Probability and bayesian modeling:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton ; London ; New York
CRC Press LLC
[2020]
|
| Series: | Chapman and Hall/CRC Texts in Statistical Science Ser
Texts in statistical science |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xiv, 537 Seiten Illustrationen, Diagramme |
| ISBN: | 9781138492561 |
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MARC
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| adam_text | Contents Preface xi 1 Probability: A Measurementof Uncertainty 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 Introduction ............................................................................... The Classical View of a Probability ........................................ The Frequency View of a Probability ..................................... The Subjective View of a Probability ..................................... The Sample Space...................................................................... Assigning Probabilities ............................................................. Events and Event Operations.................................................... The Three Probability Axioms................................................. The Complement and Addition Properties ............................ Exercises ..................................................................................... 2 Counting Methods 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Introduction: Rolling Dice, Yahtzee, and Roulette ............... Equally Likely Outcomes .......................................................... The Multiplication Counting Rule ........................................... Permutations............................................................................... Combinations ............................................................................ Arrangements of Non-Distinct Objects .................................. Playing Yahtzee ......................................................................... Exercises
............................................................ 3 Conditional Probability 3.1 Introduction: The Three Card Problem .................................. 3.2 In Everyday Life .......................................................................... 3.3 In a Two-Way Table................................................................... 3.4 Definition and the Multiplication Rule..................................... 3.5 The Multiplication Rule under Independence......................... 3.6 Learning Using Bayes’ Rule....................................................... 3.7 R Example: Learning abouta Spinner...................................... 3.8 Exercises ...................................................................................... 1 1 2 4 6 9 12 15 16 18 19 33 33 34 35 37 39 42 46 49 57 57 60 62 65 69 75 78 83 v
vi Contents 4 Discrete Distributions 4.1 4.2 4.3 4.4 4.5 4.6 Introduction: The Hat Check Problem........................... Random Variable and Probability Distribution...................... Summarizing a Probability Distribution........................ Standard Deviation of a Probability Distribution.................. Coin-Tossing Distributions ....................................................... 4.5.1 Binomial probabilities.................................................... 4.5.2 Binomial computations ................................................. 4.5.3 Mean and standard deviation of a binomial............... 4.5.4 Negative binomial experiments..................................... Exercises ..................................................................................... 5 Continuous Distributions 5.1 Introduction: A Baseball Spinner Game.................................. 5.2 The Uniform Distribution..................... 5.3 Probability Density: Waiting for a Bus .................................. 5.4 The Cumulative Distribution Function .................................. 5.5 Summarizing a Continuous Random Variable......................... 5.6 Normal Distribution................................................................... 5.7 Binomial Probabilities and the Normal Curve ...................... 5.8 Sampling Distribution of the Mean ........................................ 5.9 Exercises ..................................................................................... 6 Joint Probability Distributions 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 Introduction
............................................................................... Joint Probability Mass Function: Sampling from a Box . . . Multinomial Experiments................................. Joint Density Functions............................................................. Independence and Measuring Association............................... Flipping a Random Coin: The Beta-Binomial Distribution ............................................................................... Bivariate Normal Distribution ................................................. Exercises ..................................................................................... 7 Learning about a Binomial Probability 7.1 7.2 7.3 Introduction: Thinking Subjectively about a Proportion .................................................................................. Bayesian Inference with Discrete Priors .................................. 7.2.1 Example: students’ dining preference............................ 7.2.2 Discrete prior distributions for proportion p............... 7.2.3 Likelihood of proportion p.............................................. 7.2.4 Posterior distribution for proportion p......................... 7.2.5 Inference: students’ dining preference............................ 7.2.6 Discussion: using a discrete prior . ............................... Continuous Priors ...................................................................... 97 97 98 102 104 110 Ill 115 117 118 121 137 137 139 143 146 149 151 157 161 169 185 185 185 191 195 200 202 205 209 217 217 221 221 221 224 225 227
228 229
