Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics

Preface I Review of Probability and Distribution Theory1 Probability and Random Variables1.1 Introduction1.2 Univariate Discrete Distributions 1.2.1 The Bernoulli and Binomial Distributions1.2.2 The Poisson Distribution 1.2.3 Binomial Distribution: Normal Approximation1.3 Univariate Continuous Distributions 1.3.1 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 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 Variance2.2.4 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 Inference3 An Introduction to Likelihood Inference3.1 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 Information3.5.3 Mean and Variance of the Score Function3.5.4 Multiparameter Case 3.5.5 Cramer-Rao Lower Bound 3.6 Sufficiency 3.7 Asymptotic Properties: Single Parameter Models3.7.1 Probability of the Data Given the Parameter3.7.2 Consistency 3.7.3 Asymptotic Normality and Effciency 3.8 Asymptotic Properties: Multiparameter Models3.9 Functional Invariance 3.9.1 Illustration of FunctionalInvariance 3.9.2 Invariance in a Single Parameter Model3.9.3 Invariance in a Multiparameter Model 4 Further Topics in Likelihood Inference4.1 Introduction 4.2 Computation of Maximum Likelihood Estimates4.3 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 Parameters4.4.2 Marginal Likelihoods 4.4.3 Profile Likelihoods 4.5 Analysis of a Multinomial Distribution 4.5.1 Amount of Information per Observation4.6 Analysis of Linear Logistic Models 4.6.1 The Logistic Distribution 4.6.2 Likelihood Function under Bernoulli Sampling4.6.3 Mixed Effects Linear Logistic Model 5 An Introduction to Bayesian Inference5.1 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 Models6.1 Introduction 6.2 The Linear Regression Model 6.2.1 Inference under Uniform Improper Priors6.2.2 Inference under Conjugate Priors 6.2.3 Orthogonal Parameterization of the Model6.3 The Mixed Linear Model 6.3.1 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 Introduction 7.2 An Illustration of the Effect of Priors on Inferences7.3 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