Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics

Name: Likelihood, Bayesian, and MCMC Methods in Quantitative Genetics
Availability: InStock
ISBN: 9780387954400

ISBN-10: 0387954406

ISBN-13: 9780387954400

Edition: 2002

Authors: Daniel Sorensen, Daniel Gianola

List price: $379.99

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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,…

Book details

List price: $379.99
Copyright year: 2002
Publisher: Springer New York
Publication date: 8/12/2002
Binding: Hardcover
Pages: 740
Size: 6.14" wide x 9.21" long x 1.50" tall
Weight: 6.006
Language: English



Preface



Review of Probability and Distribution Theory



Probability and Random Variables



Introduction



Univariate Discrete Distributions



The Bernoulli and Binomial Distributions



The Poisson Distribution



Binomial Distribution: Normal Approximation



Univariate Continuous Distributions



The Uniform, Beta, Gamma, Normal, and Student-t Distributions



Multivariate Probability Distributions



The Multinomial Distribution



The Dirichlet Distribution



The d-Dimensional Uniform Distribution



The Multivariate Normal Distribution



The Chi-square Distribution



The Wishart and Inverse Wishart Distributions



The Multivariate-t Distribution



Distributions with Constrained Sample Space



Iterated Expectations



Functions of Random Variables



Introduction



Functions of a Single Random Variable



Discrete Random Variables



Continuous Random Variables



Approximating the Mean and Variance



Delta Method



Functions of Several Random Variables



Linear Transformations



Approximating the Mean and Covariance Matrix



Methods of Inference



An Introduction to Likelihood Inference



Introduction



The Likelihood Function



The Maximum Likelihood Estimator



Likelihood Inference in a Gaussian Model



Fisher's Information Measure



Single Parameter Case



Alternative Representation of Information



Mean and Variance of the Score Function



Multiparameter Case



Cramï¿½r-Rao Lower Bound



Sufficiency



Asymptotic Properties: Single Parameter Models



Probability of the Data Given the Parameter



Consistency



Asymptotic Normality and Efficiency



Asymptotic Properties: Multiparameter Models



Functional Invariance



Illustration of Functional Invariance



Invariance in a Single Parameter Model



Invariance in a Multiparameter Model



Further Topics in Likelihood Inference



Introduction



Computation of Maximum Likelihood Estimates



Evaluation of Hypotheses



Likelihood Ratio Tests



Confidence Regions



Wald's Test



Score Test



Nuisance Parameters



Loss of Efficiency Due to Nuisance Parameters



Marginal Likelihoods



Profile Likelihoods



Analysis of a Multinomial Distribution



Amount of Information per Observation



Analysis of Linear Logistic Models



The Logistic Distribution



Likelihood Function under Bernoulli Sampling



Mixed Effects Linear Logistic Model



An Introduction to Bayesian Inference



Introduction



Bayes Theorem: Discrete Case



Bayes Theorem: Continuous Case



Posterior Distributions



Bayesian Updating



Features of Posterior Distributions



Posterior Probabilities



Posterior Quantiles



Posterior Modes



Posterior Mean Vector and Covariance Matrix



Bayesian Analysis of Linear Models



Introduction



The Linear Regression Model



Inference under Uniform Improper Priors



Inference under Conjugate Priors



Orthogonal Parameterization of the Model



The Mixed Linear Model



Bayesian View of the Mixed Effects Model



Joint and Conditional Posterior Distributions



Marginal Distribution of Variance Components



Marginal Distribution of Location Parameters



The Prior Distribution and Bayesian Analysis



Introduction



An Illustration of the Effect of Priors on Inferences



A Rapid Tour of Bayesian Asymptotics



Discrete Parameter



Continuous Parameter



Statistical Information and Entropy



Information



Entropy of a Discrete Distribution



Entropy of a Joint and Conditional Distribution



Entropy of a Continuous Distribution



Information about a Parameter



Fisher's Information Revisited



Prior and Posterior Discrepancy



Priors Conveying Little Information



The Uniform Prior



Other Vague Priors



