Ldb Convergenze Parallele_trueblood_02

Modeling Human Judgments
with Quantum Probability
Theory
Jennifer S.Trueblood
University of California, Irvine
Thursday, September 5, 13

Outline
1.Comparing Quantum and Classical Probability
2.Conjunction and Disjunction Fallacies
3. Similarity Judgments
4. Order Effects in Inference

Comparing Quantum
and Classical Probability

Sets versusVectors
Classical Probability Quantum Probability
• Sample space S is a set
of N points
• Hilbert space H: spanned
by a set S of N basis vectors
• Event A ⊆ S
• If A ⊆ S and B ⊆ S
• ¬A = S/A
•A ∩ B
•A ∪ B
• Events in S form a Boolean
algebra
• Event A = span(SA ⊆ S)
• If A = span(SA ⊆ S),
B = span(SB ⊆ S)
•A⊥ = span(S/SA)
•A ⋀ B = span(SA ⋂ SB)
•A ⋁ B = span(SA ⋃ SB)
• Events form a Boolean algebra
if the basis for H is ﬁxed

Comparing Probability Functions

Conditional Probability

Distributive Axiom

Compatibility

Conjunction and
Disjunction Fallacies

Conjunction and Disjunction
Fallacies
• Story: Linda majored in philosophy, was concerned about social justice, and
was active in the anti-nuclear movement (Tversky & Kahneman, 1983)
p(feminist) > p(feminist or bank teller) > p(feminist and bank teller) > p(bank teller)
Disjunction Fallacy Conjunction Fallacy
• Task: Rate the probability of the following events (Morier & Borgida, 1984)
• Linda is a feminist (.83)
• Linda is a bank teller (.26)
• Linda is a feminist and a bank teller (.36)
• Linda is a feminist or a bank teller (.60)

Geometric Account of the
Conjunction Fallacy
• B = bank teller; F = feminist B
¯B
¯F
F
| i
Busemeyer, J. R., Pothos, E., Frano, R., & Trueblood, J. S.
(2011). A quantum theoretical explanation for probability
judgment error. Psychological Review, 118, 193-218.
| i = .16|Bi .99| ¯Bi
|Bi = .31|Fi + .95| ¯Fi
| ¯Bi = .95|Fi .31| ¯Fi
| i = 0.46| ¯Fi .89|Fi
p(F) = ( .89)2
= 0.79
p(B) = (.16)2
= 0.026
p(B|F) = (.31)2
= 0.096
p(F)p(B|F) = 0.076
{
P(F “and then” B):

Analytic Result for the
Conjunction Fallacy
• B = bank teller; F = feminist
p(B) = ||PB| i||2
= ||PB · I · | i||2
= ||PB(PF + P ¯F )| i||2
= ||PBPF | i + PBP ¯F | i||2
= ||PBPF | i||2
+ ||PBP ¯F | i||2
+ IntB
B
¯B
¯F
F
| i

Conjunction Fallacy
p(B) = ||PB| i||2
= ||PB · I · | i||2
= ||PB(PF + P ¯F )| i||2
= ||PBPF | i + PBP ¯F | i||2
= ||PBPF | i||2
+ ||PBP ¯F | i||2
+ IntB
p(F B) = p(F)p(B|F)
= ||PBPF | i||2
B
¯B
¯F
F
| i
Feminist is considered ﬁrst because
it is more representative of Linda

Conjunction Fallacy
p(B) = ||PB| i||2
= ||PB · I · | i||2
= ||PB(PF + P ¯F )| i||2
= ||PBPF | i + PBP ¯F | i||2
= ||PBPF | i||2
+ ||PBP ¯F | i||2
+ IntB
p(F B) = p(F)p(B|F)
= ||PBPF | i||2
p(F B) > p(B) =) IntB < ||PBP ¯F | i||2
B
¯B
¯F
F
| i
Same type of analysis can be used to derive
the disjunction fallacy in a completely
consistent way
Feminist is considered ﬁrst because
it is more representative of Linda

Interference
• Int = p(B) - [p(F)p(B | F) + p(~F)p(B | ~F)] < 0
• Direct route is not as effective for retrieving conclusion
as the sum of the indirect routes
• Availability type mechanism

Disjunction Fallacy
Linda is not ‘a bank teller or feminist’
iff
Linda is ‘not a bank teller’ and ‘not a feminist’
Fallacy occurs when
p(F) > p(F or B) = 1 - p(~B)p(~F | ~B)
that is when
p(~F) < p(~B)p(~F | ~B) Int < 0

