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L.A. Wasserman 531●●●●●●●●●●●●●●● ●●●●●●● ●●●●● ●● ●●●●●●●●●FIGURE 44.3The mean shift algorithm. The data points move along trajectories duringiterations until they reach the two modes marked by the two large asterisks.The mean shift algorithm (Fukunaga and Hostetler, 1974; Comaniciu andMeer, 2002) is a method for finding the modes of a density by following thesteepest ascent paths. The algorithm starts with a mesh of points and thenmoves the points along gradient ascent trajectories towards local maxima. Asimple example is shown in Figure 44.3.Given a function p : R D → R, letg(x) =∇p(x) denotethegradientatxand let H(x) denotetheHessianmatrix.Letλ 1 (x) ≥···≥λ D (x) (44.8)denote the eigenvalues of H(x) and let Λ(x) bethediagonalmatrixwhosediagonal elements are the eigenvalues. Write the spectral decomposition ofH(x) as H(x) =U(x)Λ(x)U(x) ⊤ .Fix0≤ d
532 Rise of the machinesis that the flow passing through x is a gradient ascent path moving towardshigher values of p. Unlike the paths defined by the gradient g which movetowards modes, the paths defined by G move towards ridges.The paths can be parameterized in many ways. One commonly used parameterizationis to use t ∈ [−∞, ∞] where large values of t correspond tohigher values of p. Inthiscaset = ∞ will correspond to a point on the ridge.In this parameterization we can express each integral curve in the flow as follows.A map π : R → R D is an integral curve with respect to the flow of Gifπ ′ (t) =G{π(t)} = L{π(t)}g{π(t)}. (44.10)Definition. The ridge R consists of the destinations of the integral curves:y ∈ R if lim t→∞ π(t) =y for some π satisfying (44.10).As mentioned above, the integral curves partition the space and for eachx/∈ R, thereisauniquepathπ x passing through x. The ridge points are zerosof the projected gradient: y ∈ R implies that G(y) =(0,...,0) ⊤ .Ozertemand Erdogmus (2011) derived an extension of the mean-shift algorithm, calledthe subspace constrained mean shift algorithm that finds ridges which can beapplied to the kernel density estimator. Our results can be summarized asfollows:1. Stability. We showed that if two functions are sufficiently close togetherthen their ridges are also close together (in Hausdorff distance).2. We constructed an estimator ̂R such thatH(R, ̂R) =O P( (log nn) 2)D+8(44.11)where H is the Hausdorff distance. Further, we showed that ̂R is topologicallysimilar to R. We also construct an estimator ̂R h for h>0 thatsatisfies( (log ) 1)H(R h , ̂R n2h )=O P , (44.12)nwhere R h is a smoothed version of R.3. Suppose the data are obtained by sampling points on a manifold andadding noise with small variance σ 2 .Weshowedthattheresultingdensityp has a ridge R σ such thatH(M,R σ )=O ( σ 2 log 3 (1/σ) ) (44.13)and R σ is topologically similar to M. Hence when the noise σ is small, theridge is close to M. Itthenfollowsthat( (log ) 2)H(M, ̂R) nD+8=O P + O ( σ 2 log 3 (1/σ) ) . (44.14)n
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ContentsPrefacexviiContributorsI Th
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viii15 We live in exciting times 15
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x24.6 Open areas for research . . .
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xii36 Buried treasures 405Michael A
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xiv45.3 Multi-phase inference . . .
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PrefaceStatistics is the science of
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PrefacexixPart V comprises seven ar
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xxiiNancy FlournoyUniversity of Mis
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Part IThe history of COPSS
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4 A brief history of COPSSOnchiota
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6 A brief history of COPSSThe items
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8 A brief history of COPSSTABLE 1.1
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10 A brief history of COPSSwomen of
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12 A brief history of COPSSTABLE 1.
