statisticalrethinkin..

More documents

Recommendations

Info

318 12. MONSTERS AND MIXTURESDensity0.6 0.8 1.0 1.2 1.4alpha = 1, beta = 1Density0.0 0.5 1.0 1.5alpha = 3, beta = 30.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0p ip ialpha = 4, beta = 1alpha = 0.9, beta = 0.8Density0 1 2 3 4Density1.0 1.2 1.4 1.6 1.80.0 0.2 0.4 0.6 0.8 1.00.0 0.2 0.4 0.6 0.8 1.0p ip iFIGURE 12.2. e beta distribution defines the probabilities of differentprobabilities, p i . e horizontal axis in each plot is the range of binomialprobabilities from which to draw samples. It is a parameter, but now witha different value for each case i. e vertical axis is the likelihood (density)of each probability, according to a beta distribution with shape defined bythe A and B values shown at the top of each plot.12.3.2. Beta distribution. You haven’t seen a beta distribution before, unless it was outsidethis book, so it’s worth saying a little about it. e BETA DISTRIBUTION is a probability densityfor probabilities themselves, continuous values between zero and one. It is a member ofthe exponential family, just like the Gaussian and Poisson and binomial. But I didn’t mentionit before, because it isn’t so oen used as a top-level distribution. 110 Instead, it becomesvaluable when we start stacking and mixing probability distributions.e beta distribution is very flexible, and its shape is defined by two parameters, A andB. Sadly, the conventional names for these parameters are α and β, and that’s how theyappear in most books. But since α and β are the same symbols oen used to build linearmodels, I’m going to use the “fancy” capital letter names for them. Hopefully this will reduceany confusion for now. As you’ll see a little later, we’re going to re-parameterize the betadistribution anyway.Let’s aim to understand how the distribution can take different shapes. You can see someof its characterist shapes, and how they correspond to the values of A and B, in FIGURE 12.2.
12.3. VARIABLE PROBABILITIES: BETA-BINOMIAL 319e beta distribution can take many shapes, from flat (A = 1, B = 1, top-le), to peaked inthe middle (A = 3, B = 3, top-right), to highly skewed (A = 4, B = 1, bottom-le), andto massing probability at the ends near zero and one (A = 0.9, B = 0.8, bottom-right). Youcan explore more shapes by changing the A and B values in the following code:A
Page 1:
Statistical RethinkingA BAYESIAN CO
Page 4 and 5:
4 CONTENTS5.5. Ordinary least squar
Page 7 and 8:
PrefaceMasons, when they start upon
Page 9 and 10:
HOW TO USE THIS BOOK 9least a minor
Page 11 and 12:
HOW TO USE THIS BOOK 11than initial
Page 13 and 14:
1 e Golem of PragueIn the 16th cent
Page 16:
16 1. THE GOLEM OF PRAGUEconverses
Page 19 and 20:
1.2. WRECKING PRAGUE 19model P 1B ,
Page 21 and 22:
1.2. WRECKING PRAGUE 21e dominant r
Page 23 and 24:
1.3. THREE TOOLS FOR GOLEM ENGINEER
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
2 Small Worlds and Large WorldsWhen
Page 31 and 32:
2.1. PROBABILITY IS JUST COUNTING 3
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
2.2. COLOMBO’S FIRST BAYESIAN MOD
Page 39 and 40:
W L W W W L W L W2.2. COLOMBO’S F
Page 41 and 42:
2.3. COMPONENTS OF THE MODEL 41not
Page 43 and 44:
2.3. COMPONENTS OF THE MODEL 43You
Page 45 and 46:
2.3. COMPONENTS OF THE MODEL 45e ma
Page 47 and 48:
2.4. MAKING THE MODEL GO 47other pr
Page 49 and 50:
2.4. MAKING THE MODEL GO 495 points
Page 51 and 52:
2.4. MAKING THE MODEL GO 51just the
Page 53 and 54:
