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Applied Statistical Inference

Likelihood and Bayes

verfasst von: Leonhard Held, Daniel Sabanés Bové

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book covers modern statistical inference based on likelihood with applications in medicine, epidemiology and biology. Two introductory chapters discuss the importance of statistical models in applied quantitative research and the central role of the likelihood function. The rest of the book is divided into three parts. The first describes likelihood-based inference from a frequentist viewpoint. Properties of the maximum likelihood estimate, the score function, the likelihood ratio and the Wald statistic are discussed in detail. In the second part, likelihood is combined with prior information to perform Bayesian inference. Topics include Bayesian updating, conjugate and reference priors, Bayesian point and interval estimates, Bayesian asymptotics and empirical Bayes methods. Modern numerical techniques for Bayesian inference are described in a separate chapter. Finally two more advanced topics, model choice and prediction, are discussed both from a frequentist and a Bayesian perspective.

A comprehensive appendix covers the necessary prerequisites in probability theory, matrix algebra, mathematical calculus, and numerical analysis.

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Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

This chapter introduces several examples which will be considered throughout this book. It also gives a brief discussion of different aspects of statistical inference and the role of statistical models.

Leonhard Held, Daniel Sabanés Bové

2. Likelihood

Abstract

Chapter 2 introduces the fundamental notion of the likelihood function and related quantities, such as the maximum likelihood estimate, the score function, and Fisher information. Computational algorithms are treated to compute the maximum likelihood estimate, such as optimization and the EM algorithm. The concept of sufficiency and the likelihood principle are finally discussed in some detail. Exercises are given at the end.

Leonhard Held, Daniel Sabanés Bové

3. Elements of Frequentist Inference

Abstract

This chapter discusses fundamental concepts of frequentist inference, such as unbiasedness and consistency, standard errors and confidence intervals, significance tests and P-values. There is also a section on the bootstrap method. Exercises are given at the end.

Leonhard Held, Daniel Sabanés Bové

4. Frequentist Properties of the Likelihood

Abstract

Frequentist properties of the maximum likelihood estimate of a scalar parameter are derived. The Wald, Score and Likelihood Ratio test statistics and the corresponding confidence intervals are introduced. Variance stabilizing transformations are also discussed. A case study comparing coverage and width of several confidence intervals for a proportion finishes this chapter, completed by a number of exercises at the end.

Leonhard Held, Daniel Sabanés Bové

5. Likelihood Inference in Multiparameter Models

Abstract

The concepts described in Chap. 4 are now extended to multiparameter models. The concept of profile likelihood is introduced as well as the generalized likelihood ratio statistic. The conditional likelihood, an alternative way to eliminate a nuisance parameter, is discussed. Exercises are given at the end.

Leonhard Held, Daniel Sabanés Bové

6. Bayesian Inference

Abstract

This chapter gives an introduction to Bayesian inference. Conjugate, improper and Jeffreys prior distributions are introduced as well as various Bayesian point and interval estimates. Bayesian inference in multiparameter models is discussed and some results from Bayesian asymptotics are described. Finally, empirical Bayes methods are described, completed by a number of exercises at the end.

Leonhard Held, Daniel Sabanés Bové

7. Model Selection

Abstract

This chapter describes methodology for Model selection both from a likelihood and a Bayesian perspective. In particular, AIC and BIC is discussed and its connection to cross-validation. Bayesian model selection based on the marginal likelihood is described, including Bayesian model averaging. Finally, DIC is introduced, completed by a number of exercises at the end.

Leonhard Held, Daniel Sabanés Bové

8. Numerical Methods for Bayesian Inference

Abstract

This chapter describes numerical methods for Bayesian inference in non-conjugate settings. Standard numerical techniques and the Laplace approximation provide ways to numerically compute posterior characteristics of interest. Monte Carlo methods, including Monte Carlo integration, rejection and importance sampling as well as Markov chain Monte Carlo are described. Finally, numerical computation of the marginal likelihood, necessary for Bayesian model selection, is discussed. Exercises are given at the end.

Leonhard Held, Daniel Sabanés Bové

9. Prediction

Abstract

Chapter 9 describes the statistical methodology to predict future data in the presence of unknown model parameters. Emphasis is given on probabilistic predictions, obtained with either a likelihood or Bayesian approach. Connections to the simpler plug-in prediction are also described. Finally, methods to assess the quality of probabilistic predictions, such as the Brier and the logarithmic score, are described. Exercises are given at the end.

Leonhard Held, Daniel Sabanés Bové

Backmatter

Titel: Applied Statistical Inference
verfasst von: Leonhard Held
Daniel Sabanés Bové
Copyright-Jahr: 2014
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-37887-4
Print ISBN: 978-3-642-37886-7
DOI: https://doi.org/10.1007/978-3-642-37887-4

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