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!