| Foreword 1 Prerequisites in probability calculus 1.1 Background 1.2 Formulae and Definitions 1.2.1 Alphabet, Sequence 1.2.2 Random Variables and their Distributions 1.2.3 Joint Probability Distributions 1.2.4 Conditional Probability Distributions 1.2.5 A Chain Rule 1.2.6 Independence 1.2.7 Conditional Independence 1.2.8 Probability Models with Independence 1.2.9 Multinomial Probability Distribution 1.2.10 A Weight Matrix Model for a Family of Sequences 1.2.11 Simplifying Notations 1.3 Learning and Bayes' Rule 1.3.1 Bayes' Rule 1.3.2 A Missing Information Principle and Inference 1.4 Some Distributions for DNA Analysis 1.4.1 Fragment Accuracy 1.4.2 The Distribution of the Number of Fragments 1.5 Expectation 1.6 Jensen's Inequality 1.7 Conditional Expectation 1.8 Law of Large Numbers 1.9 Exercises 1.10 References and Further Reading: 2 Information and the Kullback Distance 2.1 Introduction 2.2 Mutual Information 2.3 Properties of Mutual Information 2.3.1 Entropy 2.3.2 Some Further Formulas 2.4 Shannon's Source Coding Theorems 2.4.1 AEP 2.4.2 The Source Coding Theorem 2.4.3 Lossless Compression Codes and Entropy 2.5 Kullback Distance 2.5.1 Definition and Examples 2.5.2 Calibration 2.5.3 Properties 2.6 The Score and the Fisher Information 2.7 Exercises on Mutual Information and Codelengths 2.8 Kullback Distance and Fisher Information 2.9 References and Further Reading 3 Probabilistic Models and Learning 3.1 Introduction 3.2 Bayesian probability 3.2.1 Chance and Probability 3.2.2 Coherence 3.3 Models with Conditional Independence 3.3.1 Modelling and Learning for Tosses of a Thumb tack 3.3.2 Learning of the Multinomial Process 3.3.3 General Summary 3.4 Comparison of Model Families 3.4.1 Bayes Factor 3.4.2 Inductive Learning, Updates 3.5 Some Asymptotics for Evidence 3.6 Evidence and Bayesian Codelengths …… 4 EM Algorthm 5 Alignment and Scoring 6 Mixture Models and Profiles 7 Markov Chains 8 Learning of Markov Chains 9 Markovian Models for DNA sequences 10 Hidden Mardov Models: and Overview 11 HMM for DNA Sequences 12 Left to Right HMM for Sequences 13 Derin's Algorithm 14 Forward-Backward Algorithm 15 Baum-Welch Learning Algorthm 16 Limit Points of Baum-Welch 17 Asymptotics of Learning 18 Full probabilistic HMM Index |
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