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78 3. SAMPLING THE IMAGINARYDoes this mean that the model is bad? at depends. e model will always be wrongin some sense, be mis-specified. But whether or not the mis-specification should lead us totry other models will depend upon our specific interests. In this case, if tosses do tend toswitch from W to L and L to W, then each toss will provide less information about the truecoverage of water on the globe. In the long run, even the wrong model we’ve used throughoutthe chapter will converge on the correct proportion. But it will do so more slowly than theposterior distribution may lead us to believe.Rethinking: What does more extreme mean? A common way of measuring deviation of observationfrom model is to count up the tail area that includes the observed data and any more extreme data.Ordinary p-values are an example of such a tail-area probability. When comparing observations todistributions of simulated predictions, as in FIGURE 3.6 and FIGURE 3.7, we might wonder how farout in the tail the observed data must be before we conclude that the model is a poor one. Becausestatistical contexts vary so much, it’s impossible to give a universally useful answer.But more importantly, there are usually very many ways to view data and define “extreme.” Ordinaryp-values view the data in just the way to model expects it, and so provide a very weak form ofmodel checking. For example, the far-right plot in FIGURE 3.6 evaluates model fit in the best way forthe model. Alternative ways of defining “extreme” may provide a more serious challenge to a model.e different definitions of extreme in FIGURE 3.7 can more easily embarrass it.Model fitting remains on objective procedure—everyone and every golem conducts Bayesianupdating in a way that doesn’t depend upon personal preferences. But model checking is inherentlysubjective, and this actually allows it to be quite powerful, since subjective knowledge of an empiricaldomain provides expertise. Expertise in turn allows for imaginative checks of model performance.Since golems have terrible imaginations, we need the freedom to engage our imaginations. In thisway, the objective and subjective work together. E. T. Jaynes (1922–1998) said all of this much moresuccinctly:It would be very nice to have a formal apparatus that gives us some “optimal” wayof recognizing unusual phenomena and inventing new classes of hypotheses thatare most likely to contain the true one; but this remains an art for the creativehuman mind. 523.4. Summaryis chapter introduced the basic procedures for manipulating posterior distributions.Our fundamental tool is samples of parameter values drawn from the posterior distribution.Samples transform problems in integral calculus into problems in data summary. ese samplescan be used to produce intervals, point estimates, as well as posterior predictive checks,as well as other kinds of simulations.Posterior predictive checks combine uncertainty about parameters, as described by theposterior distribution, with uncertainty about outcomes, as described by the assumed likelihoodfunction. ese checks are useful for verifying that your soware worked correctly.ey are also useful for prospecting for ways in which your models is inadequate.Once models become more complex, posterior predictive simulations will be used fora broader range of applications. Even understanding a model oen requires simulating impliedobservations. We’ll keep working with samples from the posterior, to make these tasksas easy and customizable as possible.
3.5. PRACTICE 793.5. PracticeEasy. ese problems use the samples from the posterior distribution for the globe tossingexample. is code will give you a specific set of samples, so that you can check your answersexactly.p_grid
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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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1.3. THREE TOOLS FOR GOLEM ENGINEER
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Page 29 and 30: 2 Small Worlds and Large WorldsWhen
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Page 59 and 60: 3 Sampling the ImaginaryLots of boo
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Page 65 and 66: 3.2. SAMPLING TO SUMMARIZE 65Densit
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Page 81: 3.5. PRACTICE 813.5.17. Predict sec
Page 84 and 85: 84 4. LINEAR MODELSFIGURE 4.1. e Pt
Page 86 and 87: 86 4. LINEAR MODELSexperiment with
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Page 92 and 93: 92 4. LINEAR MODELS'data.frame': 54
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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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