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Object OrientedObject Oriented Bayesian Networks Bayesian Networks for the Analysis of Evidencefor the Analysis of Evidence
Joint Seminar
Dept. of Statistical ScienceEvidence Inference & Enquiry Programme
5 February 2007
A. Philip Dawid Amanda B. Hepler
Introduction to Wigmore ChartsIllustration (S & V Case)
Introduction to Bayesian networksIllustration (S & V Case)
Comparison
Best of both worlds…OOBN Illustration
OutlineOutlineOutlineOutline
Wigmore Chart MethodWigmore Chart MethodWigmore Chart MethodWigmore Chart Method
AnalysisDefine the ultimate and penultimate probanda
Identify relevant items of evidence (trifles)
Assign trifles to penultimate probanda
SynthesisConstructing key lists bearing upon probanda
Draw a chart showing the inferential linkages among the elements of the key list
ExampleExample*: Probanda : Probanda ExampleExample*: Probanda : 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
P1
P2
P3
* 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, 1990.4. Anonymous witness testimony to 3.5. Berardelli died at 4:00 pm on April 15, 1920.6. 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, 1920.10. S. Neal testimony to 9.
.
.
.
477. Sacco lied about his Colt and cartridges, during inquiry, to protect his friends in the anarchist movement.
478. Sacco testimony to 477.479. Sacco’s lies about his Colt had nothing to do with his radical friends.480. Sacco admission on cross-examination
Example: Key ListExample: Key ListExample: Key ListExample: Key List
Example: Abbreviated Wigmore ChartExample: Abbreviated Wigmore ChartExample: Abbreviated Wigmore ChartExample: Abbreviated Wigmore Chart
Complete Wigmore charts are located in Appendix A of Kadane and Schum.
P1 P2P3
U
1 3 5 7
2 4 6 8
11 13
914
12
10 15 1716
18 59 67 82 156 358
Charts 3 – 6
Charts 15, 16, 17, 21, 22
Chart 14
Charts 19 – 22
Chart 25
Charts 7 & 8
Observations on Wigmorean Observations on Wigmorean AnalysisAnalysis
Observations on Wigmorean Observations on Wigmorean AnalysisAnalysis
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 MethodBayesian Network MethodBayesian Network MethodBayesian 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.
SynthesisCreate nodes (unknown variables + evidentiary facts).
Connect nodes using arrows representing probabilistic dependence.
Example: Abbreviated Bayes NetExample: Abbreviated Bayes Net(Hugin)(Hugin)
Example: Abbreviated Bayes NetExample: Abbreviated Bayes Net(Hugin)(Hugin)
Observations on Bayesian NetworksObservations on Bayesian NetworksObservations on Bayesian NetworksObservations 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 FeaturesSome Desirable FeaturesSome Desirable FeaturesSome Desirable Features
Large and messy
Complex modeling process
All evidence treated at same level
Hard to interpret
““Object-Oriented”Object-Oriented”Bayesian NetworkBayesian Network““Object-Oriented”Object-Oriented”Bayesian NetworkBayesian Network
Some Undesirable Features Some Undesirable Features (BN & WC)(BN & WC)Some Undesirable Features Some Undesirable Features (BN & WC)(BN & WC)
Recall Wigmorean Analysis Recall Wigmorean Analysis Recall Wigmorean Analysis 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
Sacco is the murderer?
1st Degree Murder?
Berardelli Murdered?
Felony Committed?
Medical evidence
Payroll robbery evidence
Level 1: 1Level 1: 1stst Degree Murder? Degree Murder?Level 1: 1Level 1: 1stst Degree Murder? Degree Murder?
P1 P2
P3
U
Sacco is the Murderer?
Consciousness of Guilt?
Firearms?Opportunity?
Eyewitnesses
Cap
Murder Car
Alibi
Motive?
Level 2: Sacco is the Murderer?Level 2: Sacco is the Murderer?Level 2: Sacco is the Murderer?Level 2: Sacco is the Murderer?
P3
Sacco at Scene?
Sacco’s Cap at Scene?
Alibi?Eyewitnesses?
Pelser Constantino
Wade
Murder Car?
Level 3: OpportunityLevel 3: OpportunityLevel 3: OpportunityLevel 3: Opportunity
Level 4: Eyewitness TestimonyLevel 4: Eyewitness Testimony
Similar to Sacco?
Pelser’s Credibility
Pelser’s Testimony
Wade’s Credibility
Wade’s Testimony
Sacco at Scene?
Level 5: Generic CredibilityLevel 5: Generic Credibility
Eyewitnesses
Generic Credibility
Testimony
Competent?
Veracity?
Objectivity?
Sensation?
Event
Level 6: Attributes of CredibilityLevel 6: Attributes of Credibility
Eyewitnesses
Generic Credibility
Testimony
Competent?
Veracity?
Objectivity?
Sensation?
Event
Competent?
Sensation
Agreement?
Event
Sensation
Level 6: Attributes of CredibilityLevel 6: Attributes of Credibility
Eyewitnesses
Generic Credibility
Testimony
Competent?
Veracity?
Objectivity?
Sensation?
Event Sensation
Noisy Channel
Out
In Error?
Competent?
Sensation
Agreement?
Event
Level 4: Eyewitness TestimonyLevel 4: Eyewitness Testimony
Similar to Sacco?
Pelser’s Credibility
Pelser’s Testimony
Wade’s Credibility
Wade’s Testimony
Sacco at Scene?
Level 5: Specific CredibilityLevel 5: Specific Credibility
Eyewitnesses
Testimony
Event
Generic Credibility
Competent?
Evidence undercut by ancillary evidence
Constantino’s Testimony
Sacco is the murderer?
1st Degree Murder?
Berardelli Murdered?
Felony Committed?
Medical evidence
Payroll robbery evidence
Level 1: 1Level 1: 1stst Degree Murder? Degree Murder?Level 1: 1Level 1: 1stst Degree Murder? Degree Murder?
P1 P2
P3
U
Identification (DNA, Sacco’s cap)
Corroboration/Contradiction2 or more sources giving the same or differing statements about the same event
Convergence/ConflictTestimony by 2 or more events that lead to the same or differing conclusions about a hypothesis
Explaining AwayKnowledge of one cause lowers probability of another cause
Other Generic Modules, so far…Other Generic Modules, so far…Other Generic Modules, so far…Other Generic Modules, so far…
Y Probabilities
X
p2
Generalizationp1
XParent-Child
Y
X
True False
YTrue
False
p1 1-p2
1-p1 p2
Boolean Case
Statistical Evidence
Expert Evidence
Demystifying the NumbersDemystifying the NumbersDemystifying the NumbersDemystifying 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 LimitationsSoftware LimitationsSoftware LimitationsSoftware Limitations
Thank you!Thank you!