Description |
1 online resource (xviii, 311 pages) : illustrations |
Contents |
Machine generated contents note: 1. The EM algorithm, variational approximations and expectation propagation for mixtures / D. Michael Titterington -- 1.1. Preamble -- 1.2. The EM algorithm -- 1.2.1. Introduction to the algorithm -- 1.2.2. The E-step and the M-step for the mixing weights -- 1.2.3. The M-step for mixtures of univariate Gaussian distributions -- 1.2.4. M-step for mixtures of regular exponential family distributions formulated in terms of the natural parameters -- 1.2.5. Application to other mixtures -- 1.2.6. EM as a double expectation -- 1.3. Variational approximations -- 1.3.1. Preamble -- 1.3.2. Introduction to variational approximations -- 1.3.3. Application of variational Bayes to mixture problems -- 1.3.4. Application to other mixture problems -- 1.3.5. Recursive variational approximations -- 1.3.6. Asymptotic results -- 1.4. Expectation-propagation -- 1.4.1. Introduction -- 1.4.2. Overview of the recursive approach to be adopted |
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1.4.3. Finite Gaussian mixtures with an unknown mean parameter -- 1.4.4. Mixture of two known distributions -- 1.4.5. Discussion -- Acknowledgements -- References -- 2. Online expectation maximisation / Olivier Cappe -- 2.1. Introduction -- 2.2. Model and assumptions -- 2.3. The EM algorithm and the limiting EM recursion -- 2.3.1. The batch EM algorithm -- 2.3.2. The limiting EM recursion -- 2.3.3. Limitations of batch EM for long data records -- 2.4. Online expectation maximisation -- 2.4.1. The algorithm -- 2.4.2. Convergence properties -- 2.4.3. Application to finite mixtures -- 2.4.4. Use for batch maximum-likelihood estimation -- 2.5. Discussion -- References -- 3. The limiting distribution of the EM test of the order of a finite mixture / Pengfei Li -- 3.1. Introduction -- 3.2. The method and theory of the EM test -- 3.2.1. The definition of the EM test statistic -- 3.2.2. The limiting distribution of the EM test statistic -- 3.3. Proofs |
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3.4. Discussion -- References -- 4. Comparing Wald and likelihood regions applied to locally identifiable mixture models / Bruce G. Lindsay -- 4.1. Introduction -- 4.2. Background on likelihood confidence regions -- 4.2.1. Likelihood regions -- 4.2.2. Profile likelihood regions -- 4.2.3. Alternative methods -- 4.3. Background on simulation and visualisation of the likelihood regions -- 4.3.1. Modal simulation method -- 4.3.2. Illustrative example -- 4.4. Comparison between the likelihood regions and the Wald regions -- 4.4.1. Volume/volume error of the confidence regions -- 4.4.2. Differences in univariate intervals via worst case analysis -- 4.4.3. Illustrative example (revisited) -- 4.5. Application to a finite mixture model -- 4.5.1. Nonidentifiabilities and likelihood regions for the mixture parameters -- 4.5.2. Mixture likelihood region simulation and visualisation -- 4.5.3. Adequacy of using the Wald confidence region |
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4.6. Data analysis -- 4.7. Discussion -- References -- 5. Mixture of experts modelling with social science applications / Thomas Brendan Murphy -- 5.1. Introduction -- 5.2. Motivating examples -- 5.2.1. Voting blocs -- 5.2.2. Social and organisational structure -- 5.3. Mixture models -- 5.4. Mixture of experts models -- 5.5. A mixture of experts model for ranked preference data -- 5.5.1. Examining the clustering structure -- 5.6. A mixture of experts latent position cluster model -- 5.7. Discussion -- Acknowledgements -- References -- 6. Modelling conditional densities using finite smooth mixtures / Robert Kohn -- 6.1. Introduction -- 6.2. The model and prior -- 6.2.1. Smooth mixtures -- 6.2.2. The component models -- 6.2.3. The prior -- 6.3. Inference methodology -- 6.3.1. The general MCMC scheme -- 6.3.2. Updating & beta; and I using variable-dimension finite-step Newton proposals -- 6.3.3. Model comparison -- 6.4. Applications -- 6.4.1. A small simulation study |
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6.4.2. LIDAR data -- 6.4.3. Electricity expenditure data -- 6.5. Conclusions -- Acknowledgements -- Appendix: Implementation details for the gamma and log-normal models -- References -- 7. Nonparametric mixed membership modelling using the IBP compound Dirichlet process / David M. Blei -- 7.1. Introduction -- 7.2. Mixed membership models -- 7.2.1. Latent Dirichlet allocation -- 7.2.2. Nonparametric mixed membership models -- 7.3. Motivation -- 7.4. Decorrelating prevalence and proportion -- 7.4.1. Indian buffet process -- 7.4.2. The IBP compound Dirichlet process -- 7.4.3. An application of the ICD: focused topic models -- 7.4.4. Inference -- 7.5. Related models -- 7.6. Empirical studies -- 7.7. Discussion -- References -- 8. Discovering nonbinary hierarchical structures with Bayesian rose trees / Katherine A. Heller -- 8.1. Introduction -- 8.2. Prior work -- 8.3. Rose trees, partitions and mixtures -- 8.4. Avoiding needless cascades -- 8.4.1. Cluster models |
