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.

Preface I Review of Probability and Distribution Theory1 Probability and Random Variables1.1 Introduction1.2 Univariate Discrete Distributions1.2.1 The Bernoulli and Binomial Distributions1.2.2 The Poisson Distribution1.2.3 Binomial Distribution: Normal Approximation1.3 Univariate Continuous Distributions1.3.1 The Uniform, Beta, Gamma, Normal, and Student-t Distributions1.4 Multivariate Probability Distributions1.4.1 The Multinomial Distribution1.4.2 The Dirichlet Distribution1.4.3 The d-Dimensional Uniform Distribution1.4.4 The Multivariate Normal Distribution1.4.5 The Chi-square Distribution1.4.6 The Wishart and Inverse Wishart Distributions1.4.7 The Multivariate-t Distribution1.5 Distributions with Constrained Sample Space1.6 Iterated Expectations 2 Functions of Random Variables2.1 Introduction2.2 Functions of a Single Random Variable2.2.1 Discrete Random Variables2.2.2 Continuous Random Variables2.2.3 Approximating the Mean and Variance2.2.4 Delta Method2.3 Functions of Several Random Variables2.3.1 Linear Transformations2.3.2 Approximating the Mean and Covariance Matrix II Methods of Inference3 An Introduction to Likelihood Inference3.1 Introduction3.2 The Likelihood Function3.3 The Maximum Likelihood Estimator3.4 Likelihood Inference in a Gaussian Model3.5 Fisher¿s Information Measure3.5.1 Single Parameter Case3.5.2 Alternative Representation of Information3.5.3 Mean and Variance of the Score Function3.5.4 Multiparameter Case3.5.5 Cramer¿Rao Lower Bound3.6 Sufficiency3.7 Asymptotic Properties: Single Parameter Models3.7.1 Probability of the Data Given the Parameter3.7.2 Consistency3.7.3 Asymptotic Normality and Effciency3.8 Asymptotic Properties: Multiparameter Models3.9 Functional Invariance3.9.1 Illustration of FunctionalInvariance3.9.2 Invariance in a Single Parameter Model3.9.3 Invariance in a Multiparameter Model 4 Further Topics in Likelihood Inference4.1 Introduction4.2 Computation of Maximum Likelihood Estimates4.3 Evaluation of Hypotheses4.3.1 Likelihood Ratio Tests4.3.2 Con.dence Regions4.3.3 Wald's Test4.3.4 Score Test4.4 Nuisance Parameters4.4.1 Loss of Efficiency Due to Nuisance Parameters4.4.2 Marginal Likelihoods4.4.3 Profile Likelihoods4.5 Analysis of a Multinomial Distribution4.5.1 Amount of Information per Observation4.6 Analysis of Linear Logistic Models4.6.1 The Logistic Distribution4.6.2 Likelihood Function under Bernoulli Sampling4.6.3 Mixed Effects Linear Logistic Model 5 An Introduction to Bayesian Inference5.1 Introduction5.2 Bayes Theorem: Discrete Case5.3 Bayes Theorem: Continuous Case5.4 Posterior Distributions5.5 Bayesian Updating5.6 Features of Posterior Distributions5.6.1 Posterior Probabilities5.6.2 Posterior Quantiles5.6.3 Posterior Modes5.6.4 Posterior Mean Vector and Covariance Matrix 6 Bayesian Analysis of Linear Models6.1 Introduction6.2 The Linear Regression Model6.2.1 Inference under Uniform Improper Priors6.2.2 Inference under Conjugate Priors6.2.3 Orthogonal Parameterization of the Model6.3 The Mixed Linear Model6.3.1 Bayesian View of the Mixed Effects Model6.3.2 Joint and Conditional Posterior Distributions6.3.3 Marginal Distribution of Variance Components6.3.4 Marginal Distribution of Location Parameters 7 The Prior Distribution and Bayesian Analysis7.1 Introduction7.2 An Illustration of the Effect of Priors on Inferences7.3 A Rapid Tour of Bayesian Asymptotics7.3.1 Discrete Parameter7.3.2 Continuous Parameter7.4 Statistical Information and Entropy7.4.1 Information7.4.2 Entropy of a Discrete