The Bayesian network method. 1. Defining nodes for each variable in the problem (both unobserved and observed=evidence v
Analysis of complex patterns of evidence in legal cases: Wigmore charts vs. Bayesian networks A. P. Dawid, V. Leucari University College London
D. A. Schum George Mason University
ANALYSIS OF EVIDENCE IN A LEGAL CASE
AN EXAMPLE
EVIDENCE IN FORENSIC SCIENCE
FORMAL TOOLS FOR HANDLING EVIDENCE
A LEGAL CASE EXAMPLE An unknown number of offenders entered commercial premises
Typical features of the evidence arising from legal cases that we aim to address with our models
Marshaling and evaluating evidence are two fundamental issues in forensic science, both for constructing arguments about questions of facts and for taking final decisions. A formal approach to the analysis of evidence means
late at night through a hole which they cut in a metal grille. Inside, they were confronted by a security guard who was able to
• Complex structures: chains or webs of different sources of evidence
set off an alarm before one of the intruders punched him in the face, causing his nose to bleed . The intruders left the building
• Interactions: patterns of linkages between several items of evidence (corroboration, conflict, synergy, dependence, etc.)
• Rigorous description of both the problem and the related evidence through a model (assumptions and rules)
just as a police patrol car was arriving and they dispersed on foot. The security guard said that there were four men but the light was too poor for him to describe them and he was confused because of the blow he had received. The police in the patrol car searched
• Need for an evaluation of the evidence: accuracy, credibility, objectivity, relevance, provenance, weight
• Identification of relevant hypotheses • Quantification of prior knowledge (Bayesian approach)
the surrounding area and 10 minutes later found the suspect trying to “hot wire” a car in alley about a quarter of a mile from the incident. A tuft of red fibres was found on the jagged end of one of the edges of the grille. Blood samples were taken from the
Some references
guard and the suspect. The suspect denied having anything to do with the offence. He was wearing a jumper and jeans that were
• Dawid, A. P. and Evett, I. W. (1997). Using a graphical method to assist the evaluation of complicated patterns of evidence. Journal of Forensic Sciences 42, 226-231.
• Application of probabilistic techniques to evaluate the evidence Formal tools for handling evidence
taken for examination. A spray pattern of blood was found on the front and right sleeve of the suspect’s jumper. The blood type was different from that of the suspect, but the same as that of the guard. The tuft from the scene was found to be red acrylic . The
• Schum, D. A. (2005). A Wigmorean interpretation of the evaluation of a complicated pattern of evidence. Tech. rep.
• Wigmore charts (forensic background) • Bayesian networks (statistical background)
suspect’s jumper was red acrylic. The tuft was indistinguishable from the fibres of the jumper by eye, microspectrofluorimetry (MSF) and thin layer cromatography (TLC).
WIGMORE CHARTS AND BAYESIAN NETWORKS: A COMPARISON ULTIMATE AND PENULTIMATE PROBANDA
WIGMORE CHART ANALYSIS
WIGMORE CHART FOR PENULTIMATE PROBANDUM P1
KEY LIST FOR PENULTIMATE PROBANDUM P1 1. An unknown number of persons entered the premises of the Blackbread Brewery (BB) in the early morning hours of 1 May, 2003.
Analysis of the evidence using Wigmore charts
U
U
2. Police officer testimony to (1)
• Marshalling and organising the available evidence
3. There were four intruders who entered the BB premises 4. Security guard testimony to (3).
• Constructing arguments from evidence to penultimate probanda (describing a subjective chain of inferences)
P1
5. Upon seeing the intruders, I set off an alarm.
23
6. Guard testimony to (5)
8. Guard testimony to (7).
• Establishing the probative force of an emerging collection of evidence
9. I could not describe the intruders.
P2
P3
22
16
10. Guard testimony to (9).
P1
24
11. THe light was poor at the time of the intrusion.
P4
12. Guard testimony to (11) (ancillary evidence).
The Wigmore chart method
13. I was confused because of the blow I received from one of the intruders. 14. Guard testimony to (13) (ancillary evidence).
1. Analysis (a) Defining the ultimate probandum and the penultimate probanda
U
P1
(c) Assigning trifles to penultimate probanda
P2
15
3
21
19. The police in the patrol car saw the four men from a considerable distance away.
2
20. Police officer testimony to (19).
4
5
9
7
17
The suspect was one of the four men who broke into the premises of the Blackbread Brewery in the early hours of 1 May, 2003
23. The entry of the four men on the premises of the BB was illegal.
11
A security guard at the Blackbread Brewery was assaluted and injured during the break−in at the Blacklread Brewery on 1 May, 2003