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8.5. Greedy construction of Bayesian rose tree mixtures -- 8.5.1. Prediction -- 8.5.2. Hyperparameter optimisation -- 8.6. Bayesian hierarchical clustering, Dirichlet process models and product partition models -- 8.6.1. Mixture models and product partition models -- 8.6.2. PCluster and Bayesian hierarchical clustering -- 8.7. Results -- 8.7.1. Optimality of tree structure -- 8.7.2. Hierarchy likelihoods -- 8.7.3. Partially observed data -- 8.7.4. Psychological hierarchies -- 8.7.5. Hierarchies of Gaussian process experts -- 8.8. Discussion -- References -- 9. Mixtures of factor analysers for the analysis of high-dimensional data / Suren I. Rathnayake -- 9.1. Introduction -- 9.2. Single-factor analysis model -- 9.3. Mixtures of factor analysers -- 9.4. Mixtures of common factor analysers (MCFA) -- 9.5. Some related approaches -- 9.6. Fitting of factor-analytic models -- 9.7. Choice of the number of factors q -- 9.8. Example -- 9.9. Low-dimensional plots via MCFA approach |
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9.10. Multivariate t-factor analysers -- 9.11. Discussion -- Appendix -- References -- 10. Dealing with label switching under model uncertainty / Sylvia Fruhwirth-Schnatter -- 10.1. Introduction -- 10.2. Labelling through clustering in the point-process representation -- 10.2.1. The point-process representation of a finite mixture model -- 10.2.2. Identification through clustering in the point-process representation -- 10.3. Identifying mixtures when the number of components is unknown -- 10.3.1. The role of Dirichlet priors in overfitting mixtures -- 10.3.2. The meaning of K for overfitting mixtures -- 10.3.3. The point-process representation of overfitting mixtures -- 10.3.4. Examples -- 10.4. Overfitting heterogeneity of component-specific parameters -- 10.4.1. Overfitting heterogeneity -- 10.4.2. Using shrinkage priors on the component-specific location parameters -- 10.5. Concluding remarks -- References -- 11. Exact Bayesian analysis of mixtures / Kerrie L. Mengersen |
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11.1. Introduction -- 11.2. Formal derivation of the posterior distribution -- 11.2.1. Locally conjugate priors -- 11.2.2. True posterior distributions -- 11.2.3. Poisson mixture -- 11.2.4. Multinomial mixtures -- 11.2.5. Normal mixtures -- References -- 12. Manifold MCMC for mixtures / Mark Girolami -- 12.1. Introduction -- 12.2. Markov chain Monte Carlo Methods -- 12.2.1. Metropolis-Hastings -- 12.2.2. Gibbs sampling -- 12.2.3. Manifold Metropolis adjusted Langevin algorithm -- 12.2.4. Manifold Hamiltonian Monte Carlo -- 12.3. Finite Gaussian mixture models -- 12.3.1. Gibbs sampler for mixtures of univariate Gaussians -- 12.3.2. Manifold MCMC for mixtures of univariate Gaussians -- 12.3.3. Metric tensor -- 12.3.4. An illustrative example -- 12.4. Experiments -- 12.5. Discussion -- Acknowledgements -- Appendix -- References -- 13. How many components in a finite mixture? / Murray Aitkin -- 13.1. Introduction -- 13.2. The galaxy data -- 13.3. The normal mixture model |
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13.4. Bayesian analyses -- 13.4.1. Escobar and West -- 13.4.2. Phillips and Smith -- 13.4.3. Roeder and Wasserman -- 13.4.4. Richardson and Green -- 13.4.5. Stephens -- 13.5. Posterior distributions for K (for flat prior) -- 13.6. Conclusions from the Bayesian analyses -- 13.7. Posterior distributions of the model deviances -- 13.8. Asymptotic distributions -- 13.9. Posterior deviances for the galaxy data -- 13.10. Conclusions -- References -- 14. Bayesian mixture models: a blood-free dissection of a sheep / Graham E. Gardner -- 14.1. Introduction -- 14.2. Mixture models -- 14.2.1. Hierarchical normal mixture -- 14.3. Altering dimensions of the mixture model -- 14.4. Bayesian mixture model incorporating spatial information -- 14.4.1. Results -- 14.5. Volume calculation -- 14.6. Discussion -- References |
Summary |
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 subje |
Bibliography |
Includes bibliographical references and index |
Notes |
Print version record |
Subject |
Mixture distributions (Probability theory)
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MATHEMATICS -- Probability & Statistics -- General.
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Mixture distributions (Probability theory)
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Form |
Electronic book
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Author |
Mengersen, Kerrie L.
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Robert, Christian P., 1961-
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Titterington, D. M.
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ISBN |
9781119995678 |
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1119995671 |
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9781119995685 |
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111999568X |
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