Quantum Models of
Human Judgments
Jennifer S. Trueblood
Vanderbilt University
Outline
1. Similarity Judgments
2. Order Effects in Inference
3. Causal Reasoning
Similarity Judgments
Similarity-Distance
Hypothesis
Similarity is a decreasing
function of distance
Distance Axioms
1. D(X,Y) > 0, X ≠ Y
2. D(X,Y) = 0, X = Y
3. D(X,Y) = D(Y,X) symmetry
4. 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
sim(A, B) = ||PB PA | i||
2
Pothos, E., Busemeyer, J. R., & Trueblood, J. S. (2013). A
quantum geometric model of similarity. Psych Review.
Ko
re
a
Ko
re
a
A quantum account of
asymmetry
C hina
C hina
sim(K, C) = ||PC PK | i||
= ||PC |
K i||
2
||PK | i||
2
||PK | i|| = ||PC | i||
||PC |
K i||
2
> ||PK |
2
C i||
2
2
sim(C, K) = ||PK PC | i||
= ||PK |
C i||
2
||PC | i||
2
2
State vector is assumed to be “neutral”
2
Projection to a subspace of larger dimensionality will
preserve more of the amplitude of the state vector
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?
n
Cu
Ru ssia
ba
C
ar
ib
b
ea
C om m unist
Quantum Account of
the Triangle Inequality
Ja
m
a
c
i
a
Sim(R, C) / ||PC |
Sim(C, J) / ||PJ |
R i||
2
2
= cos ✓RC = 0.79
2
2
i||
=
cos
✓CJ = 0.79
C
Not c om m unist
No
Sim(R, J) / ||PJ |
t C
ar
ib
b
ea
n
2
2
i||
=
cos
(✓RC + ✓CJ ) = 0.33
R
Order Effects in
Inference
Order Effects
≠
Order Effects in Inference
Order effects in jury decision-making:
P(guilt | prosecution, defense) ≠ P(guilty | defense, prosecution)
The events in simple classical probability models do not contain order
information and they commute:
P (G|P, D) = P (G|D, P )
To account for order effects, classical probability models need to introduce
order information:
P (G|P \ D \ O1 ) 6= P (G|P \ D \ O2 )
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
1.Two events can be realized simultaneously
2.There is no order information
Compatibility
Compatible events
1.Two events can be realized simultaneously
2.There is no order information
Incompatible events
3.Two events cannot be realized
simultaneously
4.Events are processed sequentially
Compatibility
Compatible events
}
1.Two events can be realized simultaneously
2.There is no order information
Incompatible events
3.Two events cannot be realized
simultaneously
4.Events are processed sequentially
Classic
Probability
}
Quantum
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
Ideology Event:
1. democrat (D)
1. liberal (L)
2. republican (R)
2. conservative (C)
3. independent (I)
3. moderate (M)
Vector Space For Incompatible
Events
Represented by two basis for the same 3
dimensional vector space
D
•
L
C
R
Voting Basis:
L = liberal
R = republican
C = conservative
I = independent
M
•
Ideology Basis:
D = democrat
V = {|Di, |Ri, |Ii}
I
•
M = moderate
Id = {|Li, |Ci, |M i}
Ideology Basis is a unitary
transformation of the Voting Basis:
Id = {U |Di, U |Ri, U |Ii}
What if Voting and Ideology are
Compatible?
Classical probability representation
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
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
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
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
SD versus SP
SD versus WP
1
SP,SD
SD,SP
0.8
Probability of Guilt
Probability of Guilt
0.8
1
0.6
0.4
0.2
WP,SD
SD,WP
0.6
0.4
0.2
0
Before Either Case
After the First Case
0
Before Either Case
After Both Cases
WD versus SP
1
SP,WD
WD,SP
0.8
Probability of Guilt
Probability of Guilt
After Both Cases
WD versus WP
1
0.8
After the First Case
0.6
0.4
0.2
0
Before Either Case
WP,WD
WD,WP
0.6
0.4
0.2
After the First Case
After Both Cases
0
Before Either Case
After the First Case
After Both Cases
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 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:
Ck = Ck
1
+ wk · (s(xk )
R)
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
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
Jurors consider three points of view: neutral, prosecutor’s, and defense’s
Basis vectors for the three points of view
|Nij i
|Pij i
|Dij i
Changes in Perspective
Unitary transformations relate one point of view to another
State Revision
Suppose the prosecution presents evidence (e+) favoring guilt
2
3
2
3
2
3
!h1 ^e+
↵h1 ^e+
↵h1 ^e+
6!h1 ^e 7
6↵h1 ^e 7