25. Police officer testimony to (24).
24. The four men entered the BB premises by means of a hole they cut in a metal grille on these premises.
6
18
27. The photograph shown at trial is the same one taken by the police shortly after the break-in at the BB.
13
10
GUARD 1
NUMBER
SUSPECT GUILTY
POLICE
WIGMORE/BAYES: A COMPARISON
GUARD 2
Wigmore charts and Bayesian networks share some aspects
NUMBER Number of offenders
• Graphical methods
SUSPECT GUILTY Is the suspect guilty? yes/no
• Inference networks
• Modeling dependencies between different aspects of evidence (in terms of conditional independencies)
FIBRE ID Identity of the person who left the fibres in the grille
• Representing the logical structure of the criminal case
PUNCH ID Identity of the person who punched the guard
FIBRE
JUMPER
PUNCH ID
ID
STAIN
• Subjectivity • Models for incorporating complex evidence structures But Wigmore charts and Bayesian networks differ in nature
Evidence=observed random variables
FIBRE
BLOOD 1
GUARD 1 Guard’s testimony about the number of offenders
• Wigmore charts constructed backwards after the evidence has been observed
BLOOD 2
• Bayesian networks are a “process model”, intended to capture a complex process by which some series of events could have been generated
GUARD 2 Guard’s testimony about the punch POLICE Police officer’s testimony about the suspect
1. Defining nodes for each variable in the problem (both unobserved and observed=evidence variables)
• Wigmore charts are based on binary propositions
STAIN Shape of blood stain on the suspect’s jumper
INPUT 1) conditional independencies (e.g. FIBRE ID and PUNCH ID independent given NUMBER and SUSPECT GUILTY)
FIBRE Characteristics of the fibres found at the crime scene
3. Specifying prior knowledge (conditional probabilities)
2) conditional probabilities (e.g. probability of SUSPECT GUILTY given NUMBER) OUTPUT Prob(SUSPECT GUILTY=1|evidence)
BLOOD 1 Guard’s blood type
4. Updating probabilities for unobserved variables by entering evidence in the model (in particular, computing likelihood ratio for the hypothesis of interest)
• In order to construct a Bayesian network one needs to make assumptions about events related to the problem
BLOOD 3
JUMPER Characteristics of the suspect’s jumper
2. Defining arrows between certain pairs of nodes, representing probabilistic dependencies
14
28. The entry of the four men onto the BB premises on 1 May, 2003 was forced.
Unobserved random variables
The Bayesian network method
20
26. A photograph of the hole in the metal grille on the BB premises to be shown at trial.
RANDOM VARIABLES
• Constructing a plausible pattern of linkages among probabilistic variables
26
12
8
• Interpreting patterns of evidence which involve many variables
• Structuring and organising computations via a graphical representation
19
21. The four men were trying to avoid apprehension by the police.
It was the suspect who intentionally assaulted and injured the guard during the break−in at Balckbread Brewery on 1 May, 2003
P4
27
17. The four intruders left the BB premises just as a police patrol car arrived.
22. The four men knew that they were no authorised to enter the BB premises.
P3
25
18. Guard testimony to (17).
In the eraly morning hours of 1 May, 2003, four men unlawfully broke into the premises of the Blackbread Brewery
BAYESIAN NETWORK ANALYSIS Analysis of the evidence using Bayesian networks
16. The four men who entered the BB premises on 1 May, 2003 had no authorisation to do so.
The suspect unlawfully and intentionally assaulted and injured a security guard during a brek−in at the Blackbread Brewerypremises, in the early morning hours of 1 May, 2003
2. Synthesis (b) Drawing a chart that shows inferential linkages among elements in the key lists
1
15. The security guard believed the four men to be intruders on the BB premises.
(b) Parsing and organising the evidence into trifles, i.e. assessing relevance
(a) Constructing key lists bearing upon the probanda
28
7. The intruders were confronted by a security guard.
BLOOD 2 Suspect’s blood type
OBSERVED
BLOOD 3 Blood type on the suspect’s jumper
UNOBSERVED
Prob(SUSPECT GUILTY=0|evidence)
• Wigmore charts are chains of reasoning from the bottom (evidence) to the top (ultimate probandum) • Wigmore charts use generalisations in order to establish connections among variables • Bayesian networks are entirely probabilistic • Arrows: probabilistic dependence vs. inferential flow
COMBINING WIGMORE CHARTS AND BAYESIAN NETWORKS? OBJECT-ORIENTED BAYESIAN NETWORKS FOR MIXED EVIDENCE
TRUE ITEM
CUT IN THE GRILLE
BIASED
SPURIOUS ITEM
COIN
A more elaborate Bayesian network incorporating some of the elements in the Wigmore chart
GUARD 1
PHOTO 1
• Uncertainty about the crime (did it really happen, how did it happen)
REPORT
REPORT
POLICE 1
NUMBER OF PEOPLE
BREAK IN
DISCUSSION AND FUTURE WORK
PUNCH
BLEEDING
Remarks
GUARD 2
GUARD 3
PHOTO 2
POLICE 2
GUARD 4
• Testimonies of witnesses explicitly modeled SUSPECT AT SCENE
• More details: e.g. pictures, results of analyses • Object-oriented network: simple structures used repeatedly inside the network
PUNCH IDENTITY
STAIN
PHOTO 3
ITEM AT SCENE
MATCHEVIDENCE Identification of the source of a trace left at the crime scene (e.g. blood or DNA)
POLICE 3
SUSPECT 1
SUSPECT 2
POLICE 14
MATCHEVIDENCE
JUMPER
FIBRE IDENTITY
FIBRE 1
• Comparing different models (sensitivity analysis) • Interaction between witnesses
BERNOULLI
BLOOD 1
FIBRE 2
POLICE 4
BLOOD 2
POLICE 13
CRIME
POLICE 5
POLICE 6
MSF
TLC
• Recurrent patterns of relationships
• Clearer representation
• Is it worth combining Wigmore charts and Bayesian networks? • Using conditional independence relations in order to decompose likelihood ratios
Advantages of object-oriented Bayesian networks
• Computations are simplified
• Bayesian networks can be a powerful tool for both representing and interpreting evidence, especially when it has a complex structure Future work
SUSPECT’S ITEM
HYPOTHESIS
REPORT Output (with or without errors) from testimonies about events related to the crime or from analyses on items belonging to people involved
CONSEQUENCE Event that can be regarded as a potential consequence of another event, whose fallout is uncertain
SUSPECT GUILTY
• We have illustrated methods for a formal analysis of evidence in forensic science
POLICE 9
POLICE 10
POLICE 11
POLICE 12
• Credibility of witnesses • More on how to combine Wigmore charts and Bayesian networks • Manipulated evidence
EVENT
CONSEQUENCE
POLICE 7
POLICE 8
• Finding patterns of interactions between different sources of evidence
