Mixtures Estimation and Applications

, by Mengersen, Kerrie L.; Robert, Christian; Titterington, Mike

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ISBN: 9781119993896 | 111999389X
Cover: Hardcover
Copyright: 6/13/2011

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This book uses the EM (expectation maximization) algorithm to simultaneously estimate the missing data and unknown parameter(s) associated with a data set. The parameters describe the component distributions of the mixture; the distributions may be continuous or discrete. The editors provide a complete account of the applications, mathematical structure and statistical analysis of finite mixture distributions along with MCMC computational methods, together with a range of detailed discussions covering the applications of the methods and features chapters from the leading experts on the subject. The applications are drawn from scientific discipline, including biostatistics, computer science, ecology and finance. This area of statistics is important to a range of disciplines, and its methodology attracts interest from researchers in the fields in which it can be applied.

Preface	p. xiii
Acknowledgements	p. xv
List of contributors	p. xvii
The Em algorithm, variational approximations and expectation propagation for mixtures	p. 1
Preamble	p. 1
The Em algorithm	p. 2
Introduction to the algorithm	p. 2
The E-step and the M-step for the mixing weights	p. 3
The M-step for mixtures of univariate Gaussian distributions	p. 4
M-step for mixtures of regular exponential family distributions formulated in terms of the natural parameters	p. 6
Application to other mixtures	p. 7
Em as a double expectation	p. 8
Variational approximations	p. 8
Preamble	p. 8
Introduction to variational approximations	p. 9
Application of variational Bayes to mixture problems	p. 11
Application to other mixture problems	p. 16
Recursive variational approximations	p. 18
Asymptotic results	p. 19
Expectation-propagation	p. 20
Introduction	p. 20
Overview of the recursive approach to be adopted	p. 22
Finite Gaussian mixtures with an unknown mean parameter	p. 23
Mixture of two known distributions	p. 24
Discussion	p. 26
Acknowledgements	p. 27
References	p. 27
Online expectation maximisation	p. 31
Introduction	p. 31
Model and assumptions	p. 33
The EM algorithm and the limiting Em recursion	p. 36
The batch Em algorithm	p. 36
The limiting Em recursion	p. 37
Limitations of batch Em for long data records	p. 38
Online expectation maximisation	p. 40
The algorithm	p. 40
Convergence properties	p. 41
Application to finite mixtures	p. 45
Use for batch maximum-likelihood estimation	p. 47
Discussion	p. 50
References	p. 52
The limiting distribution of the Em test of the order of a finite mixture	p. 55
Introduction	p. 55
The method and theory of the Em test	p. 56
The definition of the Em test statistic	p. 57
The limiting distribution of the EM test statistic	p. 58
Proofs	p. 60
Discussion	p. 74
	p. 74
Comparing Wald and likelihood regions applied to locally identifiable mixture models	p. 77
Introduction	p. 77
Background on likelihood confidence regions	p. 79
Likelihood regions	p. 79
Profile likelihood regions	p. 80
Alternative methods	p. 81
Background on simulation and visualisation of the likelihood regions	p. 82
Modal simulation method	p. 82
Illustrative example	p. 84
Comparison between the likelihood regions and the Wald regions	p. 85
Volume/volume error of the confidence regions	p. 86
Differences in univariate intervals via worst case analysis	p. 87
Illustrative example (revisited)	p. 88
Application to a finite mixture model	p. 89
Nonidentifiabilities and likelihood regions for the mixture parameters	p. 90
Mixture likelihood region simulation and visualisation	p. 93
Adequacy of using the Wald confidence region	p. 94
Data analysis	p. 95
Discussion	p. 98
	p. 99
Mixture of experts modelling with social science applications	p. 101
Introduction	p. 101
Motivating examples	p. 102
Voting blocs	p. 102
Social and organisational structure	p. 103
Mixture models	p. 103
Mixture of experts models	p. 104
A mixture of experts model for ranked preference data	p. 107
Examining the clustering structure	p. 111
A mixture of experts latent position cluster model	p. 112
Discussion	p. 118
Acknowledgements	p. 118
References	p. 118
Modelling conditional densities using finite smooth mixtures	p. 123
Introduction	p. 123
The model and prior	p. 125
Smooth mixtures	p. 125
The component models	p. 126
The prior	p. 128
Inference methodology	p. 129
The general MCMC scheme	p. 129
Updating ß and I using variable-dimension finite-step Newton proposals	p. 130
