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198 6. MODEL SELECTION, COMPARISON, AND AVERAGINGR code6.15m6.9
6.4. DEVIANCE INFORMATION CRITERION 199(a)(b)deviance1.5 2.5 3.51deviance1.5 2.5 3.50.5-1 0 1 2 3mu-1 0 1 2 3muFIGURE 6.7. How DIC estimates the overfitting penalty. In each plot, samplesfrom the posterior density of µ, mu, are shown against deviance at eachvalue of µ. e vertical black dashed line marks the posterior mean, E µ.e horizontal black dashed line marks its deviance, ˆD. e blue dashedline marks the average deviance, ¯D. e difference ¯D − ˆD estimates theeffective number of parameters. is difference is shown by the gray linesegment in the right margin of each plot, labeled by its value. (a) A flatprior on µ, resulting in ¯D − ˆD ≈ 1. (b) An informative µ ∼ Normal(0, 1)prior, resulting in ¯D − ˆD ≈ 0.5.e informative prior has this effect because priors act like previously observed data. Inm6.10, the prior is equivalent to exactly one previous observation, located at y i = 0. Goahead and fit a model to one observation at y i = 0, using a flat prior, and you’ll see that theposterior density of µ will have mean 0 and standard error 1. So m6.10 updates that posteriorin light of another observation at y i = 1. Since the prior and the sample are equally strong,the posterior mean ends up right between them, at µ = 0.5. And likewise, since the priorand sample influence the posterior equally, the effective number of parameters is exactlyhalf the number of parameters. In terms of the distance ¯D − ˆD, the prior effectively doublesthe sample size, concentrating the posterior more around its mean. If the posterior is moreconcentrated, then the average deviance (blue dashed line) must be closer to the deviance ofthe mean (black dashed line).Typically, you’ll work with much larger samples, so any informative prior won’t be nearlyas influential as in the pedagogical example in FIGURE 6.7. But the phenomenon is quitegeneral. Informative priors reduce the flexibility of the model—how freely it can encode thesample. Reduced flexibility tends to mean less risk of overfitting.To see how well DIC does at estimating D test , let’s repeat the simulation strategy we developedfor AIC (FIGURE 6.6, panels c and d). Now however, we’ll use Normal(0, 1) priorson every regression slope (but not the intercept) in the multivariate models. So the models
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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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5.1. SPURIOUS ASSOCIATION 141Median
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5.1. SPURIOUS ASSOCIATION 143(a)(b)
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5.2. MASKED RELATIONSHIP 145N
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Page 161 and 162: 5.4. CATEGORICAL VARIABLES 161$ wei
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Page 173 and 174: 6 Model Selection, Comparison, and
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Page 203 and 204: 6.5. USING AIC 203helps guard again
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Page 211 and 212: 6.7. PRACTICE 211Consider by analog
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Page 216 and 217: 216 7. INTERACTIONSFIGURE 7.1. TOP:
Page 218 and 219: 218 7. INTERACTIONSlog(rgdppc_2000)
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Page 226 and 227: 226 7. INTERACTIONSInteraction mode
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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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