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16 A brief history of COPSS1.4.3 Ge
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20 A brief history of COPSSFor more
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2Reminiscences of the Columbia Univ
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I. Olkin 25Bechhofer, Allan Birnbau
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I. Olkin 27Milton Sobel was the tea
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3AcareerinstatisticsHerman Chernoff
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H. Chernoff 31given three-day basic
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H. Chernoff 33Savage tried to defen
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H. Chernoff 35In working on an arti
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H. Chernoff 37measure the loss as t
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H. Chernoff 39ReferencesAlbert, A.E
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4“. . . how wonderful the field o
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D.R. Brillinger 43He went on to be
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D.R. Brillinger 45Tukey wrote many
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D.R. Brillinger 4718. Personal comm
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50 Unorthodox journey to statistics
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6Statistics before and after my COP
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P.J. Bickel 616.3.1 Imperial Colleg
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P.J. Bickel 63make. This was first
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P.J. Bickel 65We eventually develop
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P.J. Bickel 67school interest, biol
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P.J. Bickel 69ReferencesBean, D., B
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P.J. Bickel 71Meinshausen, N., Bick
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74 Accidental biostatistics profess
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84 Passion for statisticsme is its
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86 Passion for statisticsTheir Biom
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88 Passion for statisticsthe first
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90 Passion for statisticsIn my expe
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92 Passion for statistics8.5 Job an
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94 Passion for statisticsperiod. An
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96 Passion for statisticsNeyman, J.
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98 Reflections on a statistical car
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100 Reflections on a statistical ca
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110 Science mixes it up with statis
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11Lessons from a twisted career pat
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J.S. Rosenthal 119My math professor
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J.S. Rosenthal 121contrary, it null
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J.S. Rosenthal 123of statistical re
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J.S. Rosenthal 125be open to whatev
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J.S. Rosenthal 12711.4 Final though
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130 Promoting equityand fellowship
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132 Promoting equityevaluate such e
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134 Promoting equityfor equity. But
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136 Promoting equityAssistant Secre
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Part IIIPerspectives on the fieldan
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142 Statistics in service to the na
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154 Where are the majors? ArtBachel
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156 Where are the majors?References
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158 Exciting timesCritically, Efron
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160 Exciting timesIn general I find
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162 Exciting timesI entered the Uni
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164 Exciting times15.3.3 Joining th
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166 Exciting timesvariability of a
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168 Exciting timesHabemma, J.D.F.,
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16The bright future of applied stat
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R.A. Irizarry 173FIGURE 16.1Illustr
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R.A. Irizarry 17516.4 The bright fu
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17The road travelled: From statisti
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N. Chatterjee 179the Kaplan-Meyer e
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N. Chatterjee 18117.4 Genome-wide a
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N. Chatterjee 183by any means. I ju
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N. Chatterjee 185right information
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N. Chatterjee 187The risk of cancer
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190 Journey into genetics and genom
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19Reflections on women in statistic
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M.E. Thompson 205Pearson in England
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M.E. Thompson 207TABLE 19.1The 14 C
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M.E. Thompson 209time series, the s
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M.E. Thompson 211the founding Chair
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M.E. Thompson 21319.7 The current s
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M.E. Thompson 215Yarmie, A. (2003).
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218 “The whole women thing”seem
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220 “The whole women thing”of t
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222 “The whole women thing”on a
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224 “The whole women thing”20.5
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226 “The whole women thing”Ifin
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228 “The whole women thing”Lond
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230 Reflections on diversityas a st
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232 Reflections on diversityall of
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234 Reflections on diversityforce m
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22Why does statistics have two theo
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D.A.S. Fraser 23922.2 65 years and
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D.A.S. Fraser 241function is taken
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D.A.S. Fraser 24322.4 Inference for
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D.A.S. Fraser 245that record the ef
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D.A.S. Fraser 247variable that give
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D.A.S. Fraser 249simple scalar case
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D.A.S. Fraser 251DiCiccio, T.J. and
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23Conditioning is the issueJames O.