Explaining Both Fallacies
• Conjunction fallacy requires
2 · Re[(PBPF )T
· (PBP ¯F )] < p( ¯F)p(B| ¯F)
• Disjunction fallacy requires
2 · Re[(P ¯F PB )T
· (P ¯F P ¯B )] < p( ¯B)p( ¯F| ¯B)
• Both together
p(B)p(F|B) < p(F)p(B|F)

Similarity Judgments

Similarity-Distance
Hypothesis
Similarity is a decreasing
function of distance

Distance Axioms
• D(X,Y) > 0, X ≠Y
• D(X,Y) = 0, X =Y
• D(X,Y) = D(Y,X) symmetry
• D(X,Y) + D(Y,Z) > D(X,Z) triangle
inequality

Asymmetry Finding
(Tversky, 1977)
• How similar is Red China to North Korea?
• Sim(C,K)
• How similar is North Korea to Red China?
• Sim(K,C)
• Sim(K,C) > Sim(C,K)

Tversky’s Similarity
Feature Model
• Based on differential weighting of the
common and distinctive features
• Weights are free parameters and
alternative values lead to violations of
symmetry in the observed or opposite
directions
!"#"$%&"'( !, ! = !" ! ∩ ! − !" ! − ! − !"(! − !)!

Quantum Model of
Similarity
Pothos, E., Busemeyer, J. R., & Trueblood, J. S. (in review). A
quantum geometric model of similarity
sim(A, B) = ||PBPA| i||2

A quantum account of
asymmetry
C hina
Korea
Korea
C hina
sim(C, K) = ||PKPC| i||2
= ||PK| Ci||2
||PC| i||2
sim(K, C) = ||PCPK| i||2
= ||PC| Ki||2
||PK| i||2
||PK| i||2
= ||PC| i||2
||PC| Ki||2
> ||PK| Ci||2 Projection to a subspace of larger dimensionality will
preserve more of the amplitude of the state vector
State vector is assumed to be “neutral”

Triangle Inequality
(Tversky, 1977)
• R = Russia, J = Jamaica, C = Cuba
D(R,J) < D(R,C) + D(C, J) Sim(R,J) > Sim(R, C) + Sim(C,J)
• Findings
1. Sim(R,C) is large (politically)
2. Sim(C,J) is large (geography)
3. Sim(R,J) is small
• How can Sim(R,J) be so small when Sim(R,C) and Sim(C,J) are both
large?

Quantum Account of
the Triangle Inequality
Communist
Not
c ommunist
C
aribbean
Not
C
aribbean
Russia
Jamaica
Cuba
Sim(R, J) / ||PJ | Ri||2
= cos2
(✓RC + ✓CJ ) = 0.33
Sim(C, J) / ||PJ | Ci||2
= cos2
✓CJ = 0.79
Sim(R, C) / ||PC| Ri||2
= cos2
✓RC = 0.79

Order Effects in
Inference

Order Effects
≠Thursday, September 5, 13

Order Effects in Inference
• Order effects in jury decision-making:
P(guilt | prosecution, defense) ≠ P(guilty | defense, prosecution)

• The events in simple Bayesian models do not contain order information and
they commute:
P(G|P, D) = P(G|D, P)

• The events in simple Bayesian models do not contain order information and
they commute:
• To account for order effects, Bayesian models need to introduce order
information:
• event O1 that P is presented before D
• event O2 that D is presented before P
P(G|P D O1) 6= P(G|P D O2)
P(G|P, D) = P(G|D, P)

A Quantum Explanation of
Order Effects
• Quantum probability theory provides a
natural way to model order effects
• Two key principles:
• Compatibility
• Unicity

Compatibility
• Compatible events
• Two events can be realized
simultaneously
• There is no order information

Compatibility
simultaneously
• Incompatible events
• Two events cannot be realized
simultaneously
• Events are processed sequentially

Compatibility
simultaneously
simultaneously
}Classic
Probability

Compatibility
simultaneously
simultaneously
}
}Quantum
Probability
Classic
Probability

Unicity
• Classical probability theory obeys the
principle of unicity - there is a single space
that provides a complete and exhaustive
description of all events
• Quantum probability theory allows for
multiple sample spaces
• Incompatible events are represented by
separate sample spaces that are pasted
together in a coherent way

Example
• Voting Event
1. democrat (outcome D)
2. republican (outcome R)
3. independent (outcome I)
• Ideology Event:
1. liberal (outcome L)
2. conservative (outcome C)
3. moderate (outcome M)

