Introducing Monte Carlo Methods with R

297 Pages·2010·9.13 MB·English

by  Christian Robert

Use R!
Advisors:
· ·
RobertGentleman KurtHornik GiovanniParmigiani
Forothertitlespublishedinthisseries,goto
http://www.springer.com/series/6991
·
Christian P. Robert George Casella
Introducing Monte Carlo
Methods with R
123
ChristianP.Robert GeorgeCasella
Universite´ParisDauphine DepartmentofStatistics
UMRCNRS7534 UniversityofFlorida
CEREMADE 103Griffin-FloydHall
placeduMare´chaldeLattre GainesvilleFL32611-8545
deTassigny USA
75775Pariscedex16 [email protected]fl.edu
France
[email protected]
SeriesEditors
RobertGentleman KurtHornik
DepartmentofBioinformatics DepartmentofStatistikandMathematik
andComputationalBiology Wirtshchaftsuniversita¨tWienAugasse2-6
GenentechSouthSanFrancisco A-1090Wien
CA94080 Austria
USA
GiovanniParmigiani
DepartmentofBiostatistics
andComputationalBiology
Dana-FarberCancerInstitute
44BinneyStreet
Boston,MA02115
USA
ISBN978-1-4419-1575-7 e-ISBN978-1-4419-1576-4
DOI10.1007/978-1-4419-1576-4
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2009941076
(cid:2)c SpringerScience+BusinessMedia,LLC2010
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v
To our parents, who taught us much in many ways.
vi
“What if you haven’t the data?”
“Then we shall proceed directly to the brandy and cigars.”
Lyndsay Faye
The Case of Colonel Warbuton’s Madness
Preface
“After that, it was down to attitude.”
Ian Rankin
Black & Blue
Thepurposeofthisbookistoprovideaself-containedentryintoMonteCarlo
computational techniques. First and foremost, it must not be confused with
a programming addendum to our earlier book Monte Carlo Statistical Meth-
ods whose second edition came out in 2004. The current book has a different
purpose, namely to make a general audience familiar with the programming
aspects of Monte Carlo methodology through practical implementation. Not
only have we introduced R at the core of this book, but the emphasis and
contentshavechangeddrasticallyfromMonte Carlo Statistical Methods,even
though the overall vision remains the same. Theoretical foundations are in-
tentionally avoided in the current book.
Indeed, the emphasis on practice is a major feature of this book in that
its primary audience consists of graduate students in statistics, biostatistics,
engineering, etc., who need to learn how to use simulation methods as a tool
to analyze their experiments and/or datasets. The book should appeal to
scientists in all fields, given the versatility of these Monte Carlo tools. It can
also be used for a more classical statistics audience when aimed at teaching a
quick entry into modern computational methods based on R, at the end of an
undergraduate program for example, even though this may prove challenging
for some students.
The choice of the programming language R, as opposed to faster alterna-
tives such as Matlab or C and more structured constructs such as BUGS, is
due to its pedagogical simplicity and its versatility. Readers can easily con-
duct experiments in their own programming language by translating the ex-
amplesprovidedinthisbook.(Obviously,thebookcanalsosupplementother
textbooks on Bayesian modeling at the graduate level, including our books
Bayesian Choice(Robert,2001)andMonte CarloStatistical Methods(Robert
viii Preface
and Casella, 2004).) This book can also be viewed as a companion to, rather
than a competitor of, Jim Albert’s Use R! book Bayesian Computation with
R(Albert,2009).Indeed,takenasapair,thesetwobookscanprovideafairly
thorough introduction to Monte Carlo methods and Bayesian modeling.
Westressthat,ataproductionlevel(thatis,whenusingadvancedMonte
Carlo techniques or analyzing large datasets), R cannot be recommended as
the default language, but the expertise gained from this book should make
the switch to another language seamless.
Contrary to usual practice, many exercises are interspersed within the
chapters rather than postponed until the end of each chapter. There are two
reasonsforthisstylisticchoice.First,theresultsordevelopmentscontainedin
those exercises are often relevant for upcoming points in the chapter. Second,
they signal to the student (or to any reader) that some pondering over the
previouspagesmaybeusefulbeforemovingtothefollowingtopicandsomay
act as self-checking gateways. Additional exercises are found at the end of
eachchapter,withabridgedsolutionsoftheodd-numberedexercisesprovided
on our Webpages as well as Springer’s.
Thanks
We are immensely grateful to colleagues and friends for their help with this
book,inparticulartothefollowingpeople:EdGeorgeforhiscommentsonthe
generalpurposeofthebookandsomeexercisesinparticular;JimHobertand
FernandoQuintanaforhelpfuldiscussionsontheMonteCarloEM;Alessandra
Iacobucci for signaling in due time a fatal blunder; Jean-Michel Marin for
lettingusrecyclethefirstchapterofBayesian Core(MarinandRobert,2007)
into our introduction to R and for numerous pieces of R advice, as well as
pedagogical suggestions; Antonietta Mira for pointing out mistakes during
a session of an MCMC conference in Warwick; Fran¸cois Perron for inviting
CPR to Montr´eal and thus providing him with a time window to complete
Chapter8(onlyshortenedbyanice-climbingafternooninQu´eb´ec!),andalso
Franc¸ois Perron and Cl´ementine Trimont for testing the whole book from
