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326 12. MONSTERS AND MIXTURESDensity0 2 4 6 8 10predators absentDensity0 1 2 3 4 5predators present0.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0proportion survivingproportion survivingFIGURE 12.4. Estimated distribution of survival probabilities (black lines)over the empirical distribution of survival proportions (blue), for the tadpoledata, now separated by cases in which predators were absent (le) andpresent (right). Dashed curves show 95% percentile confidence interval ofthe Beta density estimates, sampled from the posterior.R code12.33post
12.4. VARIABLE RATES: GAMMA-POISSON 327is to the binomial. In a gamma-Poisson model or process, the mean number of events perunit time in the Poisson, λ, is assumed to vary among sampling units. ese units may bedifferent vials of fruit flies or strands of DNA or soccer teams or, as you’ll model in thissection, Swedish roads.You can write down the mathematical form of the gamma-Poisson model, just like youdid for the beta-binomial before.y i ∼ GammaPoisson(µ i , s)log µ i = α + βx iI’ve gone ahead and established the link function we’ll need, as well. Let me explain thismodel, one line at a time. e top line represents the stochastic process that produces actualobservations, Poisson distributed values, y, one observed for each unique case i. is Poissonprocess is governed by a collection of rate parameters, λ i , one for each unit i in the data. esevarying rates in turn arise from the second line of the model. e second line represents adeeper stochastic process that creates variable λ values, according to a gamma distribution.e gamma is governed by two parameters, a mean µ i that is unique to each unit i and acommon dispersion parameter s (“scale”). Finally, we use a log-link to model µ i , becausethat will ensure the gamma mean is always positive. Another common choice, as you sawin Chapter 9, is the inverse link, which can misbehave but is nevertheless the natural linkfor the gamma. It’s also very common to use a log-link for the scale parameter, to ensure itdoesn’t go below zero.Why a gamma distribution? Why not use the beta again, or even a normal distribution?e first virtue of a gamma distribution is that it has the right domain for values of λ. emean number of events in a Poisson process must be a positive real value. e gamma distributionproduces positive real values. Neither the beta nor normal does. e second virtueis that the gamma is the natural sister distribution to the Poisson. If you start with a gammadistributed prior and update it with Poisson observations, the resulting posterior will alsobe a gamma distribution. is is just like the beta is to the binomial. e practical valueof this fact is that the likelihood function for gamma-Poisson model is easy to solve for andcompute. is will not necessarily be true for more complex multilevel models that you’llmeet in the next chapter.Like with the beta and the binomial, note that the gamma distribution in the modeleffectively defines the prior distribution for the parameter λ of the Poisson. e data, Poissonobservations, will allow us to estimate the shape of this prior. e parameters of the gamma,in light of the data, will have a posterior that provides a distribution of λ i values.Since you met the gamma already in Chapter 7, we can jump right into using the gamma-Poisson now. at’s what’s next.12.4.1. Example: Swedish traffic accidents. e data for this example come from an experimentconducted by the Swedish government in the early 1960’s. ey wanted to knowhow effective imposing speed limits would be for reducing traffic accidents. On some days aspeed limit was enforced, while on other days it was not. ese data have some interesting aspectswe’ll glide over for now. We’re just going to be interested in estimating the distributionof traffic accidents, using the speed limit as a predictor variable.Load the data with:
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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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270 9. BIG ENTROPY AND THE GENERALI
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10 Distance and DurationA curious t
Page 275: 10.2. GAMMA 275can be quite complic
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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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376 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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