Vector Space For Incompatible
Events
• Represented by two basis for the same 3
dimensional vector spaceD
R
I
C
M
L
• Ideology Basis:
L = liberal
C = conservative
M = moderate
• Voting Basis:
D = democrat
R = republican
I = independent
• Ideology Basis is a unitary
transformation of theVoting Basis:
Id = {U|Di, U|Ri, U|Ii}
V = {|Di, |Ri, |Ii} Id = {|Li, |Ci, |Mi}

What ifVoting and Ideology are
Compatible?
p(L) p(C) p(M)
p(D) p(D ∩ L) p(D ∩ C) p(D ∩ M)
p(R) p(R ∩ L) p(R ∩ C) p(R ∩ M)
p(I) p(I ∩ L) p(I ∩ C) p(I ∩ M)
Large nine
dimensional
sample space
Classical probability representation

Multiple Sample Spaces
• Quantum probability does not require
probabilities to be assigned to all joint
events
• Incompatible events result in a low
dimensional vector space
• Quantum probability provides a simple and
efﬁcient way to evaluate events within
human processing capabilities

When are events Compatible
versus Incompatible?
It is hypothesized, that incompatible
representations are adopted when
1. situations are uncertain and individuals
do not have a wealth of past experience
2. information is provided by different
sources with different points of view

Experiment 1: Order Effects in
Criminal Inference
• 291 participants read eight ﬁctitious criminal cases and reported the
likelihood of the defendant’s guilt (between 0 and 1):
1.Before reading the prosecution or defense
2. After reading either the prosecution or defense
3. After reading the remaining case

Criminal Inference
• Two strength levels for each case: strong (S) and weak (W)

Criminal Inference
• Two strength levels for each case: strong (S) and weak (W)
• Eight total sequential judgments (2 cases x 2 orders x 2 strengths)

Example
People
v.
Robins
Indictment:
The
defendant
Janice
Robins
is
charged

with
stealing
a
motor
vehicle.
Facts:
On
the
night
of
June
10th,
a
blue
Oldsmobile
was

stolen
from
the
Quick
Sell
car
lot.
The
defendant
was

arrested
the
following
day
aFer
the
police
received
an

anonymous
Gp.

Example
People
v.
Robins
Indictment:
The
defendant
Janice
Robins
is
charged

with
stealing
a
motor
vehicle.
Here
is
a
summary
of
the
prosecuGon’s
case:
•Security
cameras
at
the
Quick
Sell
car
lot
have
footage
of
a
woman
matching

Robin’s
descripGon
driving
the
blue
Oldsmobile
from
the
lot
on
the
night
of

June
10th.

Example
People
v.
Robins
Indictment:
The
defendant
Janice
Robins
is
charged

with
stealing
a
motor
vehicle.
Here
is
a
summary
of
the
prosecuGon’s
case:
•Security
cameras
at
the
Quick
Sell
car
lot
have
footage
of
a
woman
matching

Robin’s
descripGon
driving
the
blue
Oldsmobile
from
the
lot
on
the
night
of

June
10th.
•During
the
day
on
June
10th,
Robins
had
come
to
the
Quick
Sell
car
lot
and
had

talked
to
Vincent
Brown,
the
owner,
about
buying
the
blue
Oldsmobile
but
leF

without
purchasing
the
car.
•The
car
was
found
outside
of
the
Dollar
General.
Robins
is
an
employee
of
the

Dollar
General.

Example
People
v.
Robins
Indictment:
The
defendant
Janice
Robins
is
charged

with
stealing
a
motor
vehicle.
Here
is
a
summary
of
the
defense’s
case:
•Robins’
roommate,
Beth
Stall,
was
with
Robins
at
home
on
the
night
of
June

10th.
Stall
claims
that
Robins
never
leF
their
home.

Example
People
v.
Robins
Indictment:
The
defendant
Janice
Robins
is
charged

with
stealing
a
motor
vehicle.
Here
is
a
summary
of
the
defense’s
case:
•Robins’
roommate,
Beth
Stall,
was
with
Robins
at
home
on
the
night
of
June

10th.
Stall
claims
that
Robins
never
leF
their
home.
•Robins
recently
inherited
a
large
sum
of
money.
While
interested
in
acquiring
a

new
car,
she
has
no
reason
to
steal
one.
•Robins
has
no
criminal
convicGons.