2.4. MAKING THE MODEL GO 53between
Page 55 and 56:
2.6. PRACTICE 55Medium.2.6.5. m1. R
Page 57 and 58:
2.6. PRACTICE 57implied by the equa
Page 59 and 60:
3 Sampling the ImaginaryLots of boo
Page 61 and 62:
3. SAMPLING THE IMAGINARY 61a probl
Page 63 and 64:
3.2. SAMPLING TO SUMMARIZE 63plot(
Page 65 and 66:
3.2. SAMPLING TO SUMMARIZE 65Densit
Page 67 and 68:
3.2. SAMPLING TO SUMMARIZE 67In con
Page 69 and 70:
3.2. SAMPLING TO SUMMARIZE 69Densit
Page 71 and 72:
3.3. SAMPLING TO SIMULATE PREDICTIO
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
3.5. PRACTICE 793.5. PracticeEasy.
Page 81:
3.5. PRACTICE 813.5.17. Predict sec
Page 84 and 85:
84 4. LINEAR MODELSFIGURE 4.1. e Pt
Page 86 and 87:
86 4. LINEAR MODELSexperiment with
Page 88 and 89:
88 4. LINEAR MODELS4.1.4.2. Epistem
Page 90 and 91:
90 4. LINEAR MODELSe approach above
Page 92 and 93:
92 4. LINEAR MODELS'data.frame': 54
Page 94 and 95:
94 4. LINEAR MODELSe point isn’t
Page 96 and 97:
96 4. LINEAR MODELShave the samples
Page 98 and 99:
98 4. LINEAR MODELSere’s no troub
Page 100 and 101:
100 4. LINEAR MODELS)Note the comma
Page 102 and 103:
102 4. LINEAR MODELSpreviously obse
Page 104 and 105:
104 4. LINEAR MODELS4.4. Adding a p
Page 106 and 107:
106 4. LINEAR MODELS(1) What is the
Page 108 and 109:
108 4. LINEAR MODELSdata(Howell1)d
Page 110 and 111:
110 4. LINEAR MODELSRethinking: Wha
Page 112 and 113:
112 4. LINEAR MODELSheight140 150 1
Page 114 and 115:
Page 116 and 117:
116 4. LINEAR MODELSYou end up with
Page 118 and 119:
118 4. LINEAR MODELS(1) Use link to
Page 120 and 121:
Page 122 and 123:
122 4. LINEAR MODELSRethinking: Lin
Page 124 and 125:
Page 126 and 127:
126 4. LINEAR MODELS4.7.1. e1. In t
Page 129 and 130:
5 Multivariate Linear ModelsOne of
Page 131 and 132:
5.1. SPURIOUS ASSOCIATION 131Divorc
Page 133 and 134:
5.1. SPURIOUS ASSOCIATION 133But me
Page 135 and 136:
5.1. SPURIOUS ASSOCIATION 135abmbas
Page 137 and 138:
5.1. SPURIOUS ASSOCIATION 137this i
Page 139 and 140:
5.1. SPURIOUS ASSOCIATION 139slower
Page 141 and 142:
5.1. SPURIOUS ASSOCIATION 141Median
Page 143 and 144:
5.1. SPURIOUS ASSOCIATION 143(a)(b)
Page 145 and 146:
5.2. MASKED RELATIONSHIP 145N
Page 147 and 148:
5.2. MASKED RELATIONSHIP 147a ~ dno
Page 149 and 150:
5.2. MASKED RELATIONSHIP 149dcc$log
Page 151 and 152:
5.3. WHEN ADDING VARIABLES HURTS 15
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
5.4. CATEGORICAL VARIABLES 161$ wei
Page 163 and 164:
5.4. CATEGORICAL VARIABLES 1635.4.2
Page 165:
5.4. CATEGORICAL VARIABLES 165# sam
Page 168 and 169:
168 5. MULTIVARIATE LINEAR MODELS5.
Page 170 and 171:
170 5. MULTIVARIATE LINEAR MODELS5.
Page 173 and 174:
6 Model Selection, Comparison, and
Page 175 and 176:
6.1. THE PROBLEM WITH PARAMETERS 17
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
6.2. INFORMATION THEORY AND MODEL P
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
6.3. AKAIKE INFORMATION CRITERION 1
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
6.4. DEVIANCE INFORMATION CRITERION
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
6.5. USING AIC 203helps guard again
Page 205 and 206:
6.5. USING AIC 205compare( m6.11 ,
Page 207 and 208:
6.5. USING AIC 207e attitude this b
Page 209 and 210:
6.5. USING AIC 209kcal.per.g0.5 0.7
Page 211 and 212:
6.7. PRACTICE 211Consider by analog
Page 213:
6.7. PRACTICE 2136.7.3. e deviance
Page 216 and 217:
216 7. INTERACTIONSFIGURE 7.1. TOP:
Page 218 and 219:
218 7. INTERACTIONSlog(rgdppc_2000)
