Multivariate Bayesian Statistics: Models for Source Separation and Signal Unmixing
Features
Summary
Of the two primary approaches to the classic source separation problem, only one does not impose potentially unreasonable model and likelihood constraints: the Bayesian statistical approach. Bayesian methods incorporate the available information regarding the model parameters and not only allow estimation of the sources and mixing coefficients, but also allow inferences to be drawn from them.
Multivariate Bayesian Statistics: Models for Source Separation and Signal Unmixing offers a thorough, self-contained treatment of the source separation problem. After an introduction to the problem using the "cocktail-party" analogy, Part I provides the statistical background needed for the Bayesian source separation model. Part II considers the instantaneous constant mixing models, where the observed vectors and unobserved sources are independent over time but allowed to be dependent within each vector. Part III details more general models in which sources can be delayed, mixing coefficients can change over time, and observation and source vectors can be correlated over time. For each model discussed, the author gives two distinct ways to estimate the parameters.
Real-world source separation problems, encountered in disciplines from engineering and computer science to economics and image processing, are more difficult than they appear. This book furnishes the fundamental statistical material and up-to-date research results that enable readers to understand and apply Bayesian methods to help solve the many "cocktail party" problems they may confront in practice.
Table of Contents
Introduction
Part l: FUNDAMENTALS
STATISTICAL DISTRIBUTIONS
Scalar Distributions
Vector Distributions
Matrix Distributions
INTRODUCTORY BAYESIAN STATISTICS
Discrete Scalar Variables
Continuous Scalar Variables
Continuous Vector Variables
Continuous Matrix Variables
PRIOR DISTRIBUTIONS
Vague Priors
Conjugate Priors
Generaliz ed Priors
Correlation Priors
HYPERPARAMETER ASSESSMENT
Introduction
Binomial Likelihood
Scalar Normal Likelihood
Multivariate Normal Likelihood
Matrix Normal Likelihood
BAYESIAN ESTIMATION METHODS
Marginal Posterior Mean
Maximum a Posteriori
Advantages of ICM over Gibbs Sampling
Advantages of Gibbs Sampling over ICM
REGRESSION
Introduction
Normal Samples
Simple Linear Regression
Multiple Linear Regression
Multivariate Linear Regression
Part II: II Models
BAYESIAN REGRESSION
Introduction
The Bayesian Regression Model
Likelihood
Conjugate Priors and Posterior
Conjugate Estimation and Inference
Generalized Priors and Posterior
Generalized Estimation and Inference
Interpretation
Discussion
BAYESIAN FACTOR ANALYSIS
Introduction
The Bayesian Factor Analysis Model
Likelihood
Conjugate Priors and Posterior
Conjugate Estimation and Inference
Generalized Priors and Posterior
Generalized Estimation and Inference
Interpretation
Discussion
BAYESIAN SOURCE SEPARATION
Introduction
Source Separation Model
Source Separation Likelihood
Conjugate Priors and Posterior
Conjugate Estimation and Inference
Generalized Priors and Posterior
Generalized Estimation and Inference
Interpretation
Discussion
UNOBSERVABLE AND OBSERVABLE SOURCE SEPARATION
Introduction
Model
Likelihood
Conjugate Priors and Posterior
Conjugate Estimation and Inference
Generalized Priors and Posterior
Generalized Estimation and Inference
Interpretation
Discussion
FMRI CASE STUDY
Introduction
Model
Priors and Posterior
Estimation and Inference
Simulated FMRI Experiment
Real FMRI Experiment
FMRI Conclusion
Part III: Generalizations
DELAYED SOURCES AND DYNAMIC COEFFICIENTS
Introduction
Model
Delayed Constant Mixing
Delayed Nonconstant Mixing
Instantaneous Nonconstant Mixing
Likelihood
Conjugate Priors and Posterior
Conjugate Estimation and Inference
Generalized Priors and Posterior
Generalized Estimation and Inference
Interpretation
Discussion
CORRELATED OBSERVATION AND SOURCE VECTORS
Introduction
Model
Likelihood
Conjugate Priors and Posterior
Conjugate Estimation and Inference
Posterior Conditionals
Generalized Priors and Posterior
Generalized Estimation and Inference
Interpretation
Discussion
CONCLUSION
Appendix A FMRI Activation Determination
Appendix B FMRI Hyperparameter Assessment
Bibliography
Index
Editorial Reviews
"This book is a thorough exposition of Bayesian modeling techniques. … Overall, the book is well written and gives a detailed step-by-step approach to some widely applicable model types. … This book helps me understand how to build some complex models using a Bayesian approach with a much better understanding of what effect my decisions will have on the final model results."
- Technometrics, Feb. 2005, Vol. 47, No. 1
"A very useful and valuable book on a topic of great importance for researchers and students with interest in Bayesian techniques. Summing Up: Highly recommended."
-D.V. Chopra in CHOICE, June 2003