Exp. 1 Results
Before Either Case After the First Case After Both Cases
0
0.2
0.4
0.6
0.8
1
SD versus SP
ProbabilityofGuilt
SP,SD
SD,SP
0
0.2
0.4
0.6
0.8
1
SD versus WP
ProbabilityofGuilt
WP,SD
SD,WP
0
0.2
0.4
0.6
0.8
1
WD versus SP
ProbabilityofGuilt
SP,WD
WD,SP
0
0.2
0.4
0.6
0.8
1
WD versus WP
ProbabilityofGuilt
WP,WD
WD,WP
Trueblood, J. S. & Busemeyer, J. R. (2011). A quantum
probability account of order effects in inference.
Cognitive Science, 35, 1518-1552.

Modeling Order Effects
• Two models of order effects:
1. Belief-adjustment model (Hogarth & Einhorn, 1992)
• Accounts for order effects by either adding or averaging
evidence

Modeling Order Effects
• Two models of order effects:
1. Belief-adjustment model (Hogarth & Einhorn, 1992)
• Accounts for order effects by either adding or averaging
evidence
2. Quantum inference model (Trueblood & Busemeyer, 2011):
• Accounts for order effects by transforming a state vector
with different sequences of operators for different
orderings of information

Belief-Adjustment Model
• The belief-adjustment model assumes individuals update beliefs by a
sequence of anchoring-and-adjustment processes:
• 0 ≤ Ck ≤ 1is the degree of belief in the defendant’s guilt after case k
• s(xk) is the strength of case k
• R is a reference point
• 0 ≤ wk ≤ 1 is an adjustment weight for case k
Ck = Ck 1 + wk · (s(xk) R)

Belief-Adjustment Model
• The belief-adjustment model assumes individuals update beliefs by a
sequence of anchoring-and-adjustment processes:
• 0 ≤ Ck ≤ 1is the degree of belief in the defendant’s guilt after case k
• s(xk) is the strength of case k
• R is a reference point
• 0 ≤ wk ≤ 1 is an adjustment weight for case k
• Differences in evidence encoding result in two versions of the model:
1. adding model
2. averaging model
Ck = Ck 1 + wk · (s(xk) R)

Quantum Inference Model
• Two complementary hypotheses: h1 = guilty and h2 = not guilty
• The prosecution (P) presents evidence for guilt (e+)
• The defense (D) presents evidence for innocence (e-)

• The patterns hi ⋀ ej deﬁne a 4-D vector space

• The patterns hi ⋀ ej deﬁne a 4-D vector space
• Jurors consider three points of view: neutral, prosecutor’s, and defense’s
• Basis vectors for the three points of view
1.neutral:
2.prosecutor’s:
3.defense’s:
|Niji
|Piji
|Diji

Changes in Perspective
• Unitary transformations relate one point of view to another
and correspond to an individual’s shifts in perspective

State Revision
• Suppose the prosecution presents evidence (e+) favoring guilt
2
6
6
4
!h1ê+
!h1ê
!h2ê+
!h2ê
3
7
7
5 =)
2
6
6
4
↵h1ê+
↵h1ê
↵h2ê+
↵h2ê
3
7
7
5 =)
2
6
6
4
↵h1ê+
0
↵h2ê+
0
3
7
7
5
Neutral P rosecution P rosecution
Upn Positive Evidence

State Revision
• Suppose the prosecution presents evidence (e+) favoring guilt
2
6
6
4
!h1ê+
!h1ê
!h2ê+
!h2ê
3
7
7
5 =)
2
6
6
4
↵h1ê+
↵h1ê
↵h2ê+
↵h2ê
3
7
7
5 =)
2
6
6
4
↵h1ê+
0
↵h2ê+
0
3
7
7
5
Neutral P rosecution P rosecution
Upn Positive Evidence
• Projection is normalized to ensure that the length of the new state equals one
• When the individual is questioned about the probability of guilt, the revised
state is projected onto the guilty subspace

State Revision
• Now, suppose the defense presents evidence (e-) favoring innocence
P rosecution
Negative Evidence
2
6
6
4
↵h1ê+
0
↵h2ê+
0
3
7
7
5 =)
2
6
6
4
h1ê+
h1ê
h2ê+
h2ê
3
7
7
5 =)
2
6
6
4
0
h1ê
0
h2ê
3
7
7
5
Defense Defense
Udp

