All of Statistics， Larry Wasserman (大牛作品).pdf

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All of Statistics - A Concise Course in Statistical Inference_Larry Wasserman (Springer 2004 errata 465s)

Preface

Statistics/Data Mining Dictionary

Contents

Part I Probability

1 Probability

2 Random Variables

3 Expectation

4 Inequalities

5 Convergence of Random Variables

Part II Statistical Inference

6 Models, Statistical Inference and Learning

7 Estimating the cdf and Statistical Functionals

8 The Bootstrap

9 Parametric Inference

10 Hypothesis Testing and p-values

11 Bayesian Inference

12 Statistical Decision Theory

Part III Statistical Models and Methods

13 Linear and Logistic Regression

14 Multivariate Models

15 Inference About Independence

16 Causal Inference

17 Directed Graphs and Conditional Independence

18 Undirected Graphs

19 Log-Linear Models

20 Nonparametric Curve Estimation

21 Smoothing Using Orthogonal Functions

22 Classification

23 Probability Redux: Stochastic Processes

24 Simulation Methods

Bibliography

List of Symbols

Index

ERRATA for All of Statistics

ERRATA for All of Statistics Second Printing

Index of Larry Wasserman all-of-statistics R programs

cmu.edu

Index of /~larry/all-of-statistics/=Rprograms

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/Example15.5.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/Fig9.2.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/Fig24.1.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/ICU.data

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/Utility.functions.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/a1882_25.dat

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/band.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/bernoulli.likelihood.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/binomial.risk.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/binomial.risk.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/bloodfat.pvalue.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/bloodfat.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/boom_050801.dat

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/classify.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/cmb.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/column-names.txt

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/correlation.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/cosine.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/crime.dat

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/cv.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/dasi.dat

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/death.penalty.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/demonstrate.kernel.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/density.ci.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/density.examples.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/density.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/dose.response.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/edf.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/election.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/equilibrium.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/fdr.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/florida-election-data.txt

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/gibbs.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/glass.data

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/grades.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/graphics.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/haar.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/histogram.confidence.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/importance.sampling.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/importance.tail.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/inhomogeneous.function.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/jane.austen.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/kern.regress.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/kyphosis.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/linear.regress.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/logistic.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/loglin.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/loglin.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/loglin.sim.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/lsat.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/max_050801.dat

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/mixtures.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/multiple.regression.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/normal.density.data

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/patch.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/patch.data

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/poisson.process.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/r.by.c.table.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/react.density.marronwand.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/react.density.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/react.glass.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/react.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/row-names.txt

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/sa.data

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/spam.data

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/spam.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/stockprice.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/symmlet.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/tanh.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/tanh.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/temp.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/ten.S

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/tree.r

http://www.stat.cmu.edu/~larry/all-of-statistics/=Rprograms/tree.small.r

To Isa

Preface Taken literally, the title “All of Statistics” is an exaggeration. But in spirit, the title is apt, as the book does cover a much broader range of topics than a typical introductory book on mathematical statistics. This book is for people who want to learn probability and statistics quickly. It is suitable for graduate or advanced undergraduate students in computer science, mathematics, statistics, and related disciplines. The book includes modern topics like nonparametric curve estimation, bootstrapping, and clas- siﬁcation, topics that are usually relegated to follow-up courses. The reader is presumed to know calculus and a little linear algebra. No previous knowledge of probability and statistics is required. Statistics, data mining, and machine learning are all concerned with collecting and analyzing data. For some time, statistics research was con- ducted in statistics departments while data mining and machine learning re- search was conducted in computer science departments. Statisticians thought that computer scientists were reinventing the wheel. Computer scientists thought that statistical theory didn’t apply to their problems. Things are changing. Statisticians now recognize that computer scientists are making novel contributions while computer scientists now recognize the generality of statistical theory and methodology. Clever data mining algo- rithms are more scalable than statisticians ever thought possible. Formal sta- tistical theory is more pervasive than computer scientists had realized. Students who analyze data, or who aspire to develop new methods for analyzing data, should be well grounded in basic probability and mathematical statistics. Using fancy tools like neural nets, boosting, and support vector