Contents 7.4 7.5 7.6 7.7 7.3.1 The beta distribution and probabilities......................... 7.3.2 Choosing a beta density to represent prior opinion . . Updating the Beta Prior .......................................................... 7.4.1 Bayes’ rule calculation.................................................... 7.4.2 Prom beta prior to beta posterior: conjugate priors . . Bayesian Inferences with Continuous Priors............................ 7.5.1 Bayesian hypothesis testing........................................... 7.5.2 Bayesian credible intervals.............................................. 7.5.3 Bayesian prediction ....................................................... Predictive Checking ................................................................... Exercises . .................................................................................. 8 Modeling Measurement and Count Data 8.1 8.2 Introduction ............................................................................... Modeling Measurements............................................................. 8.2.1 Examples......................................................................... 8.2.2 The general approach.................................................... 8.2.3 Outline of chapter................................. 8.3 Bayesian Inference with Discrete Priors .................................. 8.3.1 Example: Roger Federer’s time-to-serve..................... 8.3.2 Simplification of the likelihood.................................... 8.3.3 Inference: Federer’s time-to-
serve.................................. 8.4 Continuous Priors ...................................................................... 8.4.1 The normal prior for mean μ........................................ 8.4.2 Choosing a normal prior................................................. 8.5 Updating the Normal Prior....................................................... 8.5.1 Introduction ................................................................... 8.5.2 A quick peak at the update procedure......................... 8.5.3 Bayes’ rule calculation.................................................... 8.5.4 Conjugate normal prior................................................. 8.6 Bayesian Inferences for Continuous Normal Mean ............... 8.6.1 Bayesian hypothesis testing and credible interval . . . 8.6.2 Bayesian prediction ....................................................... 8.7 Posterior Predictive Checking ................................................. 8.8 Modeling Count Data................................................................ 8.8.1 Examples........................ 8.8.2 The Poisson distribution................................................. 8.8.3 Bayesian inferences.......................................................... 8.8.4 Case study: Learning about website counts............... 8.9 Exercises .................................................................................... vii 231 234 237 238 239 242 243 244 247 250 256 267 267 267 267 269 270 271 271 275 278 278 278 279 281 281 282 285 286 288 288 290 292 294 295 296 297 300 301
Contents viii 9 Simulation by Markov Chain Monte Carlo 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Introduction ............................................................. ................. 9.1.1 The Bayesian computation problem ...................... 9.1.2 Choosing a prior............................................................. 9.1.3 The two-parameter normal problem............................ 9.1.4 Overview of the chapter................................................. Markov Chains ............................................................................. 9.2.1 Definition.......................................................................... 9.2.2 Some properties . . . ..................................................... 9.2.3 Simulating a Markov chain........................................... The Metropolis Algorithm ....................................................... 9.3.1 Example: Walking on a number line . ................... 9.3.2 The general algorithm...................... ............................. 9.3.3 A general function for the Metropolisalgorithm .... Example: Cauchy-Normal Problem........................................... 9.4.1 Choice of starting value and proposal region................ 9.4.2 Collecting the simulated draws..................................... Gibbs Sampling .......................................................................... 9.5.1 Bivariate discrete distribution........................................ 9.5.2 Beta-binomial sampling................................................. 9.5.3 Normal sampling - both
parametersunknown.............. MCMC Inputs and Diagnostics................................................. 9.6.1 Burn-in, starting values, and multiplechains ..... 9.6.2 Diagnostics....................................................................... 9.6.3 Graphs and summaries.................................................... Using JAGS ................................................................................ 9.7.1 Normal sampling model.......................... ...................... 9.7.2 Multiple chains................................................................ 9.7.3 Posterior predictive checking........................................ 9.7.4 Comparing two proportions........................................... Exercises ...................................................................................... 10 Bayesian Hierarchical Modeling 10.1 Introduction ................................................................................ 10.1.1 Observations in groups.................................................... 10.1.2 Example: standardized test scores ............................... 10.1.3 Separate estimates? ....................................................... 10.1.4 Combined estimates?........................ 10.1.5 A two-stage prior leading to compromise estimates . . 10.2 Hierarchical Normal Modeling ................................................. 10.2.1 Example: ratings of animation movies......................... 10.2.2 A hierarchical Normal model with random σ............ 10.2.3 Inference through
MCMC.............................................. 10.3 Hierarchical Beta-Binomial Modeling ..................................... 10.3.1 Example: Deaths after heart attacks............................ 313 313 313 313 315 316 317 317 318 319 320 320 323 326 326 327 329 330 330 332 333 338 338 338 339 341 342 345 347 349 354 365 365 365 366 366 367 367 369 369 370 374 381 381