the perspectives of a professor and a student, respectively; Martyn Plummer
for answering queries about coda; Jeff Rosenthal for very helpful exchanges
on amcmc; Dimitris Rizopoulos for providing Exercise 7.21; and Phil Spector
from Berkeley for making available his detailed and helpful notes and slides
on R and C, now partly published as Spector (2009). Comments from both
reviewers were especially useful in finalizing the book. We are also grateful to
John Kimmel of Springer for his advice and efficiency, as well as for creating
thevisionaryUseR!seriesandsupportingthedevelopmentoftheR language
that way. From a distance, we also wish to thank Professors Gentleman and
Ihaka for creating the R language and for doing it in open-source, as well as
the entire R community for contributing endlessly to its development.
Sceaux and Gainesville Christian P. Robert and George Casella
October 18, 2009
Contents
Preface ........................................................ vii
List of Figures................................................. xiii
List of Examples...............................................xvii
1 Basic R Programming ..................................... 1
1.1 Introduction ............................................ 2
1.2 Getting started.......................................... 3
1.3 R objects ............................................... 5
1.3.1 The vector class ................................... 6
1.3.2 The matrix, array, and factor classes.................. 9
1.3.3 The list and data.frame classes ...................... 12
1.4 Probability distributions in R ............................. 14
1.5 Basic and not-so-basic statistics ........................... 14
1.6 Graphical facilities....................................... 26
1.7 Writing new R functions.................................. 31
1.8 Input and output in R.................................... 35
1.9 Administration of R objects............................... 36
1.10 The mcsm package....................................... 36
1.11 Additional exercises...................................... 37
2 Random Variable Generation .............................. 41
2.1 Introduction ............................................ 42
2.1.1 Uniform simulation ................................ 42
2.1.2 The inverse transform.............................. 44
2.2 General transformation methods........................... 46
2.2.1 A normal generator................................ 47
2.2.2 Discrete distributions .............................. 48
2.2.3 Mixture representations ............................ 50
2.3 Accept–reject methods ................................... 51
x Contents
2.4 Additional exercises...................................... 57
3 Monte Carlo Integration................................... 61
3.1 Introduction ............................................ 62
3.2 Classical Monte Carlo integration.......................... 65
3.3 Importance sampling..................................... 69
3.3.1 An arbitrary change of reference measure............. 69
3.3.2 Sampling importance resampling .................... 75
3.3.3 Selection of the importance function ................. 78
3.4 Additional exercises...................................... 86
4 Controlling and Accelerating Convergence................. 89
4.1 Introduction ............................................ 90
4.2 Monitoring variation..................................... 91
4.3 Asymptotic variance of importance sampling estimators ...... 92
4.4 Effective sample size and perplexity........................ 98
4.5 Simultaneous monitoring .................................100
4.6 Rao–Blackwellization and deconditioning ...................107
4.7 Acceleration methods ....................................111
4.7.1 Correlated simulations .............................111
4.7.2 Antithetic variables................................113
4.7.3 Control variates...................................116
4.8 Additional exercises......................................122
5 Monte Carlo Optimization.................................125
5.1 Introduction ............................................126
5.2 Numerical optimization methods ..........................127
5.3 Stochastic search ........................................130
5.3.1 A basic solution...................................130
5.3.2 Stochastic gradient methods ........................136
5.3.3 Simulated annealing ...............................140
5.4 Stochastic approximation.................................146
5.4.1 Optimizing Monte Carlo approximations .............146
5.4.2 Missing-data models and demarginalization...........150
5.4.3 The EM algorithm ................................152
5.4.4 Monte Carlo EM ..................................157
5.5 Additional exercises......................................163
6 Metropolis–Hastings Algorithms...........................167
6.1 Introduction ............................................168
6.2 A peek at Markov chain theory............................168
6.3 Basic Metropolis–Hastingsalgorithms ......................170
6.3.1 A generic Markov chain Monte Carlo algorithm .......171
6.3.2 The independent Metropolis–Hastings algorithm ......175
6.4 A selection of candidates .................................182