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Item Detail

Hidden Markov Models for Bioinformatics

Author(s): Koski, Timo

ISBN10: 1402001355

ISBN13: 9781402001352

Cover: Hardcover

New Copy: Usually Ships in 7-10 Business Days

List Price ~~$209.00~~
Our Price $199.70
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The purpose of this book is to give a thorough and systematic introduction to probabilistic modeling in bioinformatics. The book contains a mathematically strict and extensive presentation of the kind of probabilistic models that have turned out to be useful in genome analysis. Questions of parametric inference, selection between model families, and various architectures are treated. Several examples are given of known architectures (e.g., profile HMM) used in genome analysis. Audience: This book will be of interest to advanced undergraduate and graduate students with a fairly limited background in probability theory, but otherwise well trained in mathematics and already familiar with at least some of the techniques of algorithmic sequence analysis.

Gives a thorough and systematic introduction to probabilistic bioinformatics. Of interest to advanced undergraduate and graduate students limited background in probability theory, but otherwise well trained in mathematics.

Hidden Markov Models for Bioinformatics

Foreword

Prerequisites in probability calculus

(28)

Background

(1)

Formulae and Definitions

(11)

Alphabet, Sequence

(2)

Random Variables and their Distributions

(2)

Joint Probability Distributions

(1)

Conditional Probability Distributions

(1)

A Chain Rule

(1)

Independence

(1)

Conditional Independence

(1)

Probability Models with Independence

(1)

Multinomial Probability Distribution

(2)

A Weight Matrix Model for a Family of Sequences

(1)

Simplifying Notations

(1)

Learning and Bayes' Rule

(1)

Bayes' Rule

(1)

A Missing Information Principle and Inference

(1)

Some Distributions for DNA Analysis

(2)

Fragment Accuracy

(1)

The Distribution of the Number of Fragments

(1)

Expectation

(1)

Jensen's Inequality

(1)

Conditional Expectation

(1)

Law of Large Numbers

(2)

Exercises

(6)

References and Further Reading:

(3)

Information and the Kullback Distance

(22)

Introduction

(1)

Mutual Information

(1)

Properties of Mutual Information

(2)

Entropy

(1)

Some Further Formulas

(1)

Shannon's Source Coding Theorems

(3)

AEP

(1)

The Source Coding Theorem

(1)

Lossless Compression Codes and Entropy

(1)

Kullback Distance

(4)

Definition and Examples

(2)

Calibration

(1)

Properties

(2)

The Score and the Fisher Information

(2)

Exercises on Mutual Information and Codelengths

(3)

Kullback Distance and Fisher Information

(4)

References and Further Reading:

(2)

Probabilistic Models and Learning

(32)

Introduction

(1)

Bayesian probability

(3)

Chance and Probability

(2)

Coherence

(1)

Models with Conditional Independence

(10)

Modelling and Learning for Tosses of a Thumb tack

(3)

Learning of the Multinomial Process

(6)

General Summary

(1)

Comparison of Model Families

(1)

Bayes Factor

(1)

Inductive Learning, Updates

(1)

Some Asymptotics for Evidence

(3)

Evidence and Bayesian Codelengths

(3)

Exercises

(4)

Appendix: Dirichlet Densities

(2)

Euler's Gamma

(1)

The Dirichlet density

(1)

The Beta Density

(1)

References and Further Reading:

(5)

EM Algorithm

(22)

Finite Mixtures of Distributions

(3)

Introduction

(1)

Mixtures

(1)

Likelihood and Missing Information

(1)

An Expectation Maximization Algorithm

(3)

Quasi-log Likelihood

(1)

Step E and Step M

(1)

An Explicit Form of Step M

(1)

Exponential Type of Family of Distributions

(3)

Definitions and Notations:

(1)

EM for Exponential Type of Family of Distributions

(2)

Summary of EM for Exponential Type of Families

(1)

Iterations and Convergence

(1)

A Model for Fragments with Motifs

(3)

Pattern in a Fragment

(1)

Modelling

(2)

Generating Sequences with a Motif

(1)

Learning the Model for One Motif

(1)

Exercises

(6)

References and Further Reading:

102

(3)

Alignment and Scoring

105

(20)

Summary and Overview

105

(1)

Background

105

(1)

Pairwise Alignments

106

(2)

Definitions

106

(1)

More Terminology: Gaps

107

(1)

An Evolutionary Distance

108

(3)

Metric Spaces

108

(1)

Auxiliary Terminology

109

(1)

The Sequence Metric and the Evolutionary Distance

110

(1)

