Object Oriented Bayesian 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

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

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

Example: Probanda Example * : Probanda Ultimate Probandum Sacco (and Vanzetti) were guilty of 1 st degree murder in the slaying of Berardelli during the robbery that took place in South Braintree, MA on April 15, 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 P1P1 P2P2 P3P3 * Kadane, J. B. and Schum, D. A. (1996). A probabilistic analysis of the Sacco and Vanzetti evidence. Wiley.

1.A bullet was removed from Parmenter sometime after 4:00 pm on April 15, 1920; this bullet perforated his vena cava. 2.Dr. Hunting testimony to 1. 3.Parmenter died at 5:00 am on April 16, Anonymous witness testimony to 3. 5.Berardelli died at 4:00 pm on April 15, Dr. Fraser testimony to 5. 7.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. 8.Dr. Magrath testimony to 6. 9.The Slater & Morrill payroll was delivered to Hampton House on the morning of April 15, S. Neal testimony to Sacco lied about his Colt and cartridges, during inquiry, to protect his friends in the anarchist movement. 478.Sacco testimony to Sacco’s lies about his Colt had nothing to do with his radical friends. 480.Sacco admission on cross-examination Example: Key List

Example: Abbreviated Wigmore Chart Complete Wigmore charts are located in Appendix A of Kadane and Schum. P1P1 P2P2 P3P3 U Charts 3 – 6 Charts 15, 16, 17, 21, 22 Chart 14 Charts 19 – 22 Chart 25 Charts 7 & 8

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).

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.

Example: Abbreviated Bayes Net (Hugin)

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

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) Some Desirable Features

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

Recall Wigmorean Analysis Sacco (and Vanzetti) were guilty of 1 st 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 P1P1 P2P2 P3P3

Sacco is the murderer? 1 st Degree Murder? Berardelli Murdered? Felony Committed? Medical evidence Payroll robbery evidence Level 1: 1 st Degree Murder? P1P1 P2P2 P3P3 U

Sacco is the Murderer? Consciousness of Guilt? Firearms? Opportunity? Eyewitnesses Cap Murder Car Alibi Motive? Level 2: Sacco is the Murderer? P3P3

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

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

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

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

Level 6: Attributes of Credibility Eyewitnesses Generic Credibility Testimony Competent? Veracity? Objectivity? Sensation? Event Sensation Noisy Channel Out InError? Competent? Sensation Agreement? Event

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

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

Sacco is the murderer? 1 st Degree Murder? Berardelli Murdered? Felony Committed? Medical evidence Payroll robbery evidence Level 1: 1 st Degree Murder? P1P1 P2P2 P3P3 U

Identification (DNA, Sacco’s cap) Corroboration/Contradiction 2 or more sources giving the same or differing statements about 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 Other Generic Modules, so far…

Y Probabilities X p2p2 Generalization p1p1 X Parent-Child Y X True False Y True False p1p1 1-p 2 1-p 1 p2p2 Boolean Case Statistical Evidence Expert Evidence Demystifying the Numbers

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) Software Limitations

Thank you!