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192 6. MODEL SELECTION, COMPARISON, AND AVERAGINGere are generalizations of AIC that address this problem, as well as the informativeprior problem more generally. e most common and already in wide use is DIC, the DE-VIANCE INFORMATION CRITERION. 81 More general yet is WAIC, known as the WIDELY AP-PLICABLE INFORMATION CRITERION 82 or rather the WATANABE-AKAIKE INFORMATIONCRITERION. 83 We’ll consider DIC later in this very chapter, to handle informative priors.WAIC will make an appearance much later, in Chapter 13.6.3.1.4. Other prediction frameworks. e AIC gambit entails predicting a test sampleof the same size and nature as the training sample. is most certainly does not mean AICcan only be used when we plan to predict a sample of the same size as training. For example,AIC approximates some forms of cross-validation. 84But AIC’s implied prediction task is hardly representative of everything we might wishto do with models. For example, some statisticians prefer to evaluate predictions using aPREQUENTIAL framework, in which models are judged on their accumulated learning errorover the training sample. 85 And once you start using multilevel models, “prediction” is nolonger uniquely defined, because the test sample can differ from the training sample in waysthat forbid use of some the parameter estimates. We’ll worry about that issue in Chapter 13.Rethinking: Uniformitarianism and AIC. e AIC gambit described on page 190 pulls the test samplefrom the same process as the training sample. is is a kind of uniformitarian assumption, inwhich future data are expected to come from the same process as past data and have the same roughrange of values. is can cause problems. For example, suppose we fit a regression that predictsheight using body weight. e training sample comes from a poor town, in which most people arepretty thin. e relationship between height and weight turns out to be positive and strong. Nowalso suppose our prediction goal is to guess the heights in another, much wealthier, town. Pluggingthe weights from the wealthy individuals into the model fit to the poor individuals will predictoutrageously tall people. e reason is that, once weight becomes large enough, it has essentiallyno relationship with height. AIC will not automatically recognize nor solve this problem. Nor willany other isolated procedure. But over repeated rounds of model fitting, attempts at prediction, andmodel criticism, it is possible to overcome this kind of limitation. As always, statistics is no substitutefor science.6.3.2. Simulating AIC. e gambit that leads to AIC is complicated. So let’s walk through asimulation of this procedure. Hirotugu Akaike (1927–2009) himself original did this entirelywith formal mathematics. But we’ll use an expedient simulation, so it’ll be easier to understandand to modify. is section is not necessary for using AIC. But it may be necessaryfor understanding it, both with respect to its derivation and its scope. It will also provide anexample structure for how to study forecasting for any unique modeling problem you mightencounter in the future.Let’s begin by writing a function that implements the Akaike gambit on page 190. Here’sthe code, followed by explanation.R code6.11sim.train.test
6.3. AKAIKE INFORMATION CRITERION 193}return( dev.test - dev.train )e first line just announces that sim.train.test is a function, which is a bundle ofreusable code with adjustable inputs. e adjustable input here is N, the sample size. elines of code between the braces are the reusable code. ese lines perform these jobs, inorder:(1) Simulate the training sample, stored in y.train(2) Simulate the test sample, stored in y.test(3) Fit a linear model to the training sample(4) Compute the deviance on the training sample(5) Compute the deviance on the testing sample(6) Report back as output the difference between the test and training devianceOur goal is to repeat this procedure thousands of times, to get a distribution for the differenceD test − D train . R makes this easy:diff
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Statistical RethinkingA BAYESIAN CO
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4 CONTENTS5.5. Ordinary least squar
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PrefaceMasons, when they start upon
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HOW TO USE THIS BOOK 9least a minor
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HOW TO USE THIS BOOK 11than initial
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1 e Golem of PragueIn the 16th cent
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16 1. THE GOLEM OF PRAGUEconverses
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1.2. WRECKING PRAGUE 19model P 1B ,
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1.2. WRECKING PRAGUE 21e dominant r
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1.3. THREE TOOLS FOR GOLEM ENGINEER
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2 Small Worlds and Large WorldsWhen
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2.1. PROBABILITY IS JUST COUNTING 3
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2.2. COLOMBO’S FIRST BAYESIAN MOD
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W L W W W L W L W2.2. COLOMBO’S F
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2.3. COMPONENTS OF THE MODEL 41not
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2.3. COMPONENTS OF THE MODEL 43You
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2.3. COMPONENTS OF THE MODEL 45e ma
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2.4. MAKING THE MODEL GO 47other pr
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2.4. MAKING THE MODEL GO 495 points
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2.4. MAKING THE MODEL GO 51just the
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2.4. MAKING THE MODEL GO 53between
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2.6. PRACTICE 55Medium.2.6.5. m1. R
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2.6. PRACTICE 57implied by the equa
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3 Sampling the ImaginaryLots of boo
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3. SAMPLING THE IMAGINARY 61a probl
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3.2. SAMPLING TO SUMMARIZE 63plot(
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3.2. SAMPLING TO SUMMARIZE 65Densit
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3.2. SAMPLING TO SUMMARIZE 67In con
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3.2. SAMPLING TO SUMMARIZE 69Densit
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3.3. SAMPLING TO SIMULATE PREDICTIO
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3.5. PRACTICE 793.5. PracticeEasy.