Contents ix 10.3.2 A hierarchical beta-binomial model............................... 10.3.3 Inference through MCMC.............................................. 10.4 Exercises ..................................................................................... 381 385 393 11 Simple Linear Regression 11.1 Introduction ................................................................................ 11.2 Example: Prices and Areas of House Sales ............................ 11.3 A Simple Linear Regression Model........................................... 11.4 A Weakly Informative Prior .................................................... 11.5 Posterior Analysis ...................................................................... 11.6 Inference through MCMC.......................................................... 11.7 Bayesian Inferences with Simple Linear Regression............... 11.7.1 Simulate fits from the regression model ..................... 11.7.2 Learning about the expected response........................ 11.7.3 Prediction of future response........................................ 11.7.4 Posterior predictive model checking ............................ 11.8 Informative Prior ...................................................................... 11.8.1 Standardization................................................................ 11.8.2 Prior distributions.......................................................... 11.8.3 Posterior Analysis.......................................................... 11.9 A Conditional Means
Prior....................................................... ll.lOExercises .............................. 409 409 412 413 414 415 416 420 420 421 423 425 427 428 429 431 433 437 12 Bayesian Multiple Regression and Logistic Models 12.1 Introduction ............................................................................... 12.2 Bayesian Multiple Linear Regression........................................ 12.2.1 Example: expenditures of U.S.households.................... 12.2.2 A multiple linear regression model............................... 12.2.3 Weakly informative priors and inference through MCMC............................................................................ 12.2.4 Prediction......................................................................... 12.3 Comparing Regression Models .................................................. 12.4 Bayesian Logistic Regression .................................................... 12.4.1 Example: U.S. women labor participation................... 12.4.2 A logistic regression model .................. 12.4.3 Conditional means priors and inference through MCMC 12.4.4 Prediction......................................................................... 12.5 Exercises ...................................................................................... 449 449 449 449 451 13 Case Studies 13.1 Introduction ............ ... . . . .................................................... 13.2 Federalist Papers Study.................................... 13.2.1
Introduction ................................................................... 13.2.2 Data on word use............................................................. - 13.2.3 Poisson density sampling ............................................. 487 487 488 488 488 489 453 457 459 465 465 467 469 475 477
x Contents 13.2.4 Negative binomial sampling........................................... 13.2.5 Comparison of rates for two authors . ......................... 13.2.6 Which words distinguish the two authors?................... 13.3 Career Trajectories ................................................................... 13.3.1 Introduction ................................................................... 13.3.2 Measuring hitting performance inbaseball.................. 13.3.3 A hitter’s career trajectory........................................... 13.3.4 Estimating a single trajectory........................................ 13.3.5 Estimating many trajectories by a hierarchical model 13.4 Latent Class Modeling ............................................................. 13.4.1 Two classes of test takers.............................................. 13.4.2 A latent class model with two classes . ...................... 13.4.3 Disputed authorship of the Federalist Papers............ 13.5 Exercises ......................................................... 491 494 496 497 497 498 498 499 502 505 505 509 514 517 14 Appendices 14.1 Appendix A: The constant in the beta posterior .................. 14.2 Appendix B: The posterior predictive distribution ............... 14.3 Appendix C: Comparing Bayesian models............................... 525 525 526 527 Bibliography 529 Index 531
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| spelling | Albert, Jim 1953- Verfasser (DE-588)133457834 aut Probability and bayesian modeling Boca Raton ; London ; New York CRC Press LLC [2020] xiv, 537 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Chapman and Hall/CRC Texts in Statistical Science Ser Texts in statistical science Bayes-Verfahren (DE-588)4204326-8 gnd rswk-swf Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd rswk-swf Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd rswk-swf Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd rswk-swf (DE-588)4123623-3 Lehrbuch gnd-content Wahrscheinlichkeitsrechnung (DE-588)4064324-4 s Wahrscheinlichkeitstheorie (DE-588)4079013-7 s Bayes-Verfahren (DE-588)4204326-8 s Bayes-Entscheidungstheorie (DE-588)4144220-9 s b DE-604 Hu, Jingchen Verfasser (DE-588)1204385017 aut Erscheint auch als Online-Ausgabe 978-1-351-03013-7 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=031723601&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Albert, Jim 1953- Hu, Jingchen Probability and bayesian modeling Bayes-Verfahren (DE-588)4204326-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| subject_GND | (DE-588)4204326-8 (DE-588)4064324-4 (DE-588)4144220-9 (DE-588)4079013-7 (DE-588)4123623-3 |
| title | Probability and bayesian modeling |
| title_auth | Probability and bayesian modeling |
| title_exact_search | Probability and bayesian modeling |
| title_full | Probability and bayesian modeling |
| title_fullStr | Probability and bayesian modeling |
| title_full_unstemmed | Probability and bayesian modeling |
| title_short | Probability and bayesian modeling |
| title_sort | probability and bayesian modeling |
| topic | Bayes-Verfahren (DE-588)4204326-8 gnd Wahrscheinlichkeitsrechnung (DE-588)4064324-4 gnd Bayes-Entscheidungstheorie (DE-588)4144220-9 gnd Wahrscheinlichkeitstheorie (DE-588)4079013-7 gnd |
| topic_facet | Bayes-Verfahren Wahrscheinlichkeitsrechnung Bayes-Entscheidungstheorie Wahrscheinlichkeitstheorie Lehrbuch |
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