Probability:

Modeling and Applications to Random Processes

Gregory K. Miller

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Springer, 2006 - Probabilities - 459 pages

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Finite Mixture Modeling	1

12 Finite Mixture Distributions	3

122 Some Descriptive Features of Finite Mixture Distributions	5

123 Diagnosing Similarity of Mixture Components	9

125 Statistical Modeling Based on Finite Mixture Distributions	11

13 Identifiability of a Finite Mixture Distribution	14

131 Nonidentifiability Due to Invariance to Relabeling the Components	15

132 Nonidentifiability Due to Potential Overfitting	17

823 Statistical Modeling Based on Finite Mixture of Regression Models	246

824 Outliers in a Regression Model	249

832 Bayesian Inference When the Allocations Are Known	250

833 Choosing Prior Distributions	252

834 Bayesian Inference When the Allocations Are Unknown	253

835 Bayesian Inference Using Posterior Draws	254

836 Dealing with Model Specification Uncertainty	255

84 MixedEffects Finite Mixtures of Regression Models	256

133 Formal Identifiability Constraints	19

134 Generic Identifiability	21

Statistical Inference for a Finite Mixture Model with Known Number of Components	25

22 Classification for Known Component Parameters	26

222 The Bayes Classifier for a Whole Data Set	27

23 Parameter Estimation for Known Allocation	29

232 CompleteData Maximum Likelihood Estimation	30

233 CompleteData Bayesian Estimation of the Component Parameters	31

234 CompleteData Bayesian Estimation of the Weights	35

24 Parameter Estimation When the Allocations Are Unknown	41

241 Method of Moments	42

243 A Helicopter Tour of the Mixture Likelihood Surface for Two Examples	44

244 Maximum Likelihood Estimation	49

245 Bayesian Parameter Estimation	53

246 DistanceBased Methods	54

Practical Bayesian Inference for a Finite Mixture Model with Known Number of Components	57

32 Choosing the Prior for the Parameters of a Mixture Model	58

322 Improper Priors May Cause Improper Mixture Posteriors	59

323 Conditionally Conjugate Priors	60

325 Other Priors	62

33 Some Properties of the Mixture Posterior Density	63

332 Invariance of Seemingly ComponentSpecific Functionals	64

333 The Marginal Posterior Distribution of the Allocations	65

334 Invariance of the Posterior Distribution of the Allocations	67

34 Classification Without Parameter Estimation	68

341 SingleMove Gibbs Sampling	69

342 The MetropolisHastings Algorithm	72

35 Parameter Estimation Through Data Augmentation and MCMC 73	73

352 Data Augmentation and MCMC for a Mixture of Poisson Distributions	74

353 Data Augmentation and MCMC for General Mixtures	76

354 MCMC Sampling Under Improper Priors	78

356 Permutation MCMC Sampling	81

36 Other Monte Carlo Methods Useful for Mixture Models	83

362 Importance Sampling for the Allocations	84

363 Perfect Sampling	85

372 Using Posterior Draws for Bayesian Inference	87

373 Predictive Density Estimation	89

374 Individual Parameter Inference	91

375 Inference on the Hyperparameter of a Hierarchical Prior	92

377 Model Identification	94

Statistical Inference for Finite Mixture Models Under Model Specification Uncertainty	99

42 Parameter Estimation Under Model Specification Uncertainty	100

422 Practical Bayesian Parameter Estimation for Overﬁtting Finite Mixture Models	103

423 Potential Overﬁtting	105

43 Informal Methods for Identifying the Number of Components	107

431 Mode Hunting in the Mixture Posterior	108

432 Mode Hunting in the Sample Histogram	109

433 Diagnosing Mixtures Through the Method of Moments	110

434 Diagnosing Mixtures Through Predictive Methods	112

435 Further Approaches	114

442 AIC BIC and the Schwarz Criterion	116

443 Further Approaches	117

452 Marginal Likelihoods	118

453 Bayes Factors for Model Comparison	119

454 Formal Bayesian Model Selection	121

455 Choosing Priors for Model Selection	122

456 Further Approaches	123

Computational Tools for Bayesian Inference for Finite Mixtures Models Under Model Specification Uncertainty	124

