MULTIVARIATE  BAYESIAN  STATISTICS  Models for  Source  Separation  and Signal  Unmixing
Daniel B. Rowe
 

Contents
List of Figures
List of Tables
Introduction
1.1 The Cocktail Party
1.2 The Source Separation Model
I Fundamentals
Statistical Distributions
2.1 Scalar Distributions
2.1.1 Binomial
2.1.2 Beta
2.1.3 Normal
2.1.4 Gamma and Scalar Wishart
2.1.5 Inverted Gamma and Scalar Inverted Wishart
2.1.6 Student t
2.1.7 F-Distribution
2.2 Vector Distributions
2.2.1 Multivariate Normal
2.2.2 Multivariate Student t
2.3 Matrix Distributions
2.3.1 MatrLx Normal
2.3.2 Wishart
2.3.3 Inverted Wishart
2.3.4 MatrLx T
Introductory Bayesian Statistics
3.1 Discrete Scalar Variables
3.1.1 Bayes' Rule and Two Simple Events
3.1.2 Bayes' Rule and the Law of Total Probability
3.2 Continuous Scalar Variables
3.3 Continuous Vector Variables
3.4 Continuous Matrix Variables
 2003 by Chapman & Hall/CRC
Prior Distributions
4.1 Vague Priors
4.1.1 Scalar Variates
4.1.2 Vector Variates
4.1.3 Mat rLx Variates
4.2 Conjugate Priors
4.2.1 Scalar Variates
4.2.2 Vector Variates
4.2.3 Mat rLx Variates
4.3 Generalized Priors
4.3.1 Scalar Variates
4.3.2 Vector Variates
4.3.3 Mat rLx Variates
4.4 Correlation Priors
4.4.1 Intraclass
4.4.2 Markov
Hyperparameter Assessment
5.1 Introduction
5.2 Binomial Likelihood
5.2.1 Scalar Beta
5.3 Scalar Normal Likelihood
5.3.1 Scalar Normal
5.3.2 Inverted Gamma or Scalar Inverted Wishart
5.4 Multivariate Normal Likelihood
5.4.1 Multivariate Normal
5.4.2 Inverted Wishart
5.5 Matrix Normal Likelihood
5.5.1 Mat rL, c Normal
5.5.2 Inverted Wishart
Bayesian Estimation Methods
6.1 Marginal Posterior Mean
6.1.1 Mat rL, c Integration
6.1.2 Gibbs Sampling
6.1.3 Gibbs Sampling Convergence
6.1.4 Normal Variate Generation
6.1.5 Wishart and Inverted Wishart Variate Generation
6.1.6 Factorization
6.1.7 Rejection Sampling
6.2 Maximum a Posteriori
6.2.1 Mat rL, c Differentiation
6.2.2 Iterated Conditional Modes (ICM)
6.3 Advantages of ICM over Gibbs Sampling
6.4 Advantages of Gibbs Sampling over ICM
 2003 by Chapman & Hall/CRC
II
Regression
7.1 Introduction
7.2 Normal Samples
7.3 Simple Linear Regression
7.4 Multiple Linear Regression
7.5 Multivariate Linear Regression
Models
Bayesian Regression
8.1 Introduction
8.2 The Bayesian Regression Model
8.3 Likelihood
8.4 Conjugate Priors and Posterior
8.5 Conjugate Estimation and Inference
8.5.1 Marginalization
8.5.2 Maximum a Posteriori
8.6 Generalized Priors and Posterior
8.7 Generalized Estimation and Inference
8.7.1 Marginalization
8.7.2 Posterior Conditionals
8.7.3 Gibbs Sampling
8.7.4 Maximum a Posteriori
8.8 Interpretation
8.9 Discussion
Bayesian Factor Analysis
9.1 Introduction
9.2 The Bayesian Factor Analysis Model
9.3 Likelihood
9.4 Conjugate Priors and Posterior
9.5 Conjugate Estimation and Inference
9.5.1 Posterior Conditionals
9.5.2 Gibbs Sampling
9.5.3 Maximum a Posteriori
9.6 Generalized Priors and Posterior
9.7 Generalized Estimation and Inference
9.7.1 Posterior Conditionals
9.7.2 Gibbs Sampling
9.7.3 Maximum a Posteriori
9.8 Interpretation
9.9 Discussion
 2003 by Chapman & Hall/CRC
10 Bayesian Source Separation
10.1 Introduction
10.2 Source Separation Model
10.3 Source Separation Likelihood
10.4 Conjugate Priors and Posterior
10.5 Conjugate Estimation and Inference
10.5.1 Posterior Conditionals
10.5.2 Gibbs Sampling
10.5.3 Maximum a Posteriori
10.6 Generalized Priors and Posterior
10.7 Generalized Estimation and Inference
10.7.1 Posterior Conditionals
10.7.2 Gibbs Sampling
10.7.3 Maximum a Posteriori
10.8 Interpretation
10.9 Discussion
11 Unobservable and Observable Source Separation
11.1 Introduction
11.2 Model
11.3 Likelihood
11.4 Conjugate Priors and Posterior
11.5 Conjugate Estimation and Inference
11.5.1 Posterior Conditionals
11.5.2 Gibbs Sampling
11.5.3 Maximum a PosterJori
11.6 Generalized Priors and Posterior
11.7 Generalized Estimation and Inference
11.7.1 Posterior Conditionals
11.7.2 Gibbs Sampling
11.7.3 Maximum a PosterJori
11.8 Interpretation
11.9 Discussion
12 FMRI Case Study
12.1 Introduction
12.2 Model
12.3 Priors and Posterior
12.4 Estimation and Inference
12.5 Simulated FMRI Experiment
12.6 Real FMRI Experiment
12.7 FMRI Conclusion
 2003 by Chapman & Hall/CRC
III Generalizations
13
Delayed Sources and Dynamic Coefficients
13.1 Introduction
13.2 Model
13.3 Delayed Constant Mixing
13.4 Delayed Nonconstant Mixing
13.5 Instantaneous Nonconstant Mixing
13.6 Likelihood
13.7 Conjugate Priors and Posterior
13.8 Conjugate Estimation and Inference
13.8.1 Posterior Conditionals
13.8.2 Gibbs Sampling
13.8.3 Maximum a Posteriori
13.9 Generalized Priors and Posterior
13.10 Generalized Estimation and Inference
13.10.1 Posterior Conditionals
13.10.2 Gibbs Sampling
13.10.3 Maximum a Posteriori
13.11 Interpretation
13.12 Discussion
14
Correlated Observation and Source Vectors
14.1 Introduction
14.2 Model
14.3 Likelihood
14.4 Conjugate Priors and Posterior
14.5 Conjugate Estimation and Inference
14.5.1 Posterior Conditionals
14.5.2 Gibbs Sampling
14.5.3 Maximum a Posteriori
14.6 Generalized Priors and Posterior
14.7 Generalized Estimation and Inference
14.7.1 Posterior Conditionals
14.7.2 Gibbs Sampling
14.7.3 Maximum a Posteriori
14.8 Interpretation
14.9 Discussion
15 Conclusion
Appendix A FMRI Activation Determination
A.1 Regression
A.2 Gibbs Sampling
A.3 ICM
Appendix B FMRI Hyperparameter Assessment
Bibliography
 2003 by Chapman & Hall/CRC