Springer Statistics

By: Sylvia FrÃ¼hwirth-Schnatter

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Hardcover | 8 August 2006

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WINNER OF THE 2007 DEGROOT PRIZE

The prominence of finite mixture modelling is greater than ever. Many important statistical topics like clustering data, outlier treatment, or dealing with unobserved heterogeneity involve finite mixture models in some way or other. The area of potential applications goes beyond simple data analysis and extends to regression analysis and to non-linear time series analysis using Markov switching models.

For more than the hundred years since Karl Pearson showed in 1894 how to estimate the five parameters of a mixture of two normal distributions using the method of moments, statistical inference for finite mixture models has been a challenge to everybody who deals with them. In the past ten years, very powerful computational tools emerged for dealing with these models which combine a Bayesian approach with recent Monte simulation techniques based on Markov chains. This book reviews these techniques and covers the most recent advances in the field, among them bridge sampling techniques and reversible jump Markov chain Monte Carlo methods.

It is the first time that the Bayesian perspective of finite mixture modelling is systematically presented in book form. It is argued that the Bayesian approach provides much insight in this context and is easily implemented in practice. Although the main focus is on Bayesian inference, the author reviews several frequentist techniques, especially selecting the number of components of a finite mixture model, and discusses some of their shortcomings compared to the Bayesian approach.

The aim of this book is to impart the finite mixture and Markov switching approach to statistical modelling to a wide-ranging community. This includes not only statisticians, but also biologists, economists, engineers, financial agents, market researcher, medical researchers or any other frequent user of statistical models. This book should help newcomers to the field to understand how finite mixture and Markov switching models are formulated, what structures they imply on the data, what they could be used for, and how they are estimated. Researchers familiar with the subject also will profit from reading this book. The presentation is rather informal without abandoning mathematical correctness. Previous notions of Bayesian inference and Monte Carlo simulation are useful but not needed.

Industry Reviews

From the reviews:

"At first glance, the numerous equations and formulas may seem to be daunting for psychologists with limited statistical background; however, the descriptions and explanations of the various models are actually quite reader friendly (more so than many advanced statistical textbooks). The author has done an excellent job of inviting newcomers to enter the world of mixture models, more impressively, the author did so without sacrificing mathematical and statistical rigor. Mixture models are appealing in many applications in social and psychological studies. This book not only offers a gentle introduction to mixture models but also provides more in depth coverage for those who look beyond the surface. I believe that psychologists who are interested in related models (e.g., latent class models, latent Markov models, and latent class regression models) will benefit greatly from this book. I highly recommend this book to all psychologists who are interested in mixture models." (Hsiu-Ting Yu, PSYCHOMETRIKA-VOL. 74, NO. 3, 559-560 SEPTEMBER 2009)

"The book is impressive in its mathematical and formal correctness, in generality and in details....it would be helfpful as an additional reference among a wider range of available textbooks in the area. [I]t will find many friends among experts and newcomers to the world of mixture models." (Atanu Biswas, Biometrics, Issue 63, September 2007)

"Finite mixture distributions are important for many models. Therefore they constitute a very active field of research. This book gives an up to date overview over the various models of this kind. ... The aim of this book is to impart the finite mixture and Markov switching approach to statistical modeling to a wide-ranging community. ... For the frequentists, it offers a good opportunity to explore the advantages of the Bayesian approach in the context of mixing models." (Gheorghe Pitis, Zentralblatt MATH, Vol. 1108(10), 2007)

"Readership: Statisticians, biologists, economists, engineers, financial agents, market researchers, medical researchers or any other frequent user of statistical models. The first nine chapters of the book are concerned with static mixture models, and the last four with Markov switching models. ... especially valuable for students, serving to demonstrate how different statistical techniques, which superficially appear to be unrelated, are in fact part of an integrated whole. This book struck me as being particularly clearly written - it is a pleasure to read." (David J. Hand, International Statistical Review, Vol. 75 (2), 2007)

"The book is excellent, giving a most readable overview of the topic of finite mixtures, aimed at a broad readership ... . Students will like the text because of the pedagogical writing style; researchers will definitely welcome the broad treatment of the subject. Both will benefit from the extensive and up-to-date bibliography ... as well as the well-organized index. No doubt, this book is a valuable addition to the field of statistics and will surely find its rightful place in many a statistician's library." (Valerie Chavez-Demoulin, Journal of the American Statistical Association, Vol. 104 (485), March, 2009)

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Published: 19th November 2010

