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Bayesian Networks and Argumentation in Evidence Analysis
Isaac Newton Institute for Mathematical Sciences
Monday 26th September 2016 - Thursday 29th September 2016
Bayes Nets for the evaluation of testimony
Paolo Garbolino
Dept. of Art and Architecture
IUAV University
Venice, Italy
Argumentation in evidence analysis 1
Evidential reasoning takes the form of a chain of arguments H, A, …, E, linking
evidence E with the hypothesis H, and where each step of reasoning is supported
by one or more generalizations that provide an appropriate relevance relation
between nodes of the chain.
David Schum, The Evidential Foundations of Probabilistic Reasoning, New York,
Wiley, 1994
Argumentation in evidence analysis 2
“We assert a generalization G which we believe links E and F, and then we put this
generalization to the test by collecting n items of ancillary evidence […] This
ancillary evidence together with the generalization being tested forms the basis of
our epistemic assessments of likelihoods”
(J. Kadane, D. Schum, A Probabilistic Analysis of Sacco and Vanzetti Evidence, New
York, Wiley, 1996, p. 268-9)
The analysis of testimony 1
H = the hypothesis that a certain event occurred,
E = the (observed) event that a witness says that event H occurred.
Bayesian inference requires evaluating the likelihoods
Prob(E|H) .
Prob(E|not-H)
Likelihoods can be estimated on the basis of scientific laws or well-supported
generalizations.
Look for a generalization G linking H and E, which supports a probabilistic judgment
such that:
Prob (E|H) > Prob (E|not-H)
The analysis of testimony 2
Three basic attributes of the credibility of human witnesses, according to Schum:



Observational sensitivity or accuracy: did the witness really perceive H?
Objectivity: does the witness remember well?
Veracity: does the witness says what she really believes?
Generalization: “any person who is an accurate eyewitness and who is objective
and who is veracious will usually tell the truth” (D. Schum, 1994, p. 101-2).
G: “Any person who is an accurate eyewitness and who is
objective and who is veracious will usually tell the truth”
Model of witness
An ideal person who satisfies by
definition generalization G
We apply the model to a real-world
situation by making some ‘theoretical
hypotheses’ which specify the relevant
aspects under which the model is similar
to the intended real-world situation, and
the degree of similarity
Real person
Peter Menzies, Reason and causes revisited, in D. Macarthur, M De Caro (eds.),
Naturalism and Normativity, New York, Columbia University Press, 2010, p. 14270).
Ronald Giere, Explaining Science: A Cognitive Approach, Chicago, Chicago University
Press, 1988;
Ronald Giere, Science without Laws, Chicago, Chicago University Press, 1999.
The ‘theoretical hypotheses’ 1
(i)
(ii)
(iii)
Witness X is an accurate person;
Witness X is an objective person;
Witness X is a veracious person;
‘Ancillary evidence’ is evidence that bears upon the truth of these ‘theoretical
hypotheses’.
Our subjective probabilities for hypotheses (i) – (iii), based upon ancillary evidence,
measure the ‘degree of similarity’ of the real person with the model
The ‘theoretical hypotheses’ 2
(G’)
(G’’)
(G’’’)
If X is an accurate person, then her senses give evidence of what she sees;
If X is an objective person, then she believes the evidence of her senses;
If X is a veracious person, then she says what she believes.
Let assume that the credibility attributes are independent properties (the three
theoretical hypotheses are independent)
S
H
AC
E
B
O
V
H = ‘Event H occurred’
The DAG
S = ‘X’s senses give evidence of H’
B = X believes H’
E = ‘X says that H’
AC = ‘X is accurate’
O = ‘X is objective’
V =  v1= ‘X is disinterested’, v2= ‘X is interested in asserting H’, v3= ‘X is interested in nonasserting H’
The Bayes Net 1
Let assume generalizations (i)-(iii) to be deterministic.
In case the witness is not accurate, or not objective, use a kind of ‘default
assumption’: consider her answer a ‘random answer’.
(P. Gärdenfors, B. Hansson, N.E. Sahlin, Evidentiary Value: Philosophical, Judicial
and Psychological Aspects of a Theory: Essays Dedicated to Sören Halldén, Lund,
Gleerups, 1983;
E.J. Olsson, ‘Corroborating testimony, probability and surprise’, British Journal for
the Philosophy of Science, 53, 2002, p. 273-88;
L. Bovens, B. Fitelson, S. Hartmann, J. Snyder, ‘Too odd (not) to be true? A reply to
Olsson’, British Journal for the Philosophy of Science, 53, 2002, p. 539-563.
The Bayes Net 2
Conditional probability table for the node S
H:
t
f .