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J.O. Berger 255Pedagogical example:
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J.O. Berger 257The SRP does not say
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J.O. Berger 25923.5 Conditional fre
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J.O. Berger 261conditional error pr
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J.O. Berger 263others). If x is the
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J.O. Berger 265How does a frequenti
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24Statistical inference from aDemps
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A.P. Dempster 269tion is simply une
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A.P. Dempster 271As the world of st
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A.P. Dempster 273when the X i are a
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A.P. Dempster 27524.5 Nonparametric
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A.P. Dempster 277can easily happen
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A.P. Dempster 279ReferencesBorel,
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25Nonparametric BayesDavid B. Dunso
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D.B. Dunson 283troduced, providing
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D.B. Dunson 285els in spatial stati
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D.B. Dunson 287methods as also prov
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D.B. Dunson 289ing background, and
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D.B. Dunson 291Murray, J.S., Dunson
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294 Choosing default methodsimagine
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296 Choosing default methods(a) mat
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298 Choosing default methodspartial
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300 Choosing default methodsdirecti
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27Serial correlation and Durbin-Wat
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T.W. Anderson 305The characteristic
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T.W. Anderson 307Consider the seria
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28Anon-asymptoticwalkinprobabilitya
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P. Massart 311(at least for discret
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P. Massart 31328.2.2 Non-asymptopia
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P. Massart 315where P = sµ. In oth
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P. Massart 317Hence by Talagrand’
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P. Massart 319ReferencesAkaike, H.
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P. Massart 321Talagrand, M. (1995).
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324 The present’s futuredata. In
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326 The present’s futureTABLE 29.
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328 The present’s future124356981
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330 The present’s futuremethod, t
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332 The present’s futureFIGURE 29
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334 The present’s futureIchino, M
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336 Lessons in biostatistics30.2 It
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338 Lessons in biostatisticsprognos
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340 Lessons in biostatisticstime fo
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342 Lessons in biostatisticsmeaning
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344 Lessons in biostatistics75% to
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346 Lessons in biostatisticshistolo
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31AvignetteofdiscoveryNancy Flourno
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N. Flournoy 351FIGURE 31.1Incidence
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N. Flournoy 353FIGURE 31.3Mean resp
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N. Flournoy 355FIGURE 31.4Kaplan-Me
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N. Flournoy 357As the study proceed
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32Statistics and public health rese
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R.L. Prentice 36132.2 Public health
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R.L. Prentice 363from urinary excre
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R.L. Prentice 36532.5 Clinical tria
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R.L. Prentice 367Kaplan, E.L. and M
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370 Statistics in a new erative eff
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372 Statistics in a new eraular com
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374 Statistics in a new erajoint de
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376 Statistics in a new eraof the n
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378 Statistics in a new eraLai, T.L
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34Meta-analyses: Heterogeneity can
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N.M. Laird 383into the sum of a “
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N.M. Laird 385are “approximately
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N.M. Laird 387included in the major
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N.M. Laird 389Yusuf, S., Peto, R.,
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392 Good health: Statistical challe
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406 Buried treasuresmeasurement tec
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408 Buried treasuresIn a most rewar
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410 Buried treasureslems different
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37Survey sampling: Past controversi
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R.J. Little 41537.2 Probability or
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R.J. Little 417for the “design-ba
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R.J. Little 419estimate of variance
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R.J. Little 421Example 2 continued
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R.J. Little 42337.3.4 The design-mo
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R.J. Little 42537.5 ConclusionsI am
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R.J. Little 427Isaki, C.T. and Full
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38Environmental informatics: Uncert
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N. Cressie 431and conquer” strate
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N. Cressie 433In the last 20 years,
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N. Cressie 435Once the satellite ha
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N. Cressie 437FIGURE 38.1Locations
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N. Cressie 439regional and seasonal
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N. Cressie 441ciated with the spati
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N. Cressie 443is the set of all pix
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N. Cressie 445challenge is to devel
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N. Cressie 447Gourdji, S.M., Muelle
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N. Cressie 449Wikle, C.K., Milliff,
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452 Statistical geneticsTABLE 39.1T
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454 Statistical geneticsStewart, 19
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456 Statistical geneticslocidata
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458 Statistical geneticssufficientl
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460 Statistical geneticsBrown, M.D.
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462 Statistical geneticsLange, K. a
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464 Statistical geneticsThompson, E
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466 Targeted learningods for nonpar
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468 Targeted learningdenote the obs
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470 Targeted learningSuch an estima
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472 Targeted learningliterature pro
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474 Targeted learninglihood estimat
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476 Targeted learning40.6 Some spec
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478 Targeted learningmemory challen
Page 501 and 502: 480 Targeted learningvan der Laan,
Page 503 and 504: 482 Ah-ha momentwe have just seen.