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Finite Mixture Modeling	p. 1
Introduction	p. 1
Finite Mixture Distributions	p. 3
Basic Definitions	p. 3
Some Descriptive Features of Finite Mixture Distributions	p. 5
Diagnosing Similarity of Mixture Components	p. 9
Moments of a Finite Mixture Distribution	p. 10
Statistical Modeling Based on Finite Mixture Distributions	p. 11
Identifiability of a Finite Mixture Distribution	p. 14
Nonidentifiability Due to Invariance to Relabeling the Components	p. 15
Nonidentifiability Due to Potential Overfitting	p. 17
Formal Identifiability Constraints	p. 19
Generic Identifiability	p. 21
Statistical Inference for a Finite Mixture Model with Known Number of Components	p. 25
Introduction	p. 25
Classification for Known Component Parameters	p. 26
Bayes' Rule for Classifying a Single Observation	p. 26
The Bayes' Classifier for a Whole Data Set	p. 27
Parameter Estimation for Known Allocation	p. 29
The Complete-Data Likelihood Function	p. 29
Complete-Data Maximum Likelihood Estimation	p. 30
Complete-Data Bayesian Estimation of the Component Parameters	p. 31
Complete-Data Bayesian Estimation of the Weights	p. 35
Parameter Estimation When the Allocations Are Unknown	p. 41
Method of Moments	p. 42
The Mixture Likelihood Function	p. 43
A Helicopter Tour of the Mixture Likelihood Surface for Two Examples	p. 44
Maximum Likelihood Estimation	p. 49
Bayesian Parameter Estimation	p. 53
Distance-Based Methods	p. 54
Comparing Various Estimation Methods	p. 54
Practical Bayesian Inference for a Finite Mixture Model with Known Number of Components	p. 57
Introduction	p. 57
Choosing the Prior for the Parameters of a Mixture Model	p. 58
Objective and Subjective Priors	p. 58
Improper Priors May Cause Improper Mixture Posteriors	p. 59
Conditionally Conjugate Priors	p. 60
Hierarchical Priors and Partially Proper Priors	p. 61
Other Priors	p. 62
Invariant Prior Distributions	p. 62
Some Properties of the Mixture Posterior Density	p. 63
Invariance of the Posterior Distribution	p. 63
Invariance of Seemingly Component-Specific Functionals	p. 64
The Marginal Posterior Distribution of the Allocations	p. 65
Invariance of the Posterior Distribution of the Allocations	p. 67
Classification Without Parameter Estimation	p. 68
Single-Move Gibbs Sampling	p. 69
The Metropolis-Hastings Algorithm	p. 72
Parameter Estimation Through Data Augmentation and MCMC	p. 73
Treating Mixture Models as a Missing Data Problem	p. 73
Data Augmentation and MCMC for a Mixture of Poisson Distributions	p. 74
Data Augmentation and MCMC for General Mixtures	p. 76
MCMC Sampling Under Improper Priors	p. 78
Label Switching	p. 78
Permutation MCMC Sampling	p. 81
Other Monte Carlo Methods Useful for Mixture Models	p. 83
A Metropolis-Hastings Algorithm for the Parameters	p. 83
Importance Sampling for the Allocations	p. 84
Perfect Sampling	p. 85
Bayesian Inference for Finite Mixture Models Using Posterior Draws	p. 85
Sampling Representations of the Mixture Posterior Density	p. 85
Using Posterior Draws for Bayesian Inference	p. 87
Predictive Density Estimation	p. 89
Individual Parameter Inference	p. 91
Inference on the Hyperparameter of a Hierarchical Prior	p. 92
Inference on Component Parameters	p. 92
Model Identification	p. 94
Statistical Inference for Finite Mixture Models Under Model Specification Uncertainty	p. 99
Introduction	p. 99
Parameter Estimation Under Model Specification Uncertainty	p. 100
Maximum Likelihood Estimation Under Model Specification Uncertainty	p. 100
Practical Bayesian Parameter Estimation for Overfitting Finite Mixture Models	p. 103
Potential Overfitting	p. 105
Informal Methods for Identifying the Number of Components	p. 107
Mode Hunting in the Mixture Posterior	p. 108
Mode Hunting in the Sample Histogram	p. 109
Diagnosing Mixtures Through the Method of Moments	p. 110
Diagnosing Mixtures Through Predictive Methods	p. 112
Further Approaches	p. 114
Likelihood-Based Methods	p. 114
The Likelihood Ratio Statistic	p. 114
AIC, BIC, and the Schwarz Criterion	p. 116
Further Approaches	p. 117
Bayesian Inference Under Model Uncertainty	p. 117
