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320 12. MONSTERS AND MIXTURESBut we’re going to be interested in using the solution numerically, not manipulating itanalytically. So we want the beta-binomial density function, within R. Several packages,including rethinking, provide it. e rethinking package calls it dbetabinom.12.3.4. Beta-binomial links. e fundamental parameters of the beta distribution are Aand B, as you saw above. But it’s going to usually be more convenient for us to model the betadistribution as a function of its mean, the average probability drawn from it, and dispersion,how spread out the distribution is. Under this parameterization, the mean is called ¯p (saypee-bar) and the dispersion θ. ese two parameters relate to A and B in a simple way:¯p =AA + B , θ = A + B.is way of using the beta distribution is very common, and so functions like dbetabinomalready expect input in this form, using the labels prob for ¯p and theta for θ. 111Redefining the beta distribution in this way is equivalent to using a link function. It’sjust like using a link, because neither of the parameters A nor B is the mean. But we’d like tomodel the mean, so we need a function to link these parameters to the mean. Since ¯p is themean, it defines our link function. But we’re also going to need to use log-odds again, so thatour linear model translates logically to the probability space between zero and one. So whatdoes the model look like, under this link strategy? Defining the logit of ¯p as a linear model,we get:logy i ∼ BetaBinomial(n i , ¯p i , θ)¯p i1 − ¯p i= α + βx iEffectively, we’ve pushed the logistic down one level of stochasticity, since the binomial levelis now determined by the beta probabilities. e parameters that are directly estimated inthis case will be α, β, and θ.Now, in some cases, you may need to also use a link for θ. Even if you don’t want to modelthe dispersion as a linear model—and I’m not going to encourage that in most circumstances,mainly because trying to model both ¯p and θ with linear models is a formula for stumblinginto a nonidentifiable model—even if you don’t want to do that, you have to restrict θ to begreater than zero. An easy approach to doing this is just to use a log-link, just like the meanλ of a Poisson, so that the estimates on the natural scale are always positive. is is what it’dlook like:logy i ∼ BetaBinomial(n i , ¯p i , θ)¯p i1 − ¯p i= α + βx ilog θ = τe name τ is arbitrary. It just needs to be different from θ, so you can keep in mind thatyou’ve estimated a parameter on the log scale as a stand-in for θ. en you’d estimate α, β,and τ as parameters. To get the posterior for θ, you’d just exponentiate the posterior for τ.I’ll use this kind of link in one of the model fits to come, so you’ll get to see what it looks likein code form.
12.3. VARIABLE PROBABILITIES: BETA-BINOMIAL 32112.3.5. Example: Bolker’s Reedfrogs. We’ll use an example also from Ben Bolker, mortalitydata on reed frog tadpoles variably exposed to aquatic predators at experimentally determineddensities. You can load the package and data with:library(rethinking)data(reedfrogs)d
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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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2.4. MAKING THE MODEL GO 51just the
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2.4. MAKING THE MODEL GO 53between
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2.6. PRACTICE 55Medium.2.6.5. m1. R
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2.6. PRACTICE 57implied by the equa
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3 Sampling the ImaginaryLots of boo
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3. SAMPLING THE IMAGINARY 61a probl
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3.2. SAMPLING TO SUMMARIZE 63plot(
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3.2. SAMPLING TO SUMMARIZE 65Densit
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3.2. SAMPLING TO SUMMARIZE 67In con
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3.2. SAMPLING TO SUMMARIZE 69Densit
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3.3. SAMPLING TO SIMULATE PREDICTIO
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3.5. PRACTICE 793.5. PracticeEasy.
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3.5. PRACTICE 813.5.17. Predict sec
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84 4. LINEAR MODELSFIGURE 4.1. e Pt
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86 4. LINEAR MODELSexperiment with
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88 4. LINEAR MODELS4.1.4.2. Epistem
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90 4. LINEAR MODELSe approach above
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92 4. LINEAR MODELS'data.frame': 54
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94 4. LINEAR MODELSe point isn’t
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96 4. LINEAR MODELShave the samples
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98 4. LINEAR MODELSere’s no troub
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100 4. LINEAR MODELS)Note the comma
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102 4. LINEAR MODELSpreviously obse
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104 4. LINEAR MODELS4.4. Adding a p
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106 4. LINEAR MODELS(1) What is the
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108 4. LINEAR MODELSdata(Howell1)d
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110 4. LINEAR MODELSRethinking: Wha
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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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Page 338 and 339: 338 13. MULTILEVEL MODELSrepeat obs
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Page 360 and 361: 360 15. MISSING DATA AND OTHER OPPO
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370 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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