

13.6.3 Model with Clustered RandomEffects 600

13.7 Parameterizations and the Gibbs Sampler 602

14 Threshold Models for Categorical Responses 605

14.1 Introduction 605

14.2 Analysis of a Single Polychotomous Trait 607

14.2.1 SamplingModel 607

14.2.2 Prior Distribution and Joint Posterior Density 608

14.2.3 Fully Conditional Posterior Distributions 611

14.2.4 The Gibbs Sampler 615

14.3 Analysis of a Categorical and a Gaussian Trait 615

14.3.1 SamplingModel 616

14.3.2 Prior Distribution and Joint Posterior Density 617

14.3.3 Fully Conditional Posterior Distributions 619

14.3.4 The Gibbs Sampler 625

14.3.5 Implementation with Binary Traits 626

15 Bayesian Analysis of Longitudinal Data 627

15.1 Introduction 627

15.2 Hierarchical or Multistage Models 628

15.2.1 First Stage 629

15.2.2 Second Stage 634

15.2.3 Third Stage 639

15.2.4 Joint Posterior Distribution 641

15.3 Two-Step Approximate Bayesian Analysis 642

15.3.1 Estimating Location Parameters 643

15.3.2 Estimating Dispersion Parameters 650

15.3.3 Special Case: Linear First Stage 652

15.4 Computation via Markov Chain Monte Carlo 653

15.4.1 Fully Conditional Posterior Distributions 655

15.5 Analysis with Thick-Tailed Distributions 664

15.5.1 First- and Second-Stage Models 665

15.5.2 Fully Conditional Posterior Distributions 666

16 Segregation and Quantitative Trait Loci Analysis 671

16.1 Introduction 671

16.2 Segregation Analysis Models 672

16.2.1 Notation and Model 672

16.2.2 Fully Conditional Posterior Distributions 675

16.2.3 Some Implementation Issues 677

16.3 QTLModels 679

16.3.1 Models with a Single QTL 680

16.3.2 Models with an Arbitrary Number of QTL 690

References 701

List of Citations 727

Subject Index 733

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