Dynamic Programming

111

(5)

The Recurrence Relation

111

(2)

Tabular Computation

113

(2)

The Traceback: Optimal Alignment

115

(1)

Global Similarity Alignment, Gap Penalities

116

(3)

Exercises

119

(1)

Appendix: On the Evolutionary Distance

120

(3)

The Metric Property

120

(2)

Edge Lengths of a Tree and the Related Metrics

122

(1)

References and Further Reading:

123

(2)

Mixture Models and Profiles

125

(26)

Summary and Overview

125

(3)

Prior Information for a Weight Matrix

125

(1)

Family and Superfamily Representations

126

(1)

Multiple Alignment and Priors of Mixtures of Dirichlet Densities

127

(1)

On a General Theory of Mixtures of Dirichlet Densities

127

(1)

A Finite Mixture of Prior Dirichlets

128

(6)

Multiple Alignment and Prior Information

128

(2)

A Likelihood Function for the Frequency Counts

130

(2)

The Mixture Estimation Algorithm

132

(2)

Estimates of the Symbol Probability Distributions

134

(1)

Aligning a Sequence to a Profile

135

(1)

Profile

135

(3)

Optimal Fit of a String to a Profile

136

(2)

Exercises

138

(3)

Appendix A: Dirichlet Mixtures

141

(1)

Definition of Dirichlet Mixtures

141

(1)

Appendix B: The Predictive Multinomial Mixture

142

(3)

The Predictive Multinomial Process

142

(1)

The Predictive Mixture

143

(1)

The Posterior Mixture Density

144

(1)

Mean Posterior Probability of Symbols

144

(1)

Appendix C: The Lemmas for Mixture Estimation

145

(3)

References and Further Reading:

148

(3)

Markov Chains

151

(24)

Introduction

151

(6)

Contents

151

(1)

Definitions and Notations

151

(1)

Examples

152

(3)

Markov Chains of kth Order

155

(1)

Joint Probability Distribution of an MC

156

(1)

Chapman--Kolmogorov Equations

157

(2)

Stationarity and Invariant Distributions

159

(2)

Ergodic Markov Chains

161

(7)

Definition and Properties

161

(1)

A Sufficient Condition for Ergodicity

162

(5)

Classification of the State Space

167

(1)

Exercises

168

(5)

On the Markov Property and the Invariant Distribution

168

(2)

Stationary Markov Processes

170

(1)

Entropy Rate of a Stationary Markov Chain

170

(1)

High-order Markov Chains

171

(2)

References and Further Reading:

173

(2)

Learning of Markov Chains

175

(16)

Background

175

(1)

General Summary

175

(1)

ML for Markov Chains

176

(3)

Preliminaries

176

(1)

ML of the Transition Matrix

177

(1)

An Example of Full Likelihood

178

(1)

The Whittle Distribution

179

(2)

Model Averaging

181

(2)

Posterior Distributions for Rows in the Transition Matrix

181

(1)

Predictive Probability

181

(2)

MC Order Comparison Using the Bayes Ratio

183

(1)

Consistency of the ML

184

(2)

Exercises

186

(3)

References and Further Reading:

189

(2)

Markovian Models for DNA sequences

191

(20)

DNA and Modelling

191

(2)

Markov Chains for DNA sequences

193

(2)

Frame Dependent Markov Chains

195

(2)

MTD and Interpolated Markov Models

197

(1)

Probability Computations for Words

198

(6)

First Occurrence of a Word

199

(3)

Conditional Expected Frequency of a Word

202

(2)

Exercises

204

(3)

References and Further Reading:

207

(4)

Hidden Markov Models: an Overview

211

(20)

Introduction

211

(1)

The HMM

211

(1)

The Thumb Tack Again

212

(1)

Standard HMM

212

(4)

Definition and notations

212

(4)

Generative Modelling of Sequences

216

(1)

Examples and Applications

216

(3)

Influence Diagrams, Nonstandard HMM

219

(5)

A Fourth Problem

224

(1)

Finite State Stochastic Machines

224

(2)

Notations and Terminology

224

(1)

Carlyle's Matrix Formula

225

(1)

Exercises

226

(3)

References and Further Reading:

229

(2)

HMM for DNA Sequences

231

(14)

Introduction

231

(1)

Heterogeneity and Segmentation of Natural DNA sequences

232

(1)

Gene Finding and HMM

233

(4)

Radiation Hybrid Mapping

237

(2)

Exercises

239

(2)

References and Further Reading:

241

(4)

Left to Right HMM for Sequences

245

(26)

