WILEY A John Wiley and Sons, Ltd., Publication

Transcription

1 Bayesian Networks An Introduction Timo Koski Institutionen för Matematik, Kungliga Tekniska Högskolan, Stockholm, Sweden John M. Noble Matematiska Institutionen, Linköpings Tekniska Högskola, Linköpings universitet, Linköping, Sweden WILEY A John Wiley and Sons, Ltd., Publication

2 Contents Preface ix 1 Graphical models and probabilistic reasoning Introduction Axioms of probability and basic notations The Bayes update of probability Inductive learning Bayes' rule Jeffrey's rule Pearl's method of virtual evidence Interpretations of probability and Bayesian networks Learning as inference about parameters Bayesian statistical inference Tossing a thumb-tack Multinomial sampling and the Dirichlet integral 24 Notes 28 Exercises: Probabilistic theories of causality, Bayes' rule, multinomial sampling and the Dirichlet density 31 2 Conditional independence, graphs and d -separation Joint probabilities Conditional independence Directed acyclic graphs and ^-separation Graphs Directed acyclic graphs and probability distributions The Bayes ball Illustrations Potentials Bayesian networks Object oriented Bayesian networks J-Separation and conditional independence 66

3 VI CONTENTS 2.9 Markov models and Bayesian networks /-maps and Markov equivalence The trek and a distribution without a faithful graph 72 Notes 73 Exercises: Conditional independence and ^-separation 75 3 Evidence, sufficiency and Monte Carlo methods Hard evidence Soft evidence and virtual evidence Jeffrey's rule Pearl's method of virtual evidence Queries in probabilistic inference The chest clinic problem Bucket elimination Bayesian sufficient statistics and prediction sufficiency Bayesian sufficient statistics Prediction sufficiency Prediction sufficiency for a Bayesian network Time variables A brief introduction to Markov chain Monte Carlo methods Simulating a Markov chain Irreducibility, aperiodicity and time reversibility The Metropolis-Hastings algorithm The one-dimensional discrete Metropolis algorithm 111 Notes 112 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 138 Notes 144 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 160 Notes 161 Exercises: Learning the conditional probability potentials Learning the graph structure Assigning a probability distribution to the graph structure 168

4 CONTENTS 6.2 Markov equivalence and consistency Establishing the DAG isomorphic property Reducing the size of the search The Chow-Liu tree The Chow-Liu tree: A predictive approach The K2 structural learning algorithm The MMHC algorithm Monte Carlo methods for locating the graph structure Women in mathematics 189 Notes 191 Exercises: Learning the graph structure Parameters and sensitivity Changing parameters in a network Measures of divergence between probability distributions The Chan-Darwiche distance measure Comparison with the Kullback-Leibler divergence and euclidean distance Global bounds for queries Applications to updating Parameter changes to satisfy query constraints Binary variables The sensitivity of queries to parameter changes 220 Notes 224 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 CG potentials Some results on marginalization CG regression 250 Notes 251 Exercises: Graphical models and exponential families Causality and intervention calculus Introduction Conditioning by observation and by intervention The intervention calculus for a Bayesian network 258

5 CONTENTS Establishing the model via a controlled experiment 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 270 Notes 273 Exercises: Causality and intervention calculus 275 The junction tree and probability updating Probability updating using a junction tree Potentials and the distributive law Marginalization and the distributive law Elimination and domain graphs Factorization along an undirected graph Factorizing along a junction tree Flow of messages initial illustration 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 311 Notes 313 Exercises: The junction tree and probability updating 314 Factor graphs and the sum product algorithm Factorization and local potentials Examples of factor graphs The sum product algorithm Detailed illustration of the algorithm 329 Notes 332 Exercise: Factor graphs and the sum product algorithm 333 References 335 Index 343