521 ProductSpace MCMC	126

522 Reversible Jump MCMC	129

523 Birth and Death MCMC Methods	137

53 Marginal Likelihoods for Finite Mixture Models	139

532 Choosing Priors for Selecting the Number of Components	141

533 Computation of the Marginal Likelihood for Mixture Models	143

542 SamplingBased Approximations for Mixture Models	144

543 Importance Sampling	146

544 Reciprocal Importance Sampling	147

545 Harmonic Mean Estimator	148

546 Bridge Sampling Technique	150

547 Comparison of Different SimulationBased Estimators	154

548 Dealing with Hierarchical Priors	159

552 Chibs Estimator	160

553 Laplace Approximation	164

56 Reversible Jump MCMC Versus Marginal Likelihoods?	165

Finite Mixture Models with Normal Components	169

612 Parameter Estimation for Mixtures of Normals	171

614 Applications of Mixture of Normal Distributions	176

62 Bayesian Estimation of Univariate Mixtures of Normals	177

623 The Inﬂuence of the Prior on the Variance Ratio	179

624 Bayesian Estimation Using MCMC	180

626 Introducing Prior Dependence Among the Components	185

627 Further SamplingBased Approaches	187

628 Application to the Fishery Data	188

63 Bayesian Estimation of Multivariate Mixtures of Normals	190

632 Prior Distributions Standard Prior Distributions	192

633 Bayesian Parameter Estimation Using MCMC	193

634 Application to Fishers Iris Data	195

642 Model Selection Problems for Mixtures of Normals	199

Data Analysis Based on Finite Mixtures	203

712 ModelBased Clustering Using Finite Mixture Models	204

713 The Classification Likelihood and the Bayesian MAP Approach	207

714 Choosing Clustering Criteria and the Number of Components	210

715 Model Choice for the Fishery Data	216

716 Model Choice for Fishers Iris Data	218

717 Bayesian Clustering Based on Loss Functions	220

718 Clustering for Fishers Iris Data	224

722 Bayesian Inference for Outlier Models Based on Finite Mixtures	225

723 Outlier Modeling of Darwins Data	226

724 Clustering Under Outliers and Noise	227

731 Parameter Estimation	230

732 Dealing with Unknown Number of Components	233

742 Discriminant Analysis	235

743 Combining Classified and Unclassified Observations	236

744 Density Estimation Using Finite Mixtures	237

745 Finite Mixtures as an Auxiliary Computational Tool in Bayesian Analysis	238

Finite Mixtures of Regression Models	240

82 Finite Mixture of Multiple Regression Models	242

822 Identifiability	243

843 Bayesian Parameter Estimation When the Allocations Are Known	257

844 Bayesian Parameter Estimation When the Allocations Are Unknown	258

85 Finite Mixture Models for Repeated Measurements	259

851 Pooling Information Across Similar Units	260

853 Choosing the Prior for Bayesian Estimation	265

855 Practical Bayesian Estimation Using MCMC	267

856 Dealing with Model Specification Uncertainty	269

857 Application to the Marketing Data	270

86 Further Issues	273

862 Modeling the Weight Distribution	274

Finite Mixture Models with Nonnormal Components	277

912 Bayesian Inference	278

92 Finite Mixtures of Poisson Distributions	279

922 Capturing Overdispersion in Count Data	280

923 Modeling Excess Zeros	282

924 Application to the Eye Tracking Data	283

93 Finite Mixture Models for Binary and Categorical Data	286

932 Finite Mixtures of Multinomial Distributions	288

94 Finite Mixtures of Generalized Linear Models	289

941 Finite Mixture Regression Models for Count Data	290

942 Finite Mixtures of Logit and Probit Regression Models	292

943 Parameter Estimation for Finite Mixtures of GLMs	293

944 Model Selection for Finite Mixtures of GLMs	294

951 The Basic Latent Class Model	295

952 Identification and Parameter Estimation	296

953 Extensions of the Basic Latent Class Model	297

96 Further Issues	298

962 Finite Mixtures of GLMs with Random Effects	299

Finite Markov Mixture Modeling	300

1022 Irreducible Aperiodic Markov Chains	304

1023 Moments of a Markov Mixture Distribution	308

1024 The Autocorrelation Function of a Process Generated by a Markov Mixture Distribution	310

1025 The Autocorrelation Function of the Squared Process	311

1026 The Standard Finite Mixture Distribution as a Limiting Case	312

1027 Identifiability of a Finite Markov Mixture Distribution	313

103 Statistical Modeling Based on Finite Markov Mixture Distributions	314

1032 The Markov Switching Regression Model	315

1033 Nonergodic Markov Chains	316

Statistical Inference for Markov Switching Models	319

1121 Statistical Inference About the States	320

1123 Filtering for Special Cases	323

1124 Smoothing the States	324

1125 Filtering and Smoothing for More General Models	326

113 Parameter Estimation for Known States	327

1132 CompleteData Bayesian Parameter Estimation	329

114 Parameter Estimation When the States are Unknown	330

1142 Maximum Likelihood Estimation	333