State Revision
• Now, suppose the defense presents evidence (e-) favoring innocence
P rosecution
Negative Evidence
• Normalize the project and project onto the guilty subspace
• A total of 4 parameters are used to define all of the unitary transformations
2
6
6
4
↵h1ê+
0
↵h2ê+
0
3
7
7
5 =)
2
6
6
4
h1ê+
h1ê
h2ê+
h2ê
3
7
7
5 =)
2
6
6
4
0
h1ê
0
h2ê
3
7
7
5
Defense Defense
Udp

Example Fits
0
0.2
0.4
0.6
0.8
1
Quantum Model: SD versus SP
ProbabilityofGuilt
SP,SD (data)
SD,SP (data)
SP,SD (QI)
SD,SP (QI)
0
0.2
0.4
0.6
0.8
1
Quantum Model: SD versus WP
ProbabilityofGuilt
WP,SD (data)
SD,WP (data)
WP,SD (QI)
SD,WP (QI)
0
0.2
0.4
0.6
0.8
1
Averaging Model: SD versus SP
ProbabilityofGuilt
SP,SD (data)
SD,SP (data)
SP,SD (Avg)
SD,SP (Avg)
0
0.2
0.4
0.6
0.8
1
Averaging Model: SD versus WP
ProbabilityofGuilt
WP,SD (data)
SD,WP (data)
WP,SD (Avg)
SD,WP (Avg)

Fits to the Data
• Three models (averaging, adding, and quantum) were ﬁt to twelve data points
for eight crime scenarios
• All three models have the same number of parameters (i.e., 4)
• R2 values for three models:
• Averaging: R2 = 0.76
• Adding: R2 = 0.98
• Quantum: R2 = 0.98

Quantum versus Adding
•Need
a
new
experiment
to
disGnguish

the
quantum
and
adding
models
•The
“irrefutable
defense”
experiment
•ProsecuGon
is
strong,
but
defense

is
irrefutable

Experiment 2: Irrefutable Defense
• Indictment:
The
defendant
Paul
Jackson
is
charged
with

robbing
an
art
museum.
Facts:
On
December
12th,
a
burglar
broke
into
the
Central
City
Art

Museum.
The
alarm
at
the
museum
noGﬁed
police
of
the
break-‐in

at
8:00
pm
that
night.
Paul
Jackson
was
arrested
the
next
day

when
the
police
received
an
anonymous
Gp.

Indictment:
The
defendant
Paul
Jackson
is
charged
with
robbing
an

art
museum.
Here
is
a
summary
of
the
prosecuGon’s
case:
•Jackson
frequently
visits
the
Central
City
Art
Museum,
and
a
security
guard
told

police
he
saw
a
man
matching
Jackson’s
descripGon
near
the
museum
around
8:00

pm
on
the
night
of
the
burglary.
Another
witness
told
police
they
saw
a
man

matching
the
defendants
descripGon
running
from
the
museum
a
lile
aFer
8:00

pm.

Indictment:
The
defendant
Paul
Jackson
is
charged
with
robbing
an

art
museum.
Here
is
a
summary
of
the
defense’s
case:
•Jackson
was
teaching
a
class
on
the
opposite
side
of
town
at
Central
City

University
between
7
and
9
pm
on
the
night
of
the
burglary.
There
were
ﬁFy

students
present
at
his
class
that
evening.
This
parGcular
class
meets
three
Gmes
a

week,
and
the
students
are
well
acquainted
with
Jackson.
Furthermore,
Jackson
has

an
idenGcal
twin
brother
who
has
a
criminal
record

A Priori Predictions
• Quantum model predicts that the prosecution will produce a major effect
when presented ﬁrst, but no effect when presented after the irrefutable
defense
• The adding model predicts that the prosecution will have the same effect in
both situations

A Priori Predictions
• Quantum model predicts that the prosecution will produce a major effect
when presented ﬁrst, but no effect when presented after the irrefutable
defense
• The adding model predicts that the prosecution will have the same effect in
both situations
0
2
4
6
8
10
12
14
16
18
20
Belief−Adjustment Model
ConfidenceinGuilt
P,D (data)
D,P (data)
P,D (B−A)
D,P (B−A)
0
2
4
6
8
10
12
14
16
18
20
Quantum Model
ConfidenceinGuilt
P,D (data)
D,P (data)
P,D (QI)
D,P (QI)
N = 164

ThankYou
• What’s coming next...
• Quantum Dynamics
• Disjunction Effect andViolations of Savage's Sure
Thing Principle
• Comparing Quantum and Markov Models with the
Prisoner’s Dilemma Game

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