统计模拟入门：R语言实现的蒙特卡洛方法

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"《Introducing Monte Carlo Methods with R》是一本详细介绍统计模拟中主要工具的书籍，特别是从程序员的角度出发，讲解如何用R语言实现各种蒙特卡洛模拟技术，并提供输出结果以便理解和比较。书中涵盖了从基础的R编程到复杂的马尔科夫链蒙特卡洛（MCMC）算法的多个主题。" 该书由Christian P. Robert和George Casella合著，他们是统计学领域的专家，分别来自法国巴黎大学和美国佛罗里达大学。该书的顾问包括知名统计学家Robert Gentleman、Kurt Hornik和Giovanni Parmigiani，他们都是在R编程和统计模拟领域有着深厚造诣的学者。书中内容结构如下： 1. **基本R编程**：这一章为读者提供了R语言的基础知识，这是进行蒙特卡洛模拟工作的必备技能，包括数据类型、操作符、控制流以及函数的使用等。 2. **随机变量生成**：介绍了如何在R中生成各种概率分布的随机变量，如均匀分布、正态分布、二项分布等，这是蒙特卡洛方法的核心部分。 3. **蒙特卡洛积分**：通过大量的随机抽样来近似复杂函数的积分，这是蒙特卡洛方法的基本应用之一。 4. **控制和加速收敛**：讨论如何提高模拟效率，包括重要性采样、变分法等技术，以加速蒙特卡洛方法的收敛速度。 5. **蒙特卡洛优化**：利用随机搜索策略进行参数估计和最优化问题的解决。 6. **Metropolis-Hastings算法**：这是一种广泛使用的MCMC算法，用于在高维空间中探索概率分布。 7. **Gibbs采样器**：Gibbs采样是MCMC的一种特殊形式，可以处理多变量的概率模型。 8. **MCMC算法的收敛监控与适应**：讲述了如何监测和调整算法的运行，确保其正确性和效率，例如使用 Geweke 统计量和 autocorrelation 分析。该书的出版旨在帮助读者深入理解蒙特卡洛方法，并提供实际操作的指导，适合统计学、数据科学、计算生物学等领域对模拟技术感兴趣的读者。此外，书中包含的实际代码输出将有助于读者直观地理解理论概念，从而更好地将理论应用于实践。通过阅读本书，读者不仅可以掌握R语言的模拟技巧，还能了解如何利用蒙特卡洛方法解决实际问题。

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xviii List of Examples

4.10 Continuation of Example 4.9 . . ...............................118

4.11 Logistic regression ..........................................119

5.1 Maximizing a Cauchy likelihood ..............................127

5.2 Mixture model likelihood ....................................128

5.3 A ﬁrst Monte Carlo maximization ............................131

5.4 Continuation of Example 5.1 . . ...............................131

5.5 Continuation of Example 5.4 . . ...............................133

5.6 Minimization of a complex function . . . ........................135

5.7 Continuation of Example 5.6 . . ...............................138

5.8 Continuation of Example 5.3 . . ...............................142

5.9 Simulated annealing for a normal mixture . . ...................144

5.10 Continuation of Example 5.6 . . ...............................145

5.11 Bayesian analysis of a simple probit model . ....................147

5.12 Missing-data mixture model . . . ..............................150

5.13 Censored–data likelihood ....................................151

5.14 Continuation of Example 5.13 . . ..............................153

5.15 EM for a normal mixture ....................................154

5.16 Missing–data multinomial model . . . ..........................158

5.17 Random eﬀect logit model ...................................159

6.1 Metropolis–Hastings algorithm for beta variables . . . ............172

6.2 Cauchys from normals ......................................177

6.3 Metropolis–Hastings for regression . . . .........................179

6.4 Normals from uniforms . .....................................183

6.5 Metropolis–Hastings for mixtures .............................183

6.6 Probit regression . . .........................................186

6.7 Continuation of Example 6.5 . . ...............................187

6.8 Model selection . . . .........................................188

6.9 Acceptance rates: normals from double exponentials . . ..........192

6.10 Continuation of Example 6.4 . . ...............................195

7.1 Normal bivariate Gibbs . . ...................................201

7.2 Generating beta-binomial random variables . . ..................202

7.3 Fitting a one-way hierarchical model ..........................202

7.4 Normal multivariate Gibbs . . ................................206

7.5 Extension of Example 7.3 . . .................................207

7.6 Censored-data Gibbs . . . .....................................210

7.7 Grouped multinomial data . . .................................211

7.8 More grouped multinomial data . . . ...........................213

7.9 Gibbs for normal mixtures . ..................................215

7.10 A ﬁrst slice sampler ........................................218

7.11 Logistic regression with the slice sampler . . . ...................219

7.12 A Poisson hierarchy . .......................................222

7.13 Correlation in a bivariate normal . . ...........................224

2 1 Basic R Programming

1.1 Introduction

This chapter only attempts at introducing R to newcomers in a few pages

and, as such, it should not be considered as a proper introduction to R. Entire

volumes, such as the monumental RBookby Crawley (2007), the introduction

by Dalgaard (2002), and the focused R data manipulation by Spector (2009),

are dedicated to the practice of this language, and therefore additional eﬀorts

(besides reading this chapter) will be required from the reader to suﬃciently

master the language.