1143 Bayesian Estimation	334

115 Bayesian Parameter Estimation with Known Number of States	335

1152 Some Properties of the Posterior Distribution of a Markov Switching Model	336

1153 Parameter Estimation Through Data Augmentation and MCMC	337

1154 Permutation MCMC Sampling	340

1156 Sampling Posterior Paths of the Hidden Markov Chain	342

1157 Other SamplingBased Approaches	345

116 Statistical Inference Under Model Speciﬁcation Uncertainty	346

1163 Marginal Likelihoods for Markov Switching Models	347

1164 Model Space MCMC	348

1172 Capturing Overdispersion and Autocorrelation Using Poisson Markov Mixture Models	349

1173 Application to the Lamb Data	351

Nonlinear Time Series Analysis Based on Markov Switching Models	357

122 The Markov Switching Autoregressive Model	358

1222 Model Deﬁnition	360

1223 Features of the MSAR Model	362

1224 Markov Switching Models for Nonstationary Time Series	363

1225 Parameter Estimation and Model Selection	365

123 Markov Switching Dynamic Regression Models	371

124 Prediction of Time Series Based on Markov Switching Models	372

1242 Forecasting of Markov Switching Models via SamplingBased Methods	374

1252 Capturing Features of Financial Time Series Through Markov Switching Models	377

1253 Switching ARCH Models	378

1254 Statistical Inference for Switching ARCH Models	380

1255 Switching GARCH Models	383

126 Some Extensions	384

1262 Markov Switching Models for Longitudinal and Panel Data	385

1263 Markov Switching Models for Multivariate Time Series	386

Switching State Space Models	389

1312 The Linear Gaussian State Space Form	391

1313 Multiprocess Models	393

1315 The General State Space Form	394

132 Nonlinear Time Series Analysis Based on Switching State Space Models	396

1323 Capturing Sudden Changes in Time Series	398

1324 Switching Dynamic Factor Models	400

1325 Switching State Space Models as a SemiParametric Smoothing Device	401

1331 The Filtering Problem	402

1333 Filtering for the Linear Gaussian State Space Model	404

1334 Filtering for Multiprocess Models	406

134 Parameter Estimation for Switching State Space Models	410

1341 The Likelihood Function of a State Space Model	411

1342 Maximum Likelihood Estimation	412

135 Practical Bayesian Estimation Using MCMC	415

1351 Various Data Augmentation Schemes	416

1352 Sampling the Continuous State Process from the Smoother Density	417

1353 Sampling the Discrete States for a Switching State Space Model	420

136 Further Issues	421

1362 Auxiliary Mixture Sampling for Nonlinear and Nonnormal State Space Models	422

137 Illustrative Application to Modeling Exchange Rate Data	423

Appendix A1 Summary of Probability Distributions	431

A12 The Binomial Distribution	432

A14 The Exponential Distribution	433

A16 The Gamma Distribution	434

A17 The Geometric Distribution	435

A110 The Normal Distribution	436

Al Summary of Probability Distributions 437	437

438 A Appendix	438

A2 Software	439

References	441

Index	481

Copyright

Other editions - View all

Probability: modeling and applications to random processes

Gregory K. Miller
Snippet view - 2006

Finite Mixture and Markov Switching Models

Sylvia Fr�hwirth-Schnatter
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Common terms and phrases

Algorithm allocations analysis applied assumed autocorrelation Bayes Bayes factor Bayesian estimation Bayesian Inference Chib classiﬁcation clustering coeﬃcients component parameters conjugate prior constraint covariance data augmentation deﬁned deﬁnition diﬀerent Dirichlet distribution discussed in Subsection eﬀects ﬁlter ﬁnite mixture models ﬁrst ﬁxed Fr�uhwirth-Schnatter Gaussian state space Gibbs sampling given heterogeneity hidden Markov chain hyperparameter identiﬁability importance density improper prior label switching likelihood function marginal distribution marginal likelihood marginal posterior Markov mixture model Markov switching models MCMC draws mixture density mixture likelihood function mixture of Poisson mixtures of normals mixtures of regression ML estimator multivariate mixtures multivariate normal distributions Nk(S normal distributions number of components observations obtained outliers overdispersion overﬁtting Parameter Estimation permutation Poisson distributions posterior distribution posterior probability Pr(Si Pr(St random variable regression model sampler simulation space models speciﬁc Statistical switching state space tion transition matrix unknown variance variance�covariance whereas

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Bibliographic information

Title	Probability: Modeling and Applications to Random Processes
Author	Gregory K. Miller
Publisher	Springer, 2006
ISBN	0387357688, 9780387357683
Length	459 pages

Export Citation	BiBTeX EndNote RefMan

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