Bayesian Networks : An Introduction

by Timo Koski (Royal Institute of Technology, Sweden ); John Noble (University of Linkö ping, Sweden )

• ISBN13: 9780470684030
• ISBN10: 0470684038
• Format: eBook
• Copyright: 2009-09-01
• Publisher: Wiley

Summary

Bayesian Networks: An Introduction provides a self-contained introduction to the theory and applications of Bayesian networks, a topic of interest and importance for statisticians, computer scientists and those involved in modelling complex data sets. The material has been extensively tested in classroom teaching and assumes a basic knowledge of probability, statistics and mathematics. All notions are carefully explained and feature exercises throughout. Features include: An introduction to Dirichlet Distribution, Exponential Families and their applications. A detailed description of learning algorithms and Conditional Gaussian Distributions using Junction Tree methods. A discussion of Pearl's intervention calculus, with an introduction to the notion of see and do conditioning. All concepts are clearly defined and illustrated with examples and exercises. Solutions are provided online. This book will prove a valuable resource for postgraduate students of statistics, computer engineering, mathematics, data mining, artificial intelligence, and biology. Researchers and users of comparable modelling or statistical techniques such as neural networks will also find this book of interest.

Table of Contents

Preface

Graphical models and probabilistic reasoning

Introduction

Axioms of probability and basic notations

The Bayes update of probability

Inductive learning

Interpretations of probability and Bayesian networks

Learning as inference about parameters

Bayesian statistical inference

Tossing a thumb-tack

Multinomial sampling and the Dirichlet integral

Notes

Exercises: Probabilistic theories of causality, Bayes' rule, multinomial sampling and the Dirichlet density

Conditional independence, graphs and d-separation

Joint probabilities

Conditional independence

Directed acyclic graphs and d-separation

The Bayes ball

Potentials

Bayesian networks

Object oriented Bayesian networks

d-Separation and conditional independence

Markov models and Bayesian networks

I-maps and Markov equivalence

Notes

Exercises: Conditional independence and d-separation

Evidence, sufficiency and Monte Carlo methods

Hard evidence

Soft evidence and virtual evidence

Queries in probabilistic inference

Bucket elimination

Bayesian sufficient statistics and prediction sufficiency

Time variables

A brief introduction to Markov chain Monte Carlo methods

The one-dimensional discrete Metropolis algorithm

Notes

Exercises: Evidence, sufficiency and Monte Carlo methods

Decomposable graphs and chain graphs

Definitions and notations

Decomposable graphs and triangulation of graphs

Junction trees

Markov equivalence

Markov equivalence, the essential graph and chain graphs

Notes

Exercises: Decomposable graphs and chain graphs

Learning the conditional probability potentials

Initial illustration: maximum likelihood estimate for a fork connection

The maximum likelihood estimator for multinomial sampling

MLE for the parameters in a DAG: the general setting

Updating, missing data, fractional updating

Notes

Exercises: Learning the conditional probability potentials

Learning the graph structure

Assigning a probability distribution to the graph structure

Markov equivalence and consistency

Reducing the size of the search

Monte Carlo methods for locating the graph structure

Women in mathematics

Notes

Exercises: Learning the graph structure

Parameters and sensitivity

Changing parameters in a network

Measures of divergence between probability distributions

The Chan-Darwiche distance measure

Parameter changes to satisfy query constraints

The sensitivity of queries to parameter changes

Notes

Exercises: Parameters and sensitivity

Graphical models and exponential families

Introduction to exponential families

Standard examples of exponential families

Graphical models and exponential families

Noisy 'or' as an exponential family

Properties of the log partition function

Fenchel Legendre conjugate

Kullback-Leibler divergence

Mean field theory

Conditional Gaussian distributions

Notes

Exercises: Graphical models and exponential families

Causality and intervention calculus

Introduction

Conditioning by observation and by intervention

The intervention calculus for a Bayesian network

Properties of intervention calculus

Transformations of probability

A note on the order of 'see' and 'do' conditioning

The 'Sure Thing' principle

Back door criterion, confounding and identifiability

Notes

Exercises: Causality and intervention calculus

The junction tree and probability updating

Probability updating using a junction tree

Potentials and the distributive law

Elimination and domain graphs

Factorization along an undirected graph

Factorizing along a junction tree

Local computation on junction trees

Schedules

Local and global consistency

Message passing for conditional Gaussian distributions

Using a junction tree with virtual evidence and soft evidence

Notes

Exercises: The junction tree and probability updating

Factor graphs and the sum product algorithm

Factorization and local potentials

The sum product algorithm

Detailed illustration of the algorithm

Notes

Exercise: Factor graphs and the sum product algorithm

References

Index

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