Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics

ISBN-13: 9780387954400 / Angielski / Twarda / 2002 / 740 str.

Daniel Sorenson; Daniel Gianola

Opis

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 a Research Professor in Statistical 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.

Recenzje

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)

Spis treści

Preface I Review of Probability and Distribution Theory 1 Probability and Random Variables 1.1 Introduction 1.2 Univariate Discrete Distributions 1.2.1 The Bernoulli and Binomial Distributions 1.2.2 The Poisson Distribution 1.2.3 Binomial Distribution: Normal Approximation 1.3 Univariate Continuous Distributions 1.3.1 The Uniform, Beta, Gamma, Normal, and Student-t Distributions 1.4 Multivariate Probability Distributions 1.4.1 The Multinomial Distribution 1.4.2 The Dirichlet Distribution 1.4.3 The d-Dimensional Uniform Distribution 1.4.4 The Multivariate Normal Distribution 1.4.5 The Chi-square Distribution 1.4.6 The Wishart and Inverse Wishart Distributions 1.4.7 The Multivariate-t Distribution 1.5 Distributions with Constrained Sample Space 1.6 Iterated Expectations 2 Functions of Random Variables 2.1 Introduction 2.2 Functions of a Single Random Variable 2.2.1 Discrete Random Variables 2.2.2 Continuous Random Variables 2.2.3 Approximating the Mean and Variance 2.2.4 Delta Method 2.3 Functions of Several Random Variables 2.3.1 Linear Transformations 2.3.2 Approximating the Mean and Covariance Matrix II Methods of Inference 3 An Introduction to Likelihood Inference 3.1 Introduction 3.2 The Likelihood Function 3.3 The Maximum Likelihood Estimator 3.4 Likelihood Inference in a Gaussian Model 3.5 Fisher’s Information Measure 3.5.1 Single Parameter Case 3.5.2 Alternative Representation of Information 3.5.3 Mean and Variance of the Score Function 3.5.4 Multiparameter Case 3.5.5 Cramer–Rao Lower Bound 3.6 Sufficiency 3.7 Asymptotic Properties: Single Parameter Models 3.7.1 Probability of the Data Given the Parameter 3.7.2 Consistency 3.7.3 Asymptotic Normality and Effciency 3.8 Asymptotic Properties: Multiparameter Models 3.9 Functional Invariance 3.9.1 Illustration of FunctionalInvariance 3.9.2 Invariance in a Single Parameter Model 3.9.3 Invariance in a Multiparameter Model 4 Further Topics in Likelihood Inference 4.1 Introduction 4.2 Computation of Maximum Likelihood Estimates 4.3 Evaluation of Hypotheses 4.3.1 Likelihood Ratio Tests 4.3.2 Con.dence Regions 4.3.3 Wald's Test 4.3.4 Score Test 4.4 Nuisance Parameters 4.4.1 Loss of Efficiency Due to Nuisance Parameters 4.4.2 Marginal Likelihoods 4.4.3 Profile Likelihoods 4.5 Analysis of a Multinomial Distribution 4.5.1 Amount of Information per Observation 4.6 Analysis of Linear Logistic Models 4.6.1 The Logistic Distribution 4.6.2 Likelihood Function under Bernoulli Sampling 4.6.3 Mixed Effects Linear Logistic Model 5 An Introduction to Bayesian Inference 5.1 Introduction 5.2 Bayes Theorem: Discrete Case 5.3 Bayes Theorem: Continuous Case 5.4 Posterior Distributions 5.5 Bayesian Updating 5.6 Features of Posterior Distributions 5.6.1 Posterior Probabilities 5.6.2 Posterior Quantiles 5.6.3 Posterior Modes 5.6.4 Posterior Mean Vector and Covariance Matrix 6 Bayesian Analysis of Linear Models 6.1 Introduction 6.2 The Linear Regression Model 6.2.1 Inference under Uniform Improper Priors 6.2.2 Inference under Conjugate Priors 6.2.3 Orthogonal Parameterization of the Model 6.3 The Mixed Linear Model 6.3.1 Bayesian View of the Mixed Effects Model 6.3.2 Joint and Conditional Posterior Distributions 6.3.3 Marginal Distribution of Variance Components 6.3.4 Marginal Distribution of Location Parameters 7 The Prior Distribution and Bayesian Analysis 7.1 Introduction 7.2 An Illustration of the Effect of Priors on Inferences 7.3 A Rapid Tour of Bayesian Asymptotics 7.3.1 Discrete Parameter 7.3.2 Continuous Parameter 7.4 Statistical Information and Entropy 7.4.1 Information 7.4.2 Entropy of a Discrete