However, before discouraging anyone, let us comfort

you with the fact that:

a. The syntax of R is simple and logical enough to quickly allow for a basic

understanding of simple R programs, as should become obvious in a few

paragraphs.

b. The best, and in a sense the only, way to learn R is through trial-and-

error on simple and then more complex examples. Reading the book with

a computer available nearby is therefore the best way of implementing

this recommendation.

In particular, the embedded help commands help() and help.search() are

very good starting points to gather information about a speciﬁc function or

a general issue, even though more detailed manuals are available both locally

and on-line. Note that help.start() opens a Web browser linked to the local

manual pages.

One may ﬁrst wonder why we support using R as the programming in-

terface for this introduction to Monte Carlo methods, since there exist other

languages, most (all?) of them faster than R, like Matlab, and even free, like C

or Python. We obviously have no partisan or commercial involvement in this

language. Rather, besides the ease of presentation, our main reason for this

choice is that the language combines a suﬃciently high power (for an inter-

preted language) with a very clear syntax both for statistical computation and

graphics. R is a ﬂexible language that is object-oriented and thus allows the

manipulation of complex data structures in a condensed and eﬃcient manner.

Its graphical abilities are also remarkable, with possible interfacing with a

text processor such as L

X with the package Sweave. R oﬀers the additional

advantages of being a free and open-source system under the GNU General

Public Licence principle, with constant upgrades and improvements from the

statistics community,

as well as numerous (free) Web-based tutorials and

user’s manuals,

and running on all platforms, including both Apple’s Mac

If you decide to skip this chapter, be sure to at least print the handy R Refer-

ence Card available at http://cran.r-project.org/doc/contrib/Short-refcard.pdf that

summarizes, in four pages, the major commands of R.

There is even an R newsletter, R-News, which is available on cran.r-

project.org/doc/Rnews.

This means it is unlikely that you will need to acquire an R programming book

in order to get a proper understanding of the R language, even though it may

1.2 Getting started 3

and Microsoft Windows (and, obviously, under the Linux and Unix operating

systems). R provides a powerful interface that can integrate programs written

in other languages such as C, C++, Fortran, Perl, Python, and Java. Not only

can you keep programming in your usual language, but integrating programs

written by others in an alien language then becomes (mostly) seamless, as

seen in Chapter 8 with the package amcmc. At last, it is increasingly common

to see people who develop new methodology simultaneously producing an R

package in support of their approach and to back up introductory statistics

courses with illustrations in R, as shown by the expanding Use R! Springer

series in which this book is published.

One choice we have not addressed above is “why R and not BUGS?” BUGS

(which stands for Bayesian inference using Gibbs sampling) is a Bayesian anal-

ysis software developed since the early 1990’s, mostly by researchers from the

Medical Research Council (MRC) at Cambridge University. The most common

version is WinBugs, working under Windows, but there also exists an open-

source version called OpenBugs. So, to return to the initial question, we are

not addressing the possible links and advantages of BUGS simply because the

purpose is diﬀerent. Our goal is to give a deep but intuitive understanding of

simulation tools for statistical inference, not (directly) help in the construction

of Bayesian models and the Bayesian estimation of their parameters. While

access to Monte Carlo speciﬁcations is possible in BUGS, most computing op-

erations are handled by the software itself, with the possible outcome that the

user does not bother about this side of the problem and instead concentrates

on Bayesian modeling. Thus, while R can be easily linked with BUGS and sim-

ulation can be done via BUGS, we think that a lower-level language such as R

is more eﬀective in bringing you in touch with simulation imperatives. Note

that this perspective would not hold so tightly for a book on computational

statistics, as Albert (2009).

1.2 Getting started

The R language is straightforward to install: It can be downloaded (obviously

free) from one of the numerous CRAN (Comprehensive R Archive Network)

mirror Websites around the world.

(Note that it is resident in most current

Linux kernels.)

At this stage, we do not cover installation of the R package and thus assume

that (a) R is installed on the machine you want to work with and (b) that

you have managed to launch it (in most cases, you simply have to click on

prove useful at a later stage. See http://cran.r-project.org/manuals.html for man-

uals available on-line.

The main CRAN Website is http://cran.r-project.org/.

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