viii Preface machines without understanding basic statistics is like doing brain surgery before knowing how to use a band-aid. But where can students learn basic probability and statistics quickly? Nowhere. At least, that was my conclusion when my computer science colleagues kept asking me: “Where can I send my students to get a good understanding of modern statistics quickly?” The typical mathematical statistics course spends too much time on tedious and uninspiring topics (counting methods, two di- mensional integrals, etc.) at the expense of covering modern concepts (boot- strapping, curve estimation, graphical models, etc.). So I set out to redesign our undergraduate honors course on probability and mathematical statistics. This book arose from that course. Here is a summary of the main features of this book. 1. The book is suitable for graduate students in computer science and honors undergraduates in math, statistics, and computer science. It is also useful for students beginning graduate work in statistics who need to ﬁll in their background on mathematical statistics. 2. I cover advanced topics that are traditionally not taught in a ﬁrst course. For example, nonparametric regression, bootstrapping, density estima- tion, and graphical models. 3. I have omitted topics in probability that do not play a central role in statistical inference. For example, counting methods are virtually ab- sent. 4. Whenever possible, I avoid tedious calculations in favor of emphasizing concepts. 5. I cover nonparametric inference before parametric inference. 6. I abandon the usual “First Term = Probability” and “Second Term = Statistics” approach. Some students only take the ﬁrst half and it would be a crime if they did not see any statistical theory. Furthermore, probability is more engaging when students can see it put to work in the context of statistics. An exception is the topic of stochastic processes which is included in the later material. 7. The course moves very quickly and covers much material. My colleagues joke that I cover all of statistics in this course and hence the title. The course is demanding but I have worked hard to make the material as intuitive as possible so that the material is very understandable despite the fast pace. 8. Rigor and clarity are not synonymous. I have tried to strike a good balance. To avoid getting bogged down in uninteresting technical details, many results are stated without proof. The bibliographic references at the end of each chapter point the student to appropriate sources.

Preface ix Probability Data generating process Observed data Inference and Data Mining FIGURE 1. Probability and inference. 9. On my website are ﬁles with R code which students can use for doing all the computing. The website is: http://www.stat.cmu.edu/∼larry/all-of-statistics However, the book is not tied to R and any computing language can be used. Part I of the text is concerned with probability theory, the formal language of uncertainty which is the basis of statistical inference. The basic problem that we study in probability is: Given a data generating process, what are the properties of the out- comes? Part II is about statistical inference and its close cousins, data mining and machine learning. The basic problem of statistical inference is the inverse of probability: Given the outcomes, what can we say about the process that gener- ated the data? These ideas are illustrated in Figure 1. Prediction, classiﬁcation, clustering, and estimation are all special cases of statistical inference. Data analysis, machine learning and data mining are various names given to the practice of statistical inference, depending on the context.

x Preface Part III applies the ideas from Part II to speciﬁc problems such as regres- sion, graphical models, causation, density estimation, smoothing, classiﬁca- tion, and simulation. Part III contains one more chapter on probability that covers stochastic processes including Markov chains. I have drawn on other books in many places. Most chapters contain a section called Bibliographic Remarks which serves both to acknowledge my debt to other authors and to point readers to other useful references. I would especially like to mention the books by DeGroot and Schervish (2002) and Grimmett and Stirzaker (1982) from which I adapted many examples and exercises. As one develops a book over several years it is easy to lose track of where pre- sentation ideas and, especially, homework problems originated. Some I made up. Some I remembered from my education. Some I borrowed from other books. I hope I do not oﬀend anyone if I have used a problem from their book and failed to give proper credit. As my colleague Mark Schervish wrote in his book (Schervish (1995)), “. . . the problems at the ends of each chapter have come from many sources. . . . These problems, in turn, came from various sources unknown to me . . . If I have used a problem without giving proper credit, please take it as a compliment.” I am indebted to many people without whose help I could not have written this book. First and foremost, the many students who used earlier versions of this text and provided much feedback. In particular, Liz Prather and Jen- nifer Bakal read the book carefully. Rob Reeder valiantly read through the entire book in excruciating detail and gave me countless suggestions for im- provements. Chris Genovese deserves special mention. He not only provided helpful ideas about intellectual content, but also spent many, many hours writing LATEXcode for the book. The best aspects of the book’s layout are due to his hard work; any stylistic deﬁciencies are due to my lack of expertise. David Hand, Sam Roweis, and David Scott read the book very carefully and made numerous suggestions that greatly improved the book. John Laﬀerty and Peter Spirtes also provided helpful feedback. John Kimmel has been sup- portive and helpful throughout the writing process. Finally, my wife Isabella Verdinelli has been an invaluable source of love, support, and inspiration. Larry Wasserman Pittsburgh, Pennsylvania July 2003

Statistics/Data Mining Dictionary Statisticians and computer scientists often use diﬀerent language for the same thing. Here is a dictionary that the reader may want to return to throughout the course. Preface xi Statistics estimation Computer Science learning classiﬁcation supervised learning clustering data covariates classiﬁer hypothesis conﬁdence interval unsupervised learning training sample features hypothesis — — directed acyclic graph Bayes net Bayesian inference Bayesian inference frequentist inference — large deviation bounds PAC learning putting data into groups (X1, Y1), . . . , (Xn, Yn) the Xi’s a map from covariates to outcomes subset of a parameter space Θ interval that contains an unknown quantity with given frequency multivariate distribution with given conditional independence relations statistical methods for Meaning using data to estimate an unknown quantity predicting a discrete Y from X using data to update beliefs statistical methods with guaranteed frequency behavior uniform bounds on probability of errors

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