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96 4. LINEAR MODELShave the samples you’ll produce in this subsection, you can compare them to the quadraticapproximation in the next.Unfortunately, doing the calculations here requires some technical tricks that add little,if any, conceptual insight. So I’m going to present the code here without explanation. You canexecute it and keep going for now, but later return and follow the endnote for an explanationof the algorithm. 58 For now, here are the guts of the golem:R code4.14mu.list
4.3. A GAUSSIAN MODEL OF HEIGHT 97FIGURE 4.3. Samples from the posterior distributionfor the heights data. e density of points ishighest in the center, reflecting the most plausiblecombinations of µ and σ.I reproduce this plot in FIGURE 4.3. Note that the function col.alpha is part of the rethinkingR package. All it does is make colors transparent, which helps the plot in FIG-URE 4.3 more easily show density, where samples overlap. Adjust the plot to your tastes byplaying around with cex (character expansion, the size of the points), pch (plot character),and the 0.1 transparency value.Now that you have these samples, you can describe the distribution of confidence in eachcombination of µ and σ by summarizing the samples. ink of them like data and describethem, just like in Chapter 3. For example, to characterize the shapes of the marginal posteriordensities of µ and σ, all we need to do is:dens( sample.mu )dens( sample.sigma )R code4.19e jargon “marginal” here means “averaging over the other parameters.” Execute the abovecode and inspect the plots. ese densities are very close to being normal distributions.And this is quite typical. As sample size increases, posterior densities approach the normaldistribution. If you look closely, though, you’ll notice that the density for σ has a longerrighthand tail. I’ll exaggerate this tendency a bit later, to show you that this condition is verycommon for standard deviation parameters.To summarize the widths of these densities with highest posterior density intervals (HPDIs),just like in Chapter 3:HPDI( sample.mu , prob=0.95 )HPDI( sample.sigma , prob=0.95 )R code4.20Since these samples are just vectors of numbers, you can compute any statistic from them thatyou could from ordinary data. If you want the mean or median, just use the correspondingR functions.Overthinking: Sample size and the normality of σ’s posterior. Before moving on to using quadraticapproximation (map) as shortcut to all of this inference, it is worth repeating the analysis of the heightdata above, but now with only a fraction of the original data. e reason to do this is to demonstratethat, in principle, the posterior is not always so Gaussian in shape.
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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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Page 59 and 60: 3 Sampling the ImaginaryLots of boo
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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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