AC:
t
f
t
f
__________________________________
Prob (S = t | H, AC)
1 0.5
0 0.5
Prob (S = f | H, AC)
0 0.5
1 0.5
The Bayes Net 3
Conditional probability table for the node B
S:
t
f .
O:
t
f
t
f
____________________________________
Prob (B = t | S, O)
1 0.5
0 0.5
Prob (B = f | S, O)
0 0.5
1 0.5
The Bayes Net 4
Conditional probability table for the node E
Vi :
v1
v2
v3 .
B:
t f t f
t f
______________________________________
Prob (E = t | B, Vi)
1 0
1 1
0 0
Prob (E = f | B, Vi)
0
1
0
0
1
1
Robustness 1
How ‘robust’ are the hypotheses made for drawing the net, in particular the
‘default probabilities’ introduced in the conditional probability tables?
Prob (S|H, AC = f) = 0.5
Prob (B|S, O = f) = 0.5
Let assume that the witness is for sure non accurate and non-objective, but
veracious.
Her witness should be irrelevant.
S
H
AC
E
B
O
V
Initialize the net with
Prob (AC) = Prob (O) = 0, and Prob (V = v1= disinterested) =1.
Then, if update the Bayes net with Prob (E) = 1, the prior probability of H remains
unchanged
(HUGIN 6.5).
Robustness 2
The blue bus.
Plaintiff is run down by a blue bus. Plaintiff is prepared to prove that defendant,
company Z, operates 80% of all the blue buses in town.
Company Z Company W Other owners Y
Vehicles
____________________________________________________________
Blue buses
80
20
0
100
Other vehicles
0
0
900
900
____________________________________________________________
80
20
900
1000
“Even assuming a standard of proof under which the plaintiff need only to establish
his case by a preponderance of evidence in order to succeed, the plaintiff does not
discharge that burden by showing simply that four-fifths, or indeed ninety-nine
percent, of all the blue buses belong to the defendant. For, unless there is a
satisfactory explanation for the plaintiff’s singular failure to do more than present
this sort of general statistical evidence, we might well rationally arrive at a
subjective probability of less than 0.5 that defendant’s bus was really involved in
the specific case (italics is mine)”.
(L. Tribe, ‘Trial by mathematics. Precision and ritual in the legal process’, Harvard
Law Review, 84, 1971, p. 1329-1393; p. 1349)
Robustness 2
E = ‘John says that he has been run down by a blue bus’.
H = ‘the vehicle that runned down the plaintiff was a blue bus’.
If plaintiff has been really run down by a blue bus, he has not any reason to lie,
therefore
Prob (E|H = z) = Prob (E|H = w) = 1
But what about
Prob (E|not-H = y) = ?
For satisfying the preponderance of evidence standard against the company Z, the
probability that the plaintiff lies must be less than 7%.
Robustness 2
Prob(E)=[Prob(E|H=z)x Prob(H=z)] + [Prob(E|H=w)x Prob(H=w)] + [Prob(E|not-H)x Prob(not-H)]
= (1 x 0.08) + (1 x 0.02) + (0.07 x 0.9) = 0.163
Prob (H =z|E) = Prob (E|H = z) Prob (H = z) = 0.08 = 0.49
Prob (E)
0.163
Robustness 2
S = ‘John saw a blue bus’;
B = ‘John believes he saw a blue bus’;
E = ‘John says it was a blue bus’.
Initialize the net with prior probabilities:
Prob (H = z) = 0.08; Prob (H = w) = 0.02; Prob (not-H) = 0.9
Prob (AC) = 1;
Prob (O) = 1;
Prob (V = v1) = 0.93; Prob (V =v2) = 0.07; Prob (V = v3) = 0
Then, if update with Prob (E) = 1, Prob (H = z|E) = 0.49
(HUGIN 6.5)
H
H*
AC
B
S
O
E
V
Object-oriented Bayesian network class for the analysis of the reliability of human
witnesses.
H is the output node, H*, AC, O and V are the input nodes and S, B, and E are the
internal nodes.
The analysis of a witness’ credibility starts with the prior probability of the
hypothesis of interest H as input, and the end of the analysis provides as an output
the posterior probability of the same hypothesis.
One can deal with this situation by using a ‘dummy variable’ H*:
H*
H
H*:
t
f
_________________________
Prob (H = t | H*)
1
0
Prob (H = f | H*)
0
1
Appendix: the witness who does not remember
E = ‘X says that he does not remember H’
Conditional probability table for the node E
Vi :
v1
v2
v3 .
B:
t f t f
t f
______________________________________
Prob (E = t | B, Vi)
0 1
1 1
0 0
Prob (E = f | B, Vi)
1
0
0
0
1
1