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Object Oriented Bayesian Networks for the Analysis of Evidence

Object Oriented Bayesian Networks for the Analysis of Evidence

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Object Oriented Bayesian Networks for the Analysis of Evidence

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  1. Object OrientedBayesian Networks for the Analysis of Evidence Joint Seminar Dept. of Statistical Science Evidence Inference & Enquiry Programme 5 February 2007 A. Philip Dawid Amanda B. Hepler

  2. Outline • Introduction to Wigmore Charts Illustration (S & V Case) • Introduction to Bayesian networks Illustration (S & V Case) • Comparison • Best of both worlds…OOBN Illustration

  3. Wigmore Chart Method Analysis • Define the ultimate and penultimate probanda • Identify relevant items of evidence (trifles) • Assign trifles to penultimate probanda Synthesis • Constructing key lists bearing upon probanda • Draw a chart showing the inferential linkages among the elements of the key list

  4. P1 P2 P3 Example*: Probanda Ultimate Probandum Sacco (and Vanzetti) were guilty of 1st degree murder in the slaying of Berardelli during the robbery that took place in South Braintree, MA on April 15, 1920. Penultimate Probanda Berardelli died of gunshot wounds. When he was shot, Berardelli was in possession of a payroll. Sacco intentionally fired shots that killed Berardelli. U * Kadane, J. B. and Schum, D. A. (1996). A probabilistic analysis of the Sacco and Vanzetti evidence. Wiley.

  5. Example: Key List • A bullet was removed from Parmenter sometime after 4:00 pm on April 15, 1920; this bullet perforated his vena cava. • Dr. Hunting testimony to 1. • Parmenter died at 5:00 am on April 16, 1990. • Anonymous witness testimony to 3. • Berardelli died at 4:00 pm on April 15, 1920. • Dr. Fraser testimony to 5. • Four bullets were extracted from Berardelli’s body. Dr. Magrath labelled the lethal bullet as bullet III; the other three were marked I, II, and IV. • Dr. Magrath testimony to 6. • The Slater & Morrill payroll was delivered to Hampton House on the morning of April 15, 1920. • S. Neal testimony to 9. . . . • Sacco lied about his Colt and cartridges, during inquiry, to protect his friends in the anarchist movement. • Sacco testimony to 477. • Sacco’s lies about his Colt had nothing to do with his radical friends. • Sacco admission on cross-examination

  6. Example: Abbreviated Wigmore Chart U Complete Wigmore charts are located in Appendix A of Kadane and Schum. P3 P1 P2 11 13 1 3 5 18 59 67 82 156 358 7 14 2 4 6 8 9 12 Charts 3 – 6 Chart 14 Chart 25 10 15 16 17 Charts 15, 16, 17, 21, 22 Charts 19 – 22 Charts 7 & 8

  7. Observations on Wigmorean Analysis • A graphical display organizing masses of evidence. • Events and hypotheses must be represented as binary propositions. • Intended to model argument strategies for both sides of a case. • Arrows indicate inferential flow. • Designed for qualitative analysis, although likelihood calculations can easily be derived (see Kadane and Schum).

  8. Bayesian Network Method Analysis • Define unknown variables to be represented as nodes in the network. • Identify relevant items of evidential facts to also become nodes in network. • Determine any probabilistic dependencies. Synthesis • Create nodes (unknown variables + evidentiary facts). • Connect nodes using arrows representing probabilistic dependence.

  9. Example: Abbreviated Bayes Net(Hugin)

  10. Observations on Bayesian Networks • Graphical display organizing masses of evidence • Events and hypotheses can be represented with any number of states • Intended to model probabilistic relationships among variables • Arrows indicate ‘causal’ flow • Designed for quantitative analysis, and likelihood calculations are automatic

  11. Some Desirable Features • Can handle complex cases with masses of evidence. (BN & WC) • Likelihoods can quantify probative force of the evidence. (BN) • Conditional probability tables can guide thinking when unclear about dependencies. (BN) • Listing probanda and trifles can guide thinking when unclear of relevant items to consider. (WC)

  12. “Object-Oriented”Bayesian Network Some Undesirable Features (BN & WC) • Large and messy • Complex modeling process • All evidence treated at same level • Hard to interpret

  13. Recall Wigmorean Analysis Sacco (and Vanzetti) were guilty of 1st degree murder in the slaying of Berardelli during the robbery that took place in South Braintree, MA on April 15, 1920 Berardelli died of gunshot wounds When he was shot, Berardelli was in possession of a payroll. Sacco intentionally fired shots that killed Berardelli during a robbery of the payroll. U P1 P2 P3

  14. Payroll robbery evidence Level 1: 1st Degree Murder? U 1st Degree Murder? Sacco is the murderer? Felony Committed? Berardelli Murdered? P3 Medical evidence P2 P1

  15. Opportunity? Eyewitnesses Alibi Murder Car Cap Level 2: Sacco is the Murderer? P3 Sacco is the Murderer? Consciousness of Guilt? Firearms? Motive?

  16. Eyewitnesses? Pelser Constantino Wade Level 3: Opportunity Sacco at Scene? Sacco’s Cap at Scene? Alibi? Murder Car?

  17. Level 4: Eyewitness Testimony Sacco at Scene? Similar to Sacco? Pelser’s Credibility Wade’s Credibility Pelser’s Testimony Wade’s Testimony

  18. Level 5: Generic Credibility Generic Credibility Event Competent? Eyewitnesses Sensation? Objectivity? Veracity? Testimony

  19. Event Agreement? Competent? Sensation Level 6: Attributes of Credibility Generic Credibility Sensation Event Competent? Eyewitnesses Sensation? Objectivity? Veracity? Testimony

  20. Event Agreement? Competent? Sensation In Error? Out Level 6: Attributes of Credibility Generic Credibility Sensation Event Competent? Eyewitnesses Sensation? Objectivity? Veracity? Noisy Channel Testimony

  21. Level 4: Eyewitness Testimony Sacco at Scene? Similar to Sacco? Pelser’s Credibility Wade’s Credibility Pelser’s Testimony Wade’s Testimony

  22. Evidence undercut by ancillary evidence Constantino’s Testimony Level 5: Specific Credibility Eyewitnesses Event Competent? Generic Credibility Testimony

  23. Payroll robbery evidence Level 1: 1st Degree Murder? U 1st Degree Murder? Sacco is the murderer? Felony Committed? Berardelli Murdered? P3 Medical evidence P2 P1

  24. Other Generic Modules, so far… • Identification (DNA, Sacco’s cap) • Corroboration/Contradiction 2 or more sources giving the same or differing statementsabout the same event • Convergence/Conflict Testimony by 2 or more events that lead to the same or differing conclusions about a hypothesis • Explaining Away Knowledge of one cause lowers probability of another cause

  25. X Probabilities Y Generalization p1 p2 Statistical Evidence Expert Evidence Demystifying the Numbers X Parent-Child Y Boolean Case

  26. Software Limitations • Need a program to streamline the process, incorporating concepts from both WC & BN • Hierarchical displays in HUGIN are lacking • Drag and drop from text (i.e. Rationale, Araucaria) • Would like probabilities to be randomly drawn from a distribution, facilitating sensitivity analysis • HUGIN runtime is slow for large oobns (10+ nested networks)

  27. Thank you!