DECISION ANALYSIS WITH BAYESIAN NETWORKS

Transcription

1 RISK ASSESSMENT AND DECISION ANALYSIS WITH BAYESIAN NETWORKS NORMAN FENTON MARTIN NEIL CRC Press Taylor & Francis Croup Boca Raton London NewYork CRC Press is an imprint of the Taylor Si Francis an Croup, informa business A CHAPMAN & HALL BOOK

2 Contents Foreword Preface Acknowledgments Authors xi xiii x vi i xix Chapter 1 There Is More to Assessing Risk Than Statistics Introduction Predicting Economic Growth: The Normal Distribution and Its Limitations Patterns and Randomness: From School League Tables to Siegfried and Roy Dubious Relationships: Why You Should Be Very Wary of Correlations and Their Significance Values Spurious Correlations: How You Can Always Find a Silly 'Cause' of Exam Success The Danger of Regression: Looking Back When You Need to Look Forward The Danger of Averages What Type of Average? When Averages Alone Will Never Be Sufficient for Decision Making When Simpson's Paradox Becomes More Worrisome Uncertain Information and Incomplete Information: Do Not Assume They Are Different Do Not Trust Anybody (Even Experts) to Properly Reason about Probabilities Chapter Summary Chapter 2 The Need for Causal, Explanatory Models in Risk Assessment Introduction Are You More Likely to Die in an Automobile Crash When the Weather Is Good Compared to Bad? When Ideology and Causation Collide The Limitations of Common Approaches to Risk Assessment Measuring Armageddon and Other Risks Risks and Opportunities Risk Registers and Heat Maps Thinking about Risk Using Causal Analysis Applying the Causal Framework to Armageddon Summary Chapter 3 Measuring Uncertainty: The Inevitability of Subjectivity Introduction Experiments, Outcomes, and Events Multiple Experiments Joint Experiments 57

3 j Contents Joint Events and Marginalization Frequentist versus Subjective View of Uncertainty Summary Chapter 4 The Basics of Probability Introduction Some Observations Leading to Axioms and Theorems of Probability Probability Distributions Probability Distributions with Infinite Outcomes Joint Probability Distributions and Probability of Marginalized Events Dealing with More than Two Variables Independent Events and Conditional Probability Binomial Distribution Using Simple Probability Theory to Solve Earlier Problems and Explain Widespread Misunderstandings The Birthday Problem The Monty Hall Problem When Incredible Events Are Really Mundane When Mundane Events Really Are Quite Incredible HO Summary Further 111 Reading Chapter 5 Bayes' Theorem and Conditional Probability Introduction All Probabilities Are Conditional Bayes'Theorem Using Bayes' Theorem to Debunk Some Probability Fallacies Traditional Statistical Hypothesis Testing The Prosecutor Fallacy Revisited The Defendant's Fallacy Odds Form of Bayes and the Likelihood Ratio Second-Order Probability Summary Chapter 6 From Bayes' Theorem to Bayesian Networks Introduction A Very Simple Risk Assessment Problem Accounting for Multiple Causes (and Effects) Using Propagation to Make Special Types of Reasoning Possible The Crucial Independence Assumptions Structural Properties of BNs Serial Connection: Causal and Evidential Trails Diverging Connection: Common Cause Converging Connection: Common Effect Determining Whether Any Two Nodes in a BN Are Dependent 151

4 Contents vii 6.7 Propagation in Bayesian Networks Using BNs to Explain Apparent Paradoxes Revisiting the Monty Hall Problem Simple Solution Complex Solution Revisiting Simpson's Paradox Steps in Building and Running a BN Model Building a BN Model Running a BN Model Inconsistent Evidence Summary Theoretical Underpinnings 169 BN Applications 169 Nature and Theory of Causality Uncertain Evidence (Soft and Virtual) Chapter 7 Denning the Structure of Bayesian Networks Introduction Causal Inference and Choosing the Correct Edge Direction The Idioms The Cause-Consequence Idiom Measurement Idiom Definitional/Synthesis Idiom Case 1: Definitional Relationship between Variables Case 2: Hierarchical Definitions Case 3: Combining Different Nodes Together to Reduce Effects of Combinatorial Explosion ("Divorcing") Induction Idiom The Problems of Asymmetry and How to Tackle Them Impossible Paths Mutually Exclusive Paths Distinct Causal Pathways Taxonomic Classification Multiobject Bayesian Network Models The Missing Variable Fallacy Conclusions Chapter 8 Building and Eliciting Node Probability Tables Introduction Factorial Growth in the Size of Probability Tables Labeled Nodes and Comparative Expressions Boolean Nodes and Functions The Asia Model The OR Function for Boolean Nodes The AND Function for Boolean Nodes M from N Operator 235