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3.5. PRACTICE 813.5.17. Predict sec
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84 4. LINEAR MODELSFIGURE 4.1. e Pt
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86 4. LINEAR MODELSexperiment with
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88 4. LINEAR MODELS4.1.4.2. Epistem
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90 4. LINEAR MODELSe approach above
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92 4. LINEAR MODELS'data.frame': 54
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94 4. LINEAR MODELSe point isn’t
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96 4. LINEAR MODELShave the samples
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98 4. LINEAR MODELSere’s no troub
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100 4. LINEAR MODELS)Note the comma
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102 4. LINEAR MODELSpreviously obse
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104 4. LINEAR MODELS4.4. Adding a p
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106 4. LINEAR MODELS(1) What is the
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108 4. LINEAR MODELSdata(Howell1)d
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110 4. LINEAR MODELSRethinking: Wha
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112 4. LINEAR MODELSheight140 150 1
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116 4. LINEAR MODELSYou end up with
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118 4. LINEAR MODELS(1) Use link to
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122 4. LINEAR MODELSRethinking: Lin
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126 4. LINEAR MODELS4.7.1. e1. In t
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5 Multivariate Linear ModelsOne of
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5.1. SPURIOUS ASSOCIATION 131Divorc
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5.1. SPURIOUS ASSOCIATION 133But me
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5.1. SPURIOUS ASSOCIATION 135abmbas
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5.1. SPURIOUS ASSOCIATION 137this i
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5.1. SPURIOUS ASSOCIATION 139slower
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8 Markov Chain Monte Carlo Estimati
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8.1. GOOD KING MARKOV AND HIS ISLAN
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8.2. MARKOV CHAIN MONTE CARLO 247fo
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8.2. MARKOV CHAIN MONTE CARLO 249No
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8.3. EASY HMC: MAP2STAN 251We’re
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8.3. EASY HMC: MAP2STAN 253$ cont_a
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8.3. EASY HMC: MAP2STAN 255mcmcpair
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8.3. EASY HMC: MAP2STAN 257FIGURE 8
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8.4. CARE AND FEEDING OF YOUR MARKO
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8.6. PRACTICE 265And since the chai
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268 9. BIG ENTROPY AND THE GENERALI
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270 9. BIG ENTROPY AND THE GENERALI
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10 Distance and DurationA curious t
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10.2. GAMMA 275can be quite complic
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278 11. COUNTING AND CLASSIFICATION
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304 12. MONSTERS AND MIXTUREShere?
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306 12. MONSTERS AND MIXTURESdensit
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308 12. MONSTERS AND MIXTURESfootbr
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310 12. MONSTERS AND MIXTURESWhy di
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312 12. MONSTERS AND MIXTURESm11.3
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314 12. MONSTERS AND MIXTURESResult
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316 12. MONSTERS AND MIXTURESanswer
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318 12. MONSTERS AND MIXTURESDensit
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320 12. MONSTERS AND MIXTURESBut we
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322 12. MONSTERS AND MIXTURESdisper
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324 12. MONSTERS AND MIXTURESe resu
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326 12. MONSTERS AND MIXTURESDensit
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328 12. MONSTERS AND MIXTURESR code
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330 12. MONSTERS AND MIXTURESPoisso
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332 12. MONSTERS AND MIXTURESthe ga
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334 12. MONSTERS AND MIXTURESdata=d
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336 12. MONSTERS AND MIXTURESma2 ~
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338 13. MULTILEVEL MODELSrepeat obs
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340 13. MULTILEVEL MODELSSo how do
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342 13. MULTILEVEL MODELSprobabilit
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344 13. MULTILEVEL MODELSa conseque
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346 13. MULTILEVEL MODELSCompute th
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348 13. MULTILEVEL MODELSabsolute e
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350 13. MULTILEVEL MODELS)sigma_act
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14 Multilevel Models II: Slopes14.1
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14.1. EVERYTHING CAN VARY AND PROBA
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14.1. EVERYTHING CAN VARY AND PROBA
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360 15. MISSING DATA AND OTHER OPPO
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16 Writing Statistics379
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382 ENDNOTES12. For an autopsy of t
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384 ENDNOTES45. Fisher (1925), in C
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386 ENDNOTES77. See two famous edit
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388 ENDNOTES113. Hurlbert (1984) is
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390 BibliographyFrank, S. A. (2011)
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392 BibliographyProulx, S. R. and A
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IndexAkaike (1973), 380, 383Akaike
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INDEX 397non-identifiability, 153Oc
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