Trans-Dimensional Bayesian Inference	p. 117
Marginal Likelihoods	p. 118
Bayes Factors for Model Comparison	p. 119
Formal Bayesian Model Selection	p. 121
Choosing Priors for Model Selection	p. 122
Further Approaches	p. 123
Computational Tools for Bayesian Inference for Finite Mixtures Models Under Model Specification Uncertainty	p. 125
Introduction	p. 125
Trans-Dimensional Markov Chain Monte Carlo Methods	p. 125
Product-Space MCMC	p. 126
Reversible Jump MCMC	p. 129
Birth and Death MCMC Methods	p. 137
Marginal Likelihoods for Finite Mixture Models	p. 139
Defining the Marginal Likelihood	p. 139
Choosing Priors for Selecting the Number of Components	p. 141
Computation of the Marginal Likelihood for Mixture Models	p. 143
Simulation-Based Approximations of the Marginal Likelihood	p. 143
Some Background on Monte Carlo Integration	p. 143
Sampling-Based Approximations for Mixture Models	p. 144
Importance Sampling	p. 146
Reciprocal Importance Sampling	p. 147
Harmonic Mean Estimator	p. 148
Bridge Sampling Technique	p. 150
Comparison of Different Simulation-Based Estimators	p. 154
Dealing with Hierarchical Priors	p. 159
Approximations to the Marginal Likelihood Based on Density Ratios	p. 159
The Posterior Density Ratio	p. 159
Chib's Estimator	p. 160
Laplace Approximation	p. 164
Reversible Jump MCMC Versus Marginal Likelihoods?	p. 165
Finite Mixture Models with Normal Components	p. 169
Finite Mixtures of Normal Distributions	p. 169
Model Formulation	p. 169
Parameter Estimation for Mixtures of Normals	p. 171
The Kiefer-Wolfowitz Example	p. 174
Applications of Mixture of Normal Distributions	p. 176
Bayesian Estimation of Univariate Mixtures of Normals	p. 177
Bayesian Inference When the Allocations Are Known	p. 177
Standard Prior Distributions	p. 179
The Influence of the Prior on the Variance Ratio	p. 179
Bayesian Estimation Using MCMC	p. 180
MCMC Estimation Under Standard Improper Priors	p. 182
Introducing Prior Dependence Among the Components	p. 185
Further Sampling-Based Approaches	p. 187
Application to the Fishery Data	p. 188
Bayesian Estimation of Multivariate Mixtures of Normals	p. 190
Bayesian Inference When the Allocations Are Known	p. 190
Prior Distributions	p. 192
Bayesian Parameter Estimation Using MCMC	p. 193
Application to Fisher's Iris Data	p. 195
Further Issues	p. 195
Parsimonious Finite Normal Mixtures	p. 195
Model Selection Problems for Mixtures of Normals	p. 199
Data Analysis Based on Finite Mixtures	p. 203
Model-Based Clustering	p. 203
Some Background on Cluster Analysis	p. 203
Model-Based Clustering Using Finite Mixture Models	p. 204
The Classification Likelihood and the Bayesian MAP Approach	p. 207
Choosing Clustering Criteria and the Number of Components	p. 210
Model Choice for the Fishery Data	p. 216
Model Choice for Fisher's Iris Data	p. 218
Bayesian Clustering Based on Loss Functions	p. 220
Clustering for Fisher's Iris Data	p. 224
Outlier Modeling	p. 224
Outlier Modeling Using Finite Mixtures	p. 224
Bayesian Inference for Outlier Models Based on Finite Mixtures	p. 225
Outlier Modeling of Darwin's Data	p. 226
Clustering Under Outliers and Noise	p. 227
Robust Finite Mixtures Based on the Student-t Distribution	p. 230
Parameter Estimation	p. 230
Dealing with Unknown Number of Components	p. 233
Further Issues	p. 233
Clustering High-Dimensional Data	p. 233
Discriminant Analysis	p. 235
Combining Classified and Unclassified Observations	p. 236
Density Estimation Using Finite Mixtures	p. 237
Finite Mixtures as an Auxiliary Computational Tool in Bayesian Analysis	p. 238
Finite Mixtures of Regression Models	p. 241
Introduction	p. 241
Finite Mixture of Multiple Regression Models	p. 242
Model Definition	p. 242
Identifiability	p. 243
Statistical Modeling Based on Finite Mixture of Regression Models	p. 246
Outliers in a Regression Model	p. 249
Statistical Inference for Finite Mixtures of Multiple Regression Models	p. 249
Maximum Likelihood Estimation	p. 249
Bayesian Inference When the Allocations Are Known	p. 250
Choosing Prior Distributions	p. 252
Bayesian Inference When the Allocations Are Unknown	p. 253