Page 220 and 221:
220 7. INTERACTIONSird, we may want
Page 222 and 223:
222 7. INTERACTIONSlog GDP year 200
Page 224 and 225:
Page 226 and 227:
226 7. INTERACTIONSInteraction mode
Page 228 and 229:
228 7. INTERACTIONSthis model and t
Page 230 and 231:
Page 232 and 233:
232 7. INTERACTIONSe main effect li
Page 234 and 235:
234 7. INTERACTIONSbs -38.91 34.94s
Page 236 and 237:
236 7. INTERACTIONSe primary reason
Page 238 and 239:
238 7. INTERACTIONSNow for the plot
Page 240 and 241:
240 7. INTERACTIONS7.4. Higher-orde
Page 243 and 244:
8 Markov Chain Monte Carlo Estimati
Page 245 and 246:
8.1. GOOD KING MARKOV AND HIS ISLAN
Page 247 and 248:
8.2. MARKOV CHAIN MONTE CARLO 247fo
Page 249 and 250:
8.2. MARKOV CHAIN MONTE CARLO 249No
Page 251 and 252:
8.3. EASY HMC: MAP2STAN 251We’re
Page 253 and 254:
8.3. EASY HMC: MAP2STAN 253$ cont_a
Page 255 and 256:
8.3. EASY HMC: MAP2STAN 255mcmcpair
Page 257 and 258:
8.3. EASY HMC: MAP2STAN 257FIGURE 8
Page 259 and 260:
8.4. CARE AND FEEDING OF YOUR MARKO
Page 261 and 262:
Page 263 and 264:
Page 265:
8.6. PRACTICE 265And since the chai
Page 268 and 269: 268 9. BIG ENTROPY AND THE GENERALI
Page 270 and 271: 270 9. BIG ENTROPY AND THE GENERALI
Page 273 and 274: 10 Distance and DurationA curious t
Page 275: 10.2. GAMMA 275can be quite complic
Page 278 and 279: 278 11. COUNTING AND CLASSIFICATION
Page 304 and 305: 304 12. MONSTERS AND MIXTUREShere?
Page 306 and 307: 306 12. MONSTERS AND MIXTURESdensit
Page 308 and 309: 308 12. MONSTERS AND MIXTURESfootbr
Page 310 and 311: 310 12. MONSTERS AND MIXTURESWhy di
Page 312 and 313: 312 12. MONSTERS AND MIXTURESm11.3
Page 314 and 315: 314 12. MONSTERS AND MIXTURESResult
Page 316 and 317: 316 12. MONSTERS AND MIXTURESanswer
Page 320 and 321: 320 12. MONSTERS AND MIXTURESBut we
Page 322 and 323: 322 12. MONSTERS AND MIXTURESdisper
Page 324 and 325: 324 12. MONSTERS AND MIXTURESe resu
Page 326 and 327: 326 12. MONSTERS AND MIXTURESDensit
Page 328 and 329: 328 12. MONSTERS AND MIXTURESR code
Page 330 and 331: 330 12. MONSTERS AND MIXTURESPoisso
Page 332 and 333: 332 12. MONSTERS AND MIXTURESthe ga
Page 334 and 335: 334 12. MONSTERS AND MIXTURESdata=d
Page 336 and 337: 336 12. MONSTERS AND MIXTURESma2 ~
Page 338 and 339: 338 13. MULTILEVEL MODELSrepeat obs
Page 340 and 341: 340 13. MULTILEVEL MODELSSo how do
Page 342 and 343: 342 13. MULTILEVEL MODELSprobabilit
Page 344 and 345: 344 13. MULTILEVEL MODELSa conseque
Page 346 and 347: 346 13. MULTILEVEL MODELSCompute th
Page 348 and 349: 348 13. MULTILEVEL MODELSabsolute e
Page 350 and 351: 350 13. MULTILEVEL MODELS)sigma_act
Page 353 and 354: 14 Multilevel Models II: Slopes14.1
Page 355 and 356: 14.1. EVERYTHING CAN VARY AND PROBA
Page 357: 14.1. EVERYTHING CAN VARY AND PROBA
Page 360 and 361: 360 15. MISSING DATA AND OTHER OPPO
Page 368 and 369:
368 15. MISSING DATA AND OTHER OPPO
Page 370 and 371:
Page 372 and 373:
Page 374 and 375:
Page 376 and 377:
Page 379:
16 Writing Statistics379
Page 382 and 383:
382 ENDNOTES12. For an autopsy of t
Page 384 and 385:
384 ENDNOTES45. Fisher (1925), in C
Page 386 and 387:
386 ENDNOTES77. See two famous edit
Page 388 and 389:
388 ENDNOTES113. Hurlbert (1984) is
Page 390 and 391:
390 BibliographyFrank, S. A. (2011)
Page 392 and 393:
392 BibliographyProulx, S. R. and A
Page 395 and 396:
IndexAkaike (1973), 380, 383Akaike
Page 397:
INDEX 397non-identifiability, 153Oc
show all

statisticalrethinkin..

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?