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72 3. SAMPLING THE IMAGINARYR code3.20R code3.21R code3.22R code3.23likelihood defines a distribution of possible observations that we can sample from, to simulateobservation. In this way, Bayesian models are always generative, capable of simulatingpredictions. Many non-Bayesian models are also generative, but many are not.We will call such simulated data DUMMY DATA, to indicate that it is a stand in for actualdata, like a crash test dummy. With the globe tossing model, the dummy data arises from abinomial likelihood:n!Pr(w|p, n) =w!(n − w)! pw (1 − p) n−w ,where w is an observed count of “water” and n is the number of tosses. Suppose n = 2, twotosses of the globe. en there are only three possible observations: 0 water, 1 water, 2 water.You can quickly compute the likelihood of each, for any given value of p. Let’s use p = 0.7,which is just about the true proportion of water on the Earth:dbinom( 0:2 , size=2 , prob=0.7 )[1] 0.09 0.42 0.49is means that there’s a 9% chance of observing w = 0, a 42% chance of w = 1, and a 49%chance of w = 2. If you change the value of p, you’ll get a different distribution of impliedobservations.Now we’re going to simulate observations, using these likelihoods. is is done by samplingfrom the distribution just described above. You could use sample to do this, but Rprovides convenient sampling functions for all the ordinary probability distributions, likethe binomial. So a single dummy data observation of w can be sampled with:rbinom( 1 , size=2 , prob=0.7 )[1] 1e “r” in rbinom stands for “random.” It can also generate more than one simulation at atime. A set of 10 simulations can be made by:rbinom( 10 , size=2 , prob=0.7 )[1] 2 2 2 1 2 1 1 1 0 2Let’s generate 100-thousand dummy observations, just to verify that each value (0, 1, or 2)appears in proportion to its likelihood:dummy_w
3.3. SAMPLING TO SIMULATE PREDICTION 73Frequency0 5000 15000 250000 2 4 6 8dummy water countFIGURE 3.5. Distribution of simulated sample observations from 9 tosses ofthe globe. ese samples assume the proportion of water is 0.7.dummy_w
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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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Page 29 and 30: 2 Small Worlds and Large WorldsWhen
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Page 47 and 48: 2.4. MAKING THE MODEL GO 47other pr
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Page 57 and 58: 2.6. PRACTICE 57implied by the equa
Page 59 and 60: 3 Sampling the ImaginaryLots of boo
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Page 79 and 80: 3.5. PRACTICE 793.5. PracticeEasy.
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Page 84 and 85: 84 4. LINEAR MODELSFIGURE 4.1. e Pt
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122 4. LINEAR MODELSRethinking: Lin
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124 4. LINEAR MODELSheight60 80 100
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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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5.2. MASKED RELATIONSHIP 147a ~ dno
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5.2. MASKED RELATIONSHIP 149dcc$log
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5.3. WHEN ADDING VARIABLES HURTS 15
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5.4. CATEGORICAL VARIABLES 161$ wei
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5.4. CATEGORICAL VARIABLES 1635.4.2
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5.4. CATEGORICAL VARIABLES 165# sam
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168 5. MULTIVARIATE LINEAR MODELS5.
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170 5. MULTIVARIATE LINEAR MODELS5.
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6 Model Selection, Comparison, and
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6.1. THE PROBLEM WITH PARAMETERS 17
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6.2. INFORMATION THEORY AND MODEL P
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6.3. AKAIKE INFORMATION CRITERION 1
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6.4. DEVIANCE INFORMATION CRITERION
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6.5. USING AIC 203helps guard again
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6.5. USING AIC 205compare( m6.11 ,
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6.5. USING AIC 207e attitude this b
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6.5. USING AIC 209kcal.per.g0.5 0.7
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6.7. PRACTICE 211Consider by analog
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6.7. PRACTICE 2136.7.3. e deviance
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216 7. INTERACTIONSFIGURE 7.1. TOP:
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218 7. INTERACTIONSlog(rgdppc_2000)
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220 7. INTERACTIONSird, we may want
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222 7. INTERACTIONSlog GDP year 200
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226 7. INTERACTIONSInteraction mode
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228 7. INTERACTIONSthis model and t
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232 7. INTERACTIONSe main effect li
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234 7. INTERACTIONSbs -38.91 34.94s
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236 7. INTERACTIONSe primary reason
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238 7. INTERACTIONSNow for the plot
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240 7. INTERACTIONS7.4. Higher-orde
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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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