Bayesian Inference Using Posterior Draws	p. 254
Dealing with Model Specification Uncertainty	p. 255
Mixed-Effects Finite Mixtures of Regression Models	p. 256
Model Definition	p. 256
Choosing Priors for Bayesian Estimation	p. 256
Bayesian Parameter Estimation When the Allocations Are Known	p. 257
Bayesian Parameter Estimation When the Allocations Are Unknown	p. 258
Finite Mixture Models for Repeated Measurements	p. 259
Pooling Information Across Similar Units	p. 260
Finite Mixtures of Random-Effects Models	p. 260
Choosing the Prior for Bayesian Estimation	p. 265
Bayesian Parameter Estimation When the Allocations Are Known	p. 265
Practical Bayesian Estimation Using MCMC	p. 267
Dealing with Model Specification Uncertainty	p. 269
Application to the Marketing Data	p. 270
Further Issues	p. 273
Regression Modeling Based on Multivariate Mixtures of Normals	p. 273
Modeling the Weight Distribution	p. 274
Mixtures-of-Experts Models	p. 274
Finite Mixture Models with Nonnormal Components	p. 277
Finite Mixtures of Exponential Distributions	p. 277
Model Formulation and Parameter Estimation	p. 277
Bayesian Inference	p. 278
Finite Mixtures of Poisson Distributions	p. 279
Model Formulation and Estimation	p. 279
Capturing Overdispersion in Count Data	p. 280
Modeling Excess Zeros	p. 282
Application to the Eye Tracking Data	p. 283
Finite Mixture Models for Binary and Categorical Data	p. 286
Finite Mixtures of Binomial Distributions	p. 286
Finite Mixtures of Multinomial Distributions	p. 288
Finite Mixtures of Generalized Linear Models	p. 289
Finite Mixture Regression Models for Count Data	p. 290
Finite Mixtures of Logit and Probit Regression Models	p. 292
Parameter Estimation for Finite Mixtures of GLMs	p. 293
Model Selection for Finite Mixtures of GLMs	p. 294
Finite Mixture Models for Multivariate Binary and Categorical Data	p. 294
The Basic Latent Class Model	p. 295
Identification and Parameter Estimation	p. 296
Extensions of the Basic Latent Class Model	p. 297
Further Issues	p. 298
Finite Mixture Modeling of Mixed-Mode Data	p. 298
Finite Mixtures of GLMs with Random Effects	p. 299
Finite Markov Mixture Modeling	p. 301
Introduction	p. 301
Finite Markov Mixture Distributions	p. 301
Basic Definitions	p. 302
Irreducible Aperiodic Markov Chains	p. 304
Moments of a Markov Mixture Distribution	p. 308
The Autocorrelation Function of a Process Generated by a Markov Mixture Distribution	p. 310
The Autocorrelation Function of the Squared Process	p. 311
The Standard Finite Mixture Distribution as a Limiting Case	p. 312
Identifiability of a Finite Markov Mixture Distribution	p. 313
Statistical Modeling Based on Finite Markov Mixture Distributions	p. 314
The Basic Markov Switching Model	p. 314
The Markov Switching Regression Model	p. 315
Nonergodic Markov Chains	p. 316
Relaxing the Assumptions of the Basic Markov Switching Model	p. 316
Statistical Inference for Markov Switching Models	p. 319
Introduction	p. 319
State Estimation for Known Parameters	p. 319
Statistical Inference About the States	p. 320
Filtered State Probabilities	p. 320
Filtering for Special Cases	p. 323
Smoothing the States	p. 324
Filtering and Smoothing for More General Models	p. 326
Parameter Estimation for Known States	p. 327
The Complete-Data Likelihood Function	p. 327
Complete-Data Bayesian Parameter Estimation	p. 329
Complete-Data Bayesian Estimation of the Transition Matrix	p. 329
Parameter Estimation When the States are Unknown	p. 330
The Markov Mixture Likelihood Function	p. 330
Maximum Likelihood Estimation	p. 333
Bayesian Estimation	p. 334
Alternative Estimation Methods	p. 334
Bayesian Parameter Estimation with Known Number of States	p. 335
Choosing the Prior for the Parameters of a Markov Mixture Model	p. 335
Some Properties of the Posterior Distribution of a Markov Switching Model	p. 336
Parameter Estimation Through Data Augmentation and MCMC	p. 337
Permutation MCMC Sampling	p. 340
Sampling the Unknown Transition Matrix	p. 340
Sampling Posterior Paths of the Hidden Markov Chain	p. 342
Other Sampling-Based Approaches	p. 345
Bayesian Inference Using Posterior Draws	p. 345
