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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. Outline. Introduction to Wigmore Charts Illustration (S & V Case)

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object oriented bayesian networks for the analysis of evidence

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

outline
Outline
  • Introduction to Wigmore Charts

Illustration (S & V Case)

  • Introduction to Bayesian networks

Illustration (S & V Case)

  • Comparison
  • Best of both worlds…OOBN Illustration
slide3

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
slide4

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.

slide5

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
slide6

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

slide7

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).
slide8

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.
slide10

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
slide11

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)
slide12

“Object-Oriented”Bayesian Network

Some Undesirable Features (BN & WC)

  • Large and messy
  • Complex modeling process
  • All evidence treated at same level
  • Hard to interpret
slide13

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

slide14

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

slide15

Opportunity?

Eyewitnesses

Alibi

Murder Car

Cap

Level 2: Sacco is the Murderer?

P3

Sacco is the Murderer?

Consciousness of Guilt?

Firearms?

Motive?

slide16

Eyewitnesses?

Pelser Constantino

Wade

Level 3: Opportunity

Sacco at Scene?

Sacco’s Cap at Scene?

Alibi?

Murder Car?

level 4 eyewitness testimony
Level 4: Eyewitness Testimony

Sacco at Scene?

Similar to Sacco?

Pelser’s Credibility

Wade’s Credibility

Pelser’s Testimony

Wade’s Testimony

level 5 generic credibility
Level 5: Generic Credibility

Generic Credibility

Event

Competent?

Eyewitnesses

Sensation?

Objectivity?

Veracity?

Testimony

level 6 attributes of credibility

Event

Agreement?

Competent?

Sensation

Level 6: Attributes of Credibility

Generic Credibility

Sensation

Event

Competent?

Eyewitnesses

Sensation?

Objectivity?

Veracity?

Testimony

level 6 attributes of credibility20

Event

Agreement?

Competent?

Sensation

In

Error?

Out

Level 6: Attributes of Credibility

Generic Credibility

Sensation

Event

Competent?

Eyewitnesses

Sensation?

Objectivity?

Veracity?

Noisy Channel

Testimony

level 4 eyewitness testimony21
Level 4: Eyewitness Testimony

Sacco at Scene?

Similar to Sacco?

Pelser’s Credibility

Wade’s Credibility

Pelser’s Testimony

Wade’s Testimony

level 5 specific credibility

Evidence undercut by ancillary evidence

Constantino’s Testimony

Level 5: Specific Credibility

Eyewitnesses

Event

Competent?

Generic Credibility

Testimony

slide23

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

slide24

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

slide25

X

Probabilities

Y

Generalization

p1

p2

Statistical Evidence

Expert Evidence

Demystifying the Numbers

X

Parent-Child

Y

Boolean Case

slide26

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)