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102 4. LINEAR MODELSpreviously observed 100 heights with mean value 165. at’s a pretty strong prior. In contrast, theformer Normal(165, 10) prior implies n = 1/10 2 = 0.01, a hundredth of an observation. is is anextremely weak prior. But of course exactly how strong or weak either prior is will depend upon howmuch data is used to update it.4.3.6. Sampling from a map fit. e above explains how to get a MAP quadratic approximationof the posterior, using map. But how do you then get samples from the quadraticapproximate posterior distribution? e answer is rather simple, but non-obvious, and itrequires recognizing that a quadratic approximation to a posterior distribution with morethan one parameter dimension—µ and σ each contribute one dimension—is just a multidimensionalGaussian distribution.As a consequence, when R constructs a quadratic approximation, it calculates not onlystandard deviations for all parameters, but also the covariances among all pairs of parameters.Just like a mean and standard deviation (or its square, a variance) are sufficient todescribe a one-dimensional Gaussian distribution, a list of means and a matrix of variancesand covariances is sufficient to describe a multi-dimensional Gaussian distribution. To seethis matrix of variances and covariances, for model m4.1, use:R code4.29vcov( m4.1 )mu sigmamu 0.1695256052 0.0003866944sigma 0.0003866944 0.0849067256e above is a VARIANCE-COVARIANCE matrix. It is the multi-dimensional glue of a quadraticapproximation, because it tells us how each parameter relates to every other parameterin the posterior distribution. A variance-covariance matrix can be factored into twoelements: (1) a vector of variances for the parameters and (2) a correlation matrix that tellsus how changes in any parameter lead to correlated changes in the others. is decompositionis usually easier to understand. So let’s do that now:R code4.30diag( vcov( m4.1 ) )cov2cor( vcov( m4.1 ) )mu sigma0.16952561 0.08490673mu sigmamu 1.00000000 0.00322314sigma 0.00322314 1.00000000e two-element vector in the output is the list of variances. If you take the square root of thisvector, you get the standard deviations that are shown in precis output. e two-by-twomatrix in the output is the correlation matrix. Each entry shows the correlation, boundedbetween −1 and +1, for each pair of parameters. e 1’s indicate a parameter’s correlationwith itself. If these values where anything except 1, we would be worried. e other entriesare typically closer to zero, and they are very close to zero in this example. is indicatesthat learning µ tells us nothing about σ and likewise that learning σ tells us nothing about
4.3. A GAUSSIAN MODEL OF HEIGHT 103µ. is is typical of simple Gaussian models of this kind, but it is quite rare in more generalmodeling.Okay, so how do we get samples from this multi-dimensional posterior? Now insteadof sampling single values from a simple Gaussian distribution, we sample vectors of valuesfrom a multi-dimensional Gaussian distribution. e rethinking package provides a conveniencefunction to do exactly that:library(rethinking)post
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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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Page 59 and 60: 3 Sampling the ImaginaryLots of boo
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Page 63 and 64: 3.2. SAMPLING TO SUMMARIZE 63plot(
Page 65 and 66: 3.2. SAMPLING TO SUMMARIZE 65Densit
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Page 88 and 89: 88 4. LINEAR MODELS4.1.4.2. Epistem
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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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