Contents

1 Introduction 1

2 Probability 3

2.1 Rules of Probability 3

2.1.1 Deductive and Plausible Reasoning 3

2.1.2 Statement Calculus 3

2.1.3 Conditional Probability 5

2.1.4 Product Rule and Sum Rule of Probability 6

2.1.5 Generalized Sum Rule 7

2.1.6 Axioms of Probability 9

2.1.7 Chain Rule and Independence 11

2.1.8 Bayes’ Theorem 12

2.1.9 Recursive Application of Bayes’ Theorem 16

2.2 Distributions 16

2.2.1 Discrete Distribution 17

2.2.2 Continuous Distribution 18

2.2.3 Binomial Distribution 20

2.2.4 Multidimensional Discrete and Continuous Distributions 22

2.2.5 MarginalDistribution 24

2.2.6 Conditional Distribution 26

2.2.7 Independent Random Variables and Chain Rule 28

2.2.8 Generalized Bayes’ Theorem 31

2.3 Expected Value, Variance and Covariance 37

2.3.1 Expected Value 37

2.3.2 Variance and Covariance 41

2.3.3 Expected Value of a Quadratic Form 44

2.4 Univariate Distributions 45

2.4.1 Normal Distribution 45

2.4.2 Gamma Distribution 47

2.4.3 Inverted Gamma Distribution 48

2.4.4 Beta Distribution 48

2.4.5 χ2-Distribution 48

2.4.6 F-Distribution 49

2.4.7 t-Distribution 49

2.4.8 Exponential Distribution 50

2.4.9 Cauchy Distribution 51

2.5 Multivariate Distributions 51

2.5.1 Multivariate NormalDistribution 51

2.5.3 Normal-Gamma Distribution 55

2.6 PriorDensity Functions 56

2.6.1 Noninformative Priors 56

2.6.2 Maximum Entropy Priors 57

2.6.3 Conjugate Priors 59

3 Parameter Estimation, Confidence Regions and Hypothesis

Testing 63

3.1 BayesRule 63

3.2 Point Estimation 65

3.2.1 Quadratic Loss Function 65

3.2.2 Loss Function of the Absolute Errors 67

3.2.3 Zero-One Loss 69

3.3 Estimation of Confidence Regions 71

3.3.1 Confidence Regions 71

3.3.2 Boundary of a Confidence Region 73

3.4 Hypothesis Testing 73

3.4.1 Different Hypotheses 74

3.4.2 Test of Hypotheses 75

3.4.3 Special Priors for Hypotheses 78

3.4.4 Test of the Point Null Hypothesis by Confidence Regions 82

5 Special Models and Applications 129

5.1 Prediction and Filtering 129

5.1.1 Model of Prediction and Filtering as Special Linear

5.1.2 Special Model of Prediction and Filtering 135

5.2 Variance and CovarianceComponents 139

5.2.1 Model and Likelihood Function 139

5.2.2 Noninformative Priors 143

5.2.3 Informative Priors 143

5.2.4 Variance Components 144

5.2.5 Distributions for Variance Components 148

5.2.6 Regularization 150

5.3 Reconstructing and Smoothing of Three-dimensional Images 154

5.3.1 Positron Emission Tomography 155

5.3.2 Image Reconstruction 156

5.3.3 Iterated Conditional Modes Algorithm 158

5.4 Pattern Recognition 159

5.4.1 Classification by Bayes Rule 160

5.4.2 Normal Distribution with Known and Unknown

Parameters 161

5.4.3 Parameters for Texture 163

5.5 BayesianNetworks 167

5.5.1 Systems with Uncertainties 167

5.5.2 Setup of a BayesianNetwork 169

5.5.3 Computation of Probabilities 173

5.5.4 Bayesian Network in Form of a Chain 181

5.5.5 Bayesian Network in Form of a Tree 184

5.5.6 Bayesian Network in Form of a Polytreee 187

6 Numerical Methods 193

6.1 Generating Random Values 193

6.1.1 Generating RandomNumbers 193

6.1.2 InversionMethod 194

6.1.3 RejectionMethod 196

6.1.4 Generating Values for Normally Distributed Random

Variables 197

6.2 Monte Carlo Integration 197

6.2.1 Importance Sampling and SIR Algorithm 198

6.2.2 Crude Monte Carlo Integration 201

6.2.3 Computation of Estimates, Confidence Regions and

Probabilities for Hypotheses 202

6.2.4 Computation of Marginal Distributions 204

6.2.5 Confidence Region for Robust Estimation of

Parameters as Example 207

6.3 MarkovChainMonte CarloMethods 216

6.3.1 Metropolis Algorithm 216

6.3.2 Gibbs Sampler 217

6.3.3 Computation of Estimates, Confidence Regions and

6.3.4 Computation of Marginal Distributions 222

6.3.5 Gibbs Sampler for Computing and Propagating

Large CovarianceMatrices 224

6.3.6 Continuation of the Example: Confidence Region for

Robust Estimation of Parameters 229

References 235