Page 505 and 506: 484 Ah-ha momentthe data is unchang
Page 507 and 508: 486 Ah-ha momentis well defined and
Page 509 and 510: 488 Ah-ha momentit was something of
Page 511 and 512: 490 Ah-ha momentdefines a Euclidean
Page 513 and 514: 492 Ah-ha momentparticipate in the
Page 515 and 516: 494 Ah-ha momentLee, Y., Wahba, G.,
Page 518 and 519: 42In praise of sparsity and convexi
Page 520 and 521: R.J. Tibshirani 499TABLE 42.1Asampl
Page 522 and 523: R.J. Tibshirani 501CancerEpithelial
Page 524 and 525: R.J. Tibshirani 503that the error v
Page 526: R.J. Tibshirani 505Tibshirani, R.J.
Page 529 and 530: 508 Features of Big Dataproblems en
Page 531 and 532: 510 Features of Big Datalearning st
Page 533 and 534: 512 Features of Big Data1210(a)p =
Page 535 and 536: 514 Features of Big Data25002000150
Page 537 and 538: 516 Features of Big DataAs an illus
Page 539 and 540: 518 Features of Big Datafor some gi
Page 541 and 542: 520 Features of Big Data43.7.3 Prop
Page 543 and 544: 522 Features of Big DataBühlmann,
Page 546 and 547: 44Rise of the machinesLarry A. Wass
Page 548 and 549: L.A. Wasserman 527if an idea requir
Page 550 and 551: L.A. Wasserman 529●●●● ●
Page 554 and 555: L.A. Wasserman 533FIGURE 44.4Simula
Page 556 and 557: L.A. Wasserman 53544.7 Education an
Page 558 and 559: 45Atrioofinferenceproblemsthatcould
Page 560 and 561: X.-L. Meng 539it is directly rooted
Page 562 and 563: X.-L. Meng 541exactly, but this doe
Page 564 and 565: X.-L. Meng 543That is, when we do n
Page 566 and 567: X.-L. Meng 545(a) For what classes
Page 568 and 569: X.-L. Meng 547subsequent analysts c
Page 570 and 571: X.-L. Meng 549can prove only that (
Page 572 and 573: X.-L. Meng 551In general, “What t
Page 574 and 575: X.-L. Meng 553vironmental progress
Page 576 and 577: X.-L. Meng 555which will be compare
Page 578 and 579: X.-L. Meng 557(a) Given partial kno
Page 580 and 581: X.-L. Meng 559AcknowledgementsThe m
Page 582 and 583: X.-L. Meng 561Knuth, D. (1997). The
Page 584: Part VAdvice for the nextgeneration
Page 587 and 588: 566 Inspiration, aspiration, ambiti
Page 589 and 590: 568 Inspiration, aspiration, ambiti
Page 592 and 593: 47Personal reflections on the COPSS
Page 594 and 595: R.J. Carroll 573them is, by definit
Page 596 and 597: R.J. Carroll 575(Carroll, 2003). In
Page 598 and 599: R.J. Carroll 57747.8 After the Pres
Page 600: R.J. Carroll 579Carroll, R.J., Spie
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582 Publishing without perishingthe
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584 Publishing without perishingbeg
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586 Publishing without perishingme,
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588 Publishing without perishingalr
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590 Publishing without perishingpor
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49Converting rejections into positi
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D.B. Rubin 595This pair of submissi
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D.B. Rubin 597There are several les
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D.B. Rubin 599very sound article, w
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D.B. Rubin 60149.8 ConclusionIhaveb
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D.B. Rubin 603Rubin, D.B. (1972). A
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606 Importance of mentorstwice as t
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608 Importance of mentorsexperiment
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610 Importance of mentorsposition a
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612 Importance of mentors50.7 The t
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51Never ask for or give advice, mak
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T. Speed 617overly generous. Perhap
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T. Speed 619and for finding enjoyme
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622 Thirteen rules3. Waste a lot of
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