Statistical Inference Under Model Specification Uncertainty	p. 346
Diagnosing Markov Switching Models	p. 346
Likelihood-Based Methods	p. 346
Marginal Likelihoods for Markov Switching Models	p. 347
Model Space MCMC	p. 348
Further Issues	p. 348
Modeling Overdispersion and Autocorrelation in Time Series of Count Data	p. 348
Motivating Example	p. 348
Capturing Overdispersion and Autocorrelation Using Poisson Markov Mixture Models	p. 349
Application to the Lamb Data	p. 351
Nonlinear Time Series Analysis Based on Markov Switching Models	p. 357
Introduction	p. 357
The Markov Switching Autoregressive Model	p. 358
Motivating Example	p. 358
Model Definition	p. 360
Features of the MSAR Model	p. 362
Markov Switching Models for Nonstationary Time Series	p. 363
Parameter Estimation and Model Selection	p. 365
Application to Business Cycle Analysis of the U.S. GDP Data	p. 365
Markov Switching Dynamic Regression Models	p. 371
Model Definition	p. 371
Bayesian Estimation	p. 371
Prediction of Time Series Based on Markov Switching Models	p. 372
Flexible Predictive Distributions	p. 372
Forecasting of Markov Switching Models via Sampling-Based Methods	p. 374
Markov Switching Conditional Heteroscedasticity	p. 375
Motivating Example	p. 375
Capturing Features of Financial Time Series Through Markov Switching Models	p. 377
Switching ARCH Models	p. 378
Statistical Inference for Switching ARCH Models	p. 380
Switching GARCH Models	p. 383
Some Extensions	p. 384
Time-Varying Transition Matrices	p. 384
Markov Switching Models for Longitudinal and Panel Data	p. 385
Markov Switching Models for Multivariate Time Series	p. 386
Switching State Space Models	p. 389
State Space Modeling	p. 389
The Local Level Model with and Without Switching	p. 389
The Linear Gaussian State Space Form	p. 391
Multiprocess Models	p. 393
Switching Linear Gaussian State Space Models	p. 393
The General State Space Form	p. 394
Nonlinear Time Series Analysis Based on Switching State Space Models	p. 396
ARMA Models with and Without Switching	p. 396
Unobserved Component Time Series Models	p. 397
Capturing Sudden Changes in Time Series	p. 398
Switching Dynamic Factor Models	p. 400
Switching State Space Models as a Semi-Parametric Smoothing Device	p. 401
Filtering for Switching Linear Gaussian State Space Models	p. 401
The Filtering Problem	p. 402
Bayesian Inference for a General Linear Regression Model	p. 402
Filtering for the Linear Gaussian State Space Model	p. 404
Filtering for Multiprocess Models	p. 406
Approximate Filtering for Switching Linear Gaussian State Space Models	p. 406
Parameter Estimation for Switching State Space Models	p. 410
The Likelihood Function of a State Space Model	p. 411
Maximum Likelihood Estimation	p. 412
Bayesian Inference	p. 412
Practical Bayesian Estimation Using MCMC	p. 415
Various Data Augmentation Schemes	p. 416
Sampling the Continuous State Process from the Smoother Density	p. 417
Sampling the Discrete States for a Switching State Space Model	p. 420
Further Issues	p. 421
Model Specification Uncertainty in Switching State Space Modeling	p. 421
Auxiliary Mixture Sampling for Nonlinear and Nonnormal State Space Models	p. 422
Illustrative Application to Modeling Exchange Rate Data	p. 423
Appendix	p. 431
Summary of Probability Distributions	p. 431
The Beta Distribution	p. 431
The Binomial Distribution	p. 432
The Dirichlet Distribution	p. 432
The Exponential Distribution	p. 433
The F-Distribution	p. 433
The Gamma Distribution	p. 434
The Geometric Distribution	p. 435
The Multinomial Distribution	p. 435
The Negative Binomial Distribution	p. 435
The Normal Distribution	p. 436
The Poisson Distribution	p. 437
The Student-t Distribution	p. 437
The Uniform Distribution	p. 438
The Wishart Distribution	p. 438
Software	p. 439
References	p. 441
Index	p. 481
Table of Contents provided by Ingram. All Rights Reserved.

Springer Statistics

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Hardcover

Industry Reviews

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How to return your order

Defective items

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