Modeling and Reasoning with Bayesian Networks. Adnan Darwiche University of California Los Angeles, CA

Transcription

1 Modeling and Reasoning with Bayesian Networks Adnan Darwiche University of California Los Angeles, CA June 24, 2008

2 Contents Preface 1 1 Introduction Automated Reasoning The Limits of Deduction Assumptions to the Rescue Degrees of Belief Deciding after Believing What do the Probabilities Mean? Probabilistic Reasoning Initial Reactions A Second Chance Bayesian Networks Modeling with Bayesian Networks Reasoning with Bayesian Networks What is not Covered in this Book Propositional Logic Introduction Syntax of Propositional Sentences Semantics of Propositional Sentences Worlds, Models and Events Logical Properties Logical Relationships Equivalences and Reductions The Monotonicity of Logical Reasoning Multi-Valued Variables Variable Instantiations and Related Notations Logical Forms Conditioning a Propositional Sentence Unit Resolution Converting Propositional Sentences to CNF Exercises i

3 ii CONTENTS 3 Probability Calculus Introduction Degrees of Belief Properties of Beliefs Quantifying Uncertainty Updating Beliefs Independence Conditional Independence Variable Independence Mutual Information Further Properties of Beliefs Soft Evidence The All things considered Method The Nothing else considered Method More on Specifying Soft Evidence Soft Evidence as a Noisy Sensor Continuous Variables as Soft Evidence Distribution and Density Functions The Bayes Factor of a Continuous Observation Gaussian Noise Exercises Bayesian Networks Introduction Capturing Independence Graphically Parameterizing the Independence Structure Properties of Probabilistic Independence A Graphical Test of Independence Complexity of d-separation Soundness and Completeness of d-separation Further Properties of d-separation More on DAGs and Independence Perfect MAPs Independence MAPs Blankets and Boundaries Exercises Proofs Building Bayesian Networks Introduction Reasoning with Bayesian Networks Probability of Evidence Prior and Posterior Marginals Most Probable Explanation: MPE Maximum a Posteriori Hypothesis: MAP Modeling with Bayesian Networks

4 CONTENTS iii Diagnosis I: Model from Expert Diagnosis II: Model from Expert Sensitivity Analysis Network Granularity Diagnosis III: Model from Design Reliability: Model from Design Channel Coding Commonsense Knowledge Genetic Linkage Analysis Dealing with Large CPTs Micro Models Other Representations of CPTs The Significance of Network Parameters Exercises Inference by Variable Elimination Introduction The Process of Elimination Factors Elimination as a Basis for Inference Computing Prior Marginals Choosing an Elimination Order Computing Posterior Marginals Network Structure and Complexity Query Structure and Complexity Pruning Nodes Pruning Edges Network Pruning Computing Marginals after Pruning: An Example Bucket Elimination Exercises Proofs Inference by Factor Elimination Introduction Factor Elimination Elimination Trees Separators and Clusters A Message-Passing Formulation Multiple Marginals and Message Reuse The Cost of Passing Messages Joint Marginals and Evidence The Polytree Algorithm The Jointree Connection The Jointree Algorithm: A Classical View The Shenoy-Shafer Architecture

5 iv CONTENTS Using Non-minimal Jointrees Complexity of the Shenoy-Shafer Architecture The Hugin Architecture Complexity of the Hugin Architecture Exercises Proofs Inference by Conditioning Introduction Cutset Conditioning Recursive Conditioning Recursive Decomposition Caching Computations Any-space Inference Decomposition Graphs The Cache Allocation Problem Cache Allocation by Systematic Search Greedy Cache Allocation Exercises Proofs Models for Graph Decomposition Introduction Moral Graphs Elimination Orders Elimination Heuristics Optimal Elimination Prefixes Lower Bounds on Treewidth Optimal Elimination Orders From Jointrees to Elimination Orders From Dtrees to Elimination Orders Jointrees From Elimination Orders to Jointrees From Dtrees to Jointrees Jointrees as a Factorization Tool Dtrees From Elimination Orders to Dtrees Constructing Dtrees by Hypergraph Partitioning Balancing Dtrees Triangulated Graphs Exercises Lemmas Proofs

