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62 3. SAMPLING THE IMAGINARYDensity0.0 0.5 1.0 1.5 2.0 2.50.0 0.2 0.4 0.6 0.8 1.0proportion water (p)FIGURE 3.1. Sampling parameter values from the posterior distribution.Le: 10-thousand samples from the posterior implied by the globe tossingdata and model. Right: e density of samples (vertical) at each parametervalue (horizontal).3.1. Sampling from a grid-approximate posteriorBefore beginning to work with samples, we need to generate them. Here’s a reminderfor how to compute the posterior for the globe tossing model, using grid approximation:R code3.2p_grid
3.2. SAMPLING TO SUMMARIZE 63plot( samples )R code3.4In this plot, it’s as if you are flying over the posterior distribution, looking down on it. ereare many more samples from the dense region near 0.6 and very few samples below 0.25. Onthe right, the plot shows the density estimate computed from these samples.library(rethinking)dens( samples )R code3.5You can see that the estimated density is very similar to ideal posterior you computed viagrid approximation. If you draw even more samples, maybe 1e5 or 1e6, the density estimatewill get more and more similar to the ideal.All you’ve done so far is crudely replicate the posterior density you had already computed.So next it is time to use these samples to describe and understand the posterior.3.2. Sampling to summarizeOnce your model produces a posterior distribution, the model’s work is done. But yourwork has just begun. It is necessary to summarize the posterior distribution. Exactly how itis summarized depends upon your purpose. But common questions include:• How much posterior probability lies below some parameter value?• How much posterior probability lies between two parameter values?• Which parameter value marks the lower 5% of the posterior probability?• Which range of parameter values contains 90% of the posterior probability?• Which parameter value has highest posterior probability?ese simple questions can be usefully divided to into questions about (1) intervals of definedboundaries, (2) questions about intervals of defined probability mass, and (3) questions aboutpoint estimates. We’ll see how to approach these questions using samples from the posterior.3.2.1. Intervals of defined boundaries. Suppose I ask you for the posterior probability thatthe proportion of water is less than 0.5. Using the grid-approximate posterior, you can justadd up all of the probabilities, where the corresponding parameter value is less than 0.5:# add up posterior probability where p < 0.5sum( posterior[ p_grid < 0.5 ] )R code3.6[1] 0.1718746So about 17% of the posterior probability is below 0.5. Couldn’t be easier. But since grid approximationisn’t practical in general, it won’t always be so easy. Once there is more than oneparameter in the posterior distribution (wait until the next chapter for that complication),even this simple sum is no longer very simple.So let’s see how to perform the same calculation, using samples from the posterior. isapproach does generalize to complex models with many parameters, and so you can useit everywhere. All you have to do is similarly add up all of the samples below 0.5, but alsodivide the resulting count by the total number of samples. In other words, find the frequencyof parameter values below 0.5:
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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
Page 11 and 12: HOW TO USE THIS BOOK 11than initial
Page 13 and 14: 1 e Golem of PragueIn the 16th cent
Page 16: 16 1. THE GOLEM OF PRAGUEconverses
Page 19 and 20: 1.2. WRECKING PRAGUE 19model P 1B ,
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Page 23 and 24: 1.3. THREE TOOLS FOR GOLEM ENGINEER
Page 29 and 30: 2 Small Worlds and Large WorldsWhen
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Page 41 and 42: 2.3. COMPONENTS OF THE MODEL 41not
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Page 49 and 50: 2.4. MAKING THE MODEL GO 495 points
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Page 55 and 56: 2.6. PRACTICE 55Medium.2.6.5. m1. R
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Page 59 and 60: 3 Sampling the ImaginaryLots of boo
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Page 94 and 95: 94 4. LINEAR MODELSe point isn’t
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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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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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