Introduction

245

(1)

Contents

245

(1)

Left to Right HMM

246

(1)

Motif-Based Hidden Markov Models

246

(5)

The Approximate Common Substring Problem

246

(3)

Profile HMM

249

(2)

Plan 7 in HMMER

251

(1)

Multiple Alignment by profile HMM

251

(2)

Scoring with HMM

253

(2)

HMM Alignment to a Profile

253

(1)

Log Odds Score

253

(1)

A Null Model for Scoring

254

(1)

Fragment Lengths

255

(7)

Transience

255

(2)

The Probability Distribution for the Length of Generated Sequence

257

(2)

Expected Length of a Generated Sequence

259

(2)

Quasi stationarity?

261

(1)

Exercises

262

(5)

ACS

262

(1)

Duration of Stay and Absorption

262

(2)

A Project on the Yaglom Limit

264

(2)

Expected length

266

(1)

References and Further Reading:

267

(4)

Derin's Algorithm

271

(22)

Introduction

271

(1)

Derin's Formula

272

(1)

Various Sufficiency Properties

273

(4)

Lumping of Markov Chains

277

(2)

Factorizations

279

(6)

Final Time Sufficiency

285

(1)

Proof of Derin's Backwards Recursion

286

(2)

Exercises:

288

(3)

References and Further Reading:

291

(2)

Forward--Backward Algorithm

293

(24)

Introduction

293

(1)

Forward--Backward Algorithm and Evaluation

294

(5)

An Outline

294

(1)

Forward Recursion

295

(2)

Backward Recursion

297

(1)

The Scoring (Evaluation) Problem

298

(1)

Recursions for Posterior Probabilities

299

(5)

Filtering, Smoothing, Prediction

299

(2)

Scaled Recursions

301

(2)

Implementation

303

(1)

The Alignment Problem and the Viterbi Algorithm

304

(5)

Formulation

304

(1)

The Bellman Optimality Principle

305

(2)

Dynamic Programming for the Alignment (Decoding) Problem

307

(1)

Viterbi Learning

308

(1)

Exercises

309

(6)

References and Further Reading:

315

(2)

Baum--Welch Learning Algorithm

317

(28)

The Learning Problem

317

(2)

Statement of the Learning Problem

317

(1)

Plan for the Study of the Learning Problem

318

(1)

The Quasi-log likelihood for HMM

319

(6)

A Lower Bound for the Log Likelihood Ratio

319

(2)

Expressions for the Quasi-log likelihood Function

321

(2)

Maximization of the Quasi-log likelihood Function

323

(1)

An Intermediate Summary of Re-estimation

324

(1)

Forward and Backward Probabilities

325

(8)

Smoothing Probabilities

326

(5)

Re-estimates in Forward and Backward Variables

331

(2)

Baum--Welch Learning for MAP

333

(3)

Prior Density

333

(1)

MAP re-estimates

334

(2)

Exercises

336

(7)

References and Further Reading:

343

(2)

Limit Points of Baum--Welch

345

(22)

Introduction

345

(2)

A Summary of Re-estimates

345

(1)

Plan for the Further Study

346

(1)

The Theorem on Limit Points of Re-estimates

347

(1)

On the Stationary Points of the Likelihood

347

(6)

A Formal Optimization Calculus

347

(1)

Some Computational Details

348

(4)

Re-estimation Formulas Rediscovered

352

(1)

The Growth Transformation

353

(5)

Introduction

353

(1)

Polynomials Homogeneous in Degree n

354

(1)

Baum--Eagon Inequality

355

(3)

Baum--Welch Learning and Maximum Likelihood

358

(2)

The Global Convergence Theorem

360

(1)

Zangwill's Theory of Algorithms

360

(1)

Exercises

361

(3)

Appendix: Results in Mathematical Analysis

364

(1)

Geometric Mean--Arithmetic Mean Inequality

364

(1)

Holder's inequality

365

(1)

Euler's Theorem on Homogeneous Functions

365

(1)

References and Further Reading:

365

(2)

Asymptotics of Learning

367

(18)

Introduction

367

(1)

Additional Assumptions

368

(2)

Steps of the Proof: An Outline

370

(1)

Some Auxiliary Statements

371

(8)

The Consistency Result

379

(3)

Exercises

382

(1)

References and Further Reading:

383

(2)

Full Probabilistic HMM

385

(4)

Introduction

385

(1)

Revision of Local Alignment

385

(1)

Probability Defined by Alignment

386

(1)

Scoring

387

(1)

References and Further Reading:

388

(1)

Index

389

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