6 CONTENTS v 10 Most Likely Instantiations Introduction Computing MPE Instantiations Computing MPE by Variable Elimination Computing MPE by Systematic Search Reduction to Weighted MAXSAT Computing MAP Instantiations Computing MAP by Variable Elimination Computing MAP by Systematic Search Computing MAP by Local Search Exercises Proofs The Complexity of Probabilistic Inference Introduction Complexity Classes Showing Hardness Showing Membership Complexity of MAP on Polytrees Reducing Probability of Evidence to Weighted Model Counting The First Encoding The Second Encoding Reducing MPE to Weighted MAXSAT Exercises Proofs Compiling Bayesian Networks Introduction Circuit Semantics Circuit Propagation Evaluation and Differentiation Passes Computing Most Probable Explanations Circuit Compilation The Circuits of Variable Elimination Circuits Embedded in a Jointree The Circuits of CNF Encodings Exercises Proofs Inference with Local Structure Introduction The Impact of Local Structure on Inference Complexity Context-Specific Independence Determinism Evidence Exposing Local Structure

7 vi CONTENTS 13.3 CNF Encodings with Local Structure Encoding Network Structure Encoding Local Structure Encoding Evidence Conditioning with Local Structure Context-Specific Decomposition Context-Specific Caching Determinism Elimination with Local Structure Algebraic Decision Diagrams ADD Operations Variable Elimination using ADDs Compiling Arithmetic Circuits using ADDs Exercises Approximate Inference by Belief Propagation Introduction The Belief Propagation Algorithm Iterative Belief Propagation The Semantics of IBP The Kullback-Leibler Divergence Optimizing the KL-Divergence Generalized Belief Propagation Joingraphs Iterative Joingraph Propagation Edge-Deletion Semantics of Belief Propagation Edge Parameters Deleting Multiple Edges Searching for Edge Parameters Choosing Edges to Delete (or Recover) Approximating the Probability of Evidence Exercises Proofs Approximate Inference by Stochastic Sampling Introduction Simulating a Bayesian Network Expectations Probability as an Expectation Estimating an Expectation: Monte Carlo Simulation Direct Sampling Bounds on the Absolute Error Bounds on the Relative Error Rao-Blackwell Sampling Estimating a Conditional Probability Importance Sampling

8 CONTENTS vii Likelihood Weighting Particle Filtering Markov Chain Simulation Markov Chains Gibbs Sampling A Gibbs Sampler for Bayesian Networks Exercises Proofs Sensitivity Analysis Introduction Query Robustness Network-Independent Robustness Network-Specific Robustness Robustness of Most Probable Explanations Query Control Single Parameter Changes Multiple Parameter Changes Exercises Proofs Learning: The Maximum Likelihood Approach Introduction Estimating Parameters from Complete Data Estimating Parameters from Incomplete Data Expectation Maximization Gradient Ascent The Missing-Data Mechanism Learning Network Structure Learning Tree Structures Learning DAG Structures Searching for Network Structure Local Search Constraining the Search Space Exercises Proofs Learning: The Bayesian Approach Introduction Meta Networks Prior Knowledge Data as Evidence Parameter Independence Learning with Discrete Parameter Sets Computing Bayesian Estimates Closed Forms for Complete Data

9 viii CONTENTS Dealing with Incomplete Data Learning with Continuous Parameter Sets Dirichlet Priors The Semantics of Continuous Parameter Sets Bayesian Learning Computing Bayesian Estimates Closed Forms for Complete Data Dealing with Incomplete Data Learning Network Structure The BD Score The BDe Score Searching for a Network Structure Exercises Proofs A Notation 641 B Concepts from Information Theory 645 C Fixed Point Iterative Methods 649 D Constrained Optimization 653