Model comparison	p. 132
Applications	p. 133
A small simulation study	p. 133
Lidar data	p. 135
Electricity expenditure data	p. 137
Conclusions	p. 141
Acknowledgements	p. 142
Appendix: Implementation details for the gamma and log-normal models	p. 142
References	p. 143
Nonparametric mixed membership modelling using the IBP compound Dirichlet process	p. 145
Introduction	p. 145
Mixed membership models	p. 146
Latent Dirichlet allocation	p. 147
Nonparametric mixed membership models	p. 147
Motivation	p. 148
Decorrelating prevalence and proportion	p. 150
Indian buffet process	p. 150
The IBP compound Dirichlet process	p. 150
An application of the ICD: focused topic models	p. 153
Inference	p. 154
Related models	p. 155
Empirical studies	p. 156
Discussion	p. 158
References	p. 159
Discovering nonbinary hierarchical structures with Bayesian rose trees	p. 161
Introduction	p. 161
Prior work	p. 163
Rose trees, partitions and mixtures	p. 165
Avoiding needless cascades	p. 169
Cluster models	p. 171
Greedy construction of Bayesian rose tree mixtures	p. 172
Prediction	p. 175
Hyperparameter optimisation	p. 176
Bayesian hierarchical clustering, Dirichlet process models and product partition models	p. 176
Mixture models and product partition models	p. 177
PCluster and Bayesian hierarchical clustering	p. 178
Results	p. 179
Optimality of tree structure	p. 179
Hierarchy likelihoods	p. 180
Partially observed data	p. 182
Psychological hierarchies	p. 182
Hierarchies of Gaussian process experts	p. 182
Discussion	p. 183
References	p. 185
Mixtures of factor analysers for the analysis of high-dimensional data	p. 189
Introduction	p. 189
Single-factor analysis model	p. 191
Mixtures of factor analysers	p. 192
"Mixtures of common factor analysers (MCFA)	p. 193
Some related approaches	p. 196
Fitting of factor-analytic models	p. 197
Choice of the number of factors q	p. 199
Example	p. 199
Low-dimensional plots via MCFA approach	p. 200
Multivariate t-factor analysers	p. 202
Discussion	p. 205
Appendix	p. 206
References	p. 210
Dealing with label switching under model uncertainty	p. 213
Introduction	p. 213
Labelling through clustering in the point-process representation	p. 214
The point-process representation of a finite mixture model	p. 214
Identification through clustering in the point-process representation	p. 217
Identifying mixtures when the number of components is unknown	p. 220
The role of Dirichlet priors in overfitting mixtures	p. 221
The meaning of K for overfitting mixtures	p. 222
The point-process representation of overfitting mixtures	p. 224
Examples	p. 227
Overfitting heterogeneity of component-specific parameters	p. 231
Overfitting heterogeneity	p. 231
Using shrinkage priors on the component-specific location parameters	p. 233
Concluding remarks	p. 237
References	p. 237
Exact Bayesian analysis of mixtures	p. 241
Introduction	p. 241
Formal derivation of the posterior distribution	p. 242
Locally conjugate priors	p. 242
True posterior distributions	p. 244
Poisson mixture	p. 246
Multinomial mixtures	p. 250
Normal mixtures	p. 252
References	p. 254
ManifoId MCMC for mixtures	p. 255
Introduction	p. 255
Markov chain Monte Carlo Methods	p. 257
Metropolis-Hastings	p. 257
Gibbs sampling	p. 257
Manifold Metropolis adjusted Langevin algorithm	p. 258
Manifold Hamiltonian Monte Carlo	p. 259
Finite Gaussian mixture models	p. 259
Gibbs sampler for mixtures of univariate Gaussians	p. 261
Manifold MCMC for mixtures of univariate Gaussians	p. 261
Metric tensor	p. 262
An illustrative example	p. 263
Experiments	p. 266
Discussion	p. 272
Acknowledgements	p. 273
Appendix	p. 273
References	p. 274
How many components in a finite mixture?	p. 277
Introduction	p. 277
The galaxy data	p. 278
The normal mixture model	p. 279
Bayesian analyses	p. 280
Escobar and West	p. 281
Phillips and Smith	p. 281
Roeder and Wasserman	p. 282
Richardson and Green	p. 282
Stephens	p. 283
Posterior distributions for K (for flat prior)	p. 283
Conclusions from the Bayesian analyses	p. 285
Posterior distributions of the model deviances	p. 286
Asymptotic distributions	p. 286
Posterior deviances for the galaxy data	p. 287
Conclusions	p. 291
References	p. 291
Bayesian mixture models: a blood-free dissection of a sheep	p. 293
Introduction	p. 293
Mixture models	p. 294
Hierarchical normal mixture	p. 294
Altering dimensions of the mixture model	p. 296
Bayesian mixture model incorporating spatial information	p. 298
Results	p. 301
Volume calculation	p. 302
Discussion	p. 307
References	p. 307
Index	p. 309
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Mixtures Estimation and Applications

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