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254 8. MARKOV CHAIN MONTE CARLO ESTIMATIONFIGURE 8.5. Pairs plot of the samples produced by Stan. e diagonal showsa density estimate for each parameter. Below the diagonal, correlations betweenparameters are shown.Note that post here is a list, not a data.frame. is fact will be very useful to use lateron, when we encounter multilevel models. For now, if it doesn’t behave like you expect it to,you can coerce it to a data frame with post
8.3. EASY HMC: MAP2STAN 255mcmcpairs(post)R code8.8FIGURE 8.5 shows the resulting plot. is is a pairs plot, so it’s still a matrix of bivariate scatterplots. But now along the diagonal the smoothed histogram of each parameter is shown, alongwith its name. And in the lower triangle of the matrix, below the diagonal, the correlationbetween each pair of parameters is shown, with stronger correlations indicated by relativesize.For this model and these data, the resulting posterior distribution is quite nearly multivariateGaussian. e density for sigma is certainly skewed in the expected direction. Butotherwise the quadratic approximation does almost as well as Hamiltonian Monte Carlo.is is a very simple kind of model structure of course, with Gaussian priors, so an approximatelyquadratic posterior should be no surprise. Later, we’ll see some more exotic posteriordistributions.Overthinking: Stan messages. When you fit a model using map2stan, R will first translate yourmodel formula into a Stan language model. en it sends that model to Stan. e messages yousee in your R console are status updates from Stan. Stan first again translates the model, this timeinto C++ code. at code is then sent to a C++ compiler, to build an executable file is a specializedsampling engine for your model. en Stan feeds the data and starting values to that executable file,and if all goes well, sampling begins. You will see Stan count through the iterations. During sampling,you might occasionally see a scary looking warning something like this:Informational Message:The current Metropolis proposal is about to berejected because of the following issue: Error in function stan::prob::multi_normal_log(N4stan5Covariance matrixCovariance matrix is not positive definite.Covariancematrix(0,0) is 0:0. If this warning occurs sporadically, such as forhighly constrained variable types like covariance matrices, then thesampler is fine, but if this warning occurs often then your model may beeither severely ill-conditioned or misspecified.Severely ill-conditioned or misspecified? at certainly sounds bad. But rarely does this messageindicate a serious problem. As long as it happens only a handful of times, and especially if it onlyhappens during warmup, then odds are very good the chain is fine. You should still always check thechain for problems, of course. Just don’t panic when you see this message.8.3.4. Using the samples. Once you have samples in an object like post, you work withthem just as you’ve already learned to do. If you have the samples from the posterior and youknow the model, you can do anything: simulate predictions, compute differences betweenparameters, and calculate DIC.Actually, by default map2stan computes DIC for you. You can extract it’s value withDIC(m8.1stan), for example. DIC is also reported and broken down by its components onthe default show output for a map2stan model fit.show(m8.1stan)R code8.9map2stan model fitFormula:log_gdp ~ dnorm(mu, sigma)
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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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Page 203 and 204: 6.5. USING AIC 203helps guard again
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Page 257 and 258: 8.3. EASY HMC: MAP2STAN 257FIGURE 8
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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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