Limit search to available items
Book Cover
E-book

Title Mixtures : estimation and applications / edited by Kerrie L. Mengersen, Christian P. Robert, D. Michael Titterington
Published Hoboken, N.J. : Wiley, 2011

Copies

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
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
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
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
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
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
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
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
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)
MATHEMATICS -- Probability & Statistics -- General.
Mixture distributions (Probability theory)
Form Electronic book
Author Mengersen, Kerrie L.
Robert, Christian P., 1961-
Titterington, D. M.
ISBN 9781119995678
1119995671
9781119995685
111999568X