5 viii Contents NoisyOR Function for Boolean Nodes Weighted Averages Ranked Nodes Background Solution: Ranked Nodes with the TNormal Distribution Alternative Weighted Functions for Ranked Nodes Hints and Tips When Working with Ranked Nodes and NPTs Tip 1: Use the Weighted Functions as Far as Possible Tip 2: Make Use of the Fact That a Ranked Node Parent Has an Underlying Numerical Scale Tip 3: Do Not Forget the Importance of the Variance in the TNormal Distribution Tip 4: Change the Granularity of a Ranked Scale without Having to Make Any Other Changes Tip 5: Do Not Create Large, Deep, Hierarchies Consisting of Rank Nodes Elicitation Elicitation Protocols and Cognitive Biases Scoring Rules and Validation Sensitivity Analysis Summary Chapter 9 Numeric Variables and Continuous Distribution Functions Introduction Some Theory on Functions and Continuous Distributions Static Discretization Dynamic Discretization Using Dynamic Discretization Prediction Using Dynamic Discretization Conditioning on Discrete Evidence Parameter Learning (Induction) Using Dynamic Discretization Classical versus Bayesian Modeling Bayesian Hierarchical Model Using Beta-Binomial Avoiding Common Problems When Using Numeric Nodes Unintentional Negative Values in a Node's State Range Potential Division by Zero Using Unbounded Distributions on a Bounded Range Observations with Very Low Probability Summary Chapter 10 Hypothesis Testing and Confidence Intervals Introduction Hypothesis Testing Bayes Factors Testing for Hypothetical Differences Comparing Bayesian and Classical Hypothesis Testing 311

6 Contents ix Model Comparison: Choosing the Best Predictive Model Accommodating Expert Judgments about Hypotheses Distribution Fitting as Hypothesis Testing Bayesian Model Comparison and Complex Causal Hypotheses Confidence Intervals The Fallacy of Frequentist Confidence Intervals The Bayesian Alternative to Confidence Intervals Summary Chapter 11 Modeling Operational Risk Introduction The Swiss Cheese Model for Rare Catastrophic Events Bow Ties and Hazards Fault Tree Analysis (FTA) Event Tree Analysis (ETA) Soft Systems, Causal Models, and Risk Arguments KUUUB Factors Operational Risk in Finance Modeling the Operational Loss Generation Process Scenarios and Stress Testing Summary Chapter 12 Systems Reliability Modeling Introduction Probability of Failure on Demand for Discrete Use Systems Time to Failure for Continuous Use Systems System Failure Diagnosis and Dynamic Bayesian Networks Dynamic Fault Trees (DFTs) Software Defect Prediction Summary Chapter 13 Bayes and the Law Introduction The Case for Bayesian Reasoning about Legal Evidence Building Legal Arguments Using Idioms The Evidence Idiom The Evidence Accuracy Idiom Idioms to Deal with the Key Notions of "Motive" and "Opportunity" Idiom for Modeling Dependency between Different Pieces of Evidence Alibi Evidence Idiom Explaining away Idiom Putting it All Together: Vole Example Using BNs to Expose Further Fallacies of Legal Reasoning 433

7 x Contents The Jury Observation Fallacy The "Crimewatch UK" Fallacy Summary Appendix A: The Basics of Counting 441 Appendix B: The Algebra of Node Probability Tables 449 Appendix C: Junction Tree Algorithm 455 Appendix D: Dynamic Discretization 465 Appendix E: Statistical Distributions 483 Index 495