Maximum Entropy Prior Distributions



Reference Prior Distributions



Bayesian Assessment of Hypotheses and Models



Introduction



Bayes Factors



Definition



Interpretation



The Bayes Factor and Hypothesis Testing



Influence of the Prior Distribution



Nested Models



Approximations to the Bayes Factor



Partial and Intrinsic Bayes Factors



Estimating the Marginal Likelihood



Goodness of Fit and Model Complexity



Goodness of Fit and Predictive Ability of a Model



Analysis of Residuals



Predictive Ability and Predictive Cross-Validation



Bayesian Model Averaging



General



Definitions



Predictive Ability of BMA



Approximate Inference Via the EM Algorithm



Introduction



Complete and Incomplete Data



The EM Algorithm



Form of the Algorithm



Derivation



Monotonic Increase of ln p (ï¿½y)



The Missing Information Principle



Complete, Observed and Missing Information



Rate of Convergence of the EM Algorithm



EM Theory for Exponential Families



Standard Errors and Posterior Standard Deviations



The Method of Louis



Supplemented EM Algorithm (SEM)



The Method of Oakes



Examples



Markov Chain Monte Carlo Methods



An Overview of Discrete Markov Chains



Introduction



Definitions



State of the System after n-Steps



Long-Term Behavior of the Markov Chain



Stationary Distribution



Aperiodicity and Irreducibility



Reversible Markov Chains



Limiting Behavior



Markov Chain Monte Carlo



Introduction



Preliminaries



Notation



Transition Kernels



Varying Dimensionality



An Overview of Markov Chain Monte Carlo



The Metropolis-Hastings Algorithm



An Informal Derivation



A More Formal Derivation



The Gibbs Sampler



Fully Conditional Posterior Distributions



The Gibbs Sampling Algorithm



Langevin-Hastings Algorithm



Reversible Jump MCMC



The Invariant Distribution



Generating the Proposal



Specifying the Reversibility Condition



Derivation of the Acceptance Probability



Deterministic Proposals



Generating Proposals via the Identity Mapping



Data Augmentation



Implementation and Analysis of MCMC Samples



Introduction



A Single Long Chain or Several Short Chains?



Convergence Issues



Effect of Posterior Correlation on Convergence



Monitoring Convergence



Inferences from the MCMC Output



Estimators of Posterior Quantities



Monte Carlo Variance



Sensitivity Analysis



Applications in Quantitative Genetics



Gaussian and Thick-Tailed Linear Models



Introduction



The Univariate Linear Additive Genetic Model



A Gibbs Sampling Algorithm



Additive Genetic Model with Maternal Effects



Fully Conditional Posterior Distributions



The Multivariate Linear Additive Genetic Model



Fully Conditional Posterior Distributions



A Blocked Gibbs Sampler for Gaussian Linear Models



Linear Models with Thick-Tailed Distributions



Motivation



A Student-t Mixed Effects Model



Model with Clustered Random Effects



Parameterizations and the Gibbs Sampler



Threshold Models for Categorical Responses



Introduction



Analysis of a Single Polychotomous Trait



Sampling Model



Prior Distribution and Joint Posterior Density



Fully Conditional Posterior Distributions



The Gibbs Sampler



Analysis of a Categorical and a Gaussian Trait



Sampling Model



Prior Distribution and Joint Posterior Density



Fully Conditional Posterior Distributions



The Gibbs Sampler



Implementation with Binary Traits



Bayesian Analysis of Longitudinal Data



Introduction



Hierarchical or Multistage Models



First Stage



Second Stage



Third Stage



Joint Posterior Distribution



Two-Step Approximate Bayesian Analysis



Estimating Location Parameters



Estimating Dispersion Parameters



Special Case: Linear First Stage



Computation via Markov Chain Monte Carlo



Fully Conditional Posterior Distributions



Analysis with Thick-Tailed Distributions



First- and Second-Stage Models



Fully Conditional Posterior Distributions



Segregation and Quantitative Trait Loci Analysis



Introduction



Segregation Analysis Models



Notation and Model



Fully Conditional Posterior Distributions



Some Implementation Issues



QTL Models



Models with a Single QTL



Models with an Arbitrary Number of QTL


References


List of Citations


Subject Index