6 0 7
7 =) 6
7
6
7 =) 6
4!h2 ^e+ 5
4↵h2 ^e+ 5
4↵h2 ^e+ 5
0
!h2 ^e
↵h2 ^e
Upn
Positive Evidence
N eutral
P rosecution
P rosecution
State Revision
Suppose the prosecution presents evidence (e+) favoring guilt
2
3
2
3
2
3
!h1 ^e+
↵h1 ^e+
↵h1 ^e+
6!h1 ^e 7
6↵h1 ^e 7
6 0 7
7 =) 6
7
6
7 =) 6
4!h2 ^e+ 5
4↵h2 ^e+ 5
4↵h2 ^e+ 5
0
!h2 ^e
↵h2 ^e
Upn
Positive Evidence
N eutral
•
P rosecution
P rosecution
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
2
3
2
↵h1 ^e+
6 0 7
6
6
7 =) 6
4↵h2 ^e+ 5
4
0
Udp
P rosecution
h1 ^e+
3
2
7
6
7 =) 6
5
4
h2 ^e+
h1 ^e
h2 ^e
0
h1 ^e
0
Negative Evidence
Def ense
h2 ^e
3
7
7
5
Def ense
State Revision
Now, suppose the defense presents evidence (e-) favoring innocence
2
3
2
↵h1 ^e+
6 0 7
6
6
7 =) 6
4↵h2 ^e+ 5
4
0
Udp
P rosecution
•
h1 ^e+
3
2
7
6
7 =) 6
5
4
h2 ^e+
h1 ^e
h2 ^e
0
h1 ^e
0
Negative Evidence
Def ense
h2 ^e
3
7
7
5
Def ense
Normalize the project and project onto the guilty subspace
•A
total of 4 parameters are used to define all of the unitary transformations
Example Fits
Quantum Model: SD versus WP
1
0.8
0.8
0.6
0.4
SP,SD (data)
SD,SP (data)
SP,SD (QI)
SD,SP (QI)
0.2
0
Before Either Case
After the First Case
Probability of Guilt
Probability of Guilt
Quantum Model: SD versus SP
1
0.6
0.4
0.2
0
Before Either Case
After Both Cases
0.8
0.8
0.4
SP,SD (data)
SD,SP (data)
SP,SD (Avg)
SD,SP (Avg)
0.2
0
Before Either Case
After the First Case
After Both Cases
After the First Case
After Both Cases
Averaging Model: SD versus WP
1
Probability of Guilt
Probability of Guilt
Averaging Model: SD versus SP
1
0.6
WP,SD (data)
SD,WP (data)
WP,SD (QI)
SD,WP (QI)
0.6
0.4
WP,SD (data)
SD,WP (data)
WP,SD (Avg)
SD,WP (Avg)
0.2
0
Before Either Case
After the First Case
After Both Cases
Fits to the Data
Two models (averaging and quantum) were fit to twelve data points for eight crime
scenarios
Both models have the same number of parameters (i.e., 4)
R2 values for the models:
1. Averaging: R2 = 0.76
2. Quantum: R2 = 0.98
Coffee Break
Causal Reasoning
Causal Reasoning
•
Causal Learning •
Causal reasoning from verbal descripGons l
a
c ity
i
s ioln Learning the effect of a cs
hemical b 2y000) a & Sahanks, DNA mutaGons (Lober l
C ob or
e
r
Causal r
easoning f
rom h
P T staGsGcal descripGons
•
Imagine you exercise hard in April. How likely is it that you weight less in May? (Fernbach et al., 2010)
Causal Bayes Nets
Account for violations of classical probability
theory by adding hidden variables (e.g., Rehder,
2014)
Z"
X"
Y"
Common%%Cause%%%%%%%%
Z"
Z"
X"
Y"
W"
X"
Z"
Y"
W"
X"
Y"
W"
Quantum Approach: A Hierarchy of Mental RepresentaGons •  Perhaps(people(use(
different(mental(
representa2ons(in(
different(situa2ons(
X
Classical
Probability
Theory
Y
E
Quantum
Probability
Theory
A"Hierarchy"of"Mental"Representa4ons"
"
"
2"Dimensional"
Representa4on"
"
•  All"variables"are"
incompa4ble"
"
"
4"Dimensional"
Representa4on"
"
•  Single"cause@effect"
rela4onships"are"
compa4ble"
"
"
8"Dimensional"
Representa4on"
"
•  All"variables"are"
compa4ble"
•  Classical"probability"
model"
2D Model
y0#
x0#
e0#
X"
| i
y1#
x1#
e1#
Three%free%parameters:%
%
✓x Rota/on%for%x%basis%
✓y
Rota/on%for%y%basis%
↵1
Loca/on%of%state%vector%
Y"
E"
4D Model
4D vector space with two bases: $
x1
x1
x0
x0
e1$
e0$
e1$
e0$
$
y1
y0
y0
y1
e1$
e0$
e1$
e0$
6 free parameters:
-­‐Three rotaGon parameters ✓1 , ✓2 , ✓3
-­‐Three state vector locaGon parameters ↵1 , ↵2 , ↵3
!
!
8D Model
8D vector space with a single bases: x1 y1 e1,& &x1 y1 e0,& &x1 y0 e1,& &x1 y0 e0&
x0 y1 e1,& &x0 y0 e0,& &x0 y0 e1,& &x0 y1 e0&
No rotaGons 7 free parameters to define the locaGon of the state vector !
Causal Power
•  Parameteriza)on,of,the,8D,model,by,causal,power,theory,
(Cheng,,1997),
•  Each,cause,has,a,power,parameter,w"
•  Also,allows,for,unknown,alterna)ve,causes,wa"
•  Condi)onal,probabili)es,are,calculated,by,a,“noisyGor”,
equa)on:,
p(e1 |xj , yk ) = 1 (1 wx )j (1 wy )k (1 wa ).
,
•  Joint,probabili)es,are,calculated,by,combining,condi)onal,
probabili)es,with,prior,probabili)es,of,the,causes:,
p(e1 , x1 , y1 ) = [1
(1
wx )(1
wy )(1
wa )]p(x1 )p(y1 )
•  Associate)coordinates)of)the)state)vector)with)the)joint)
probabili5es)from)causal)power)
•  5)free)parameters:)
–  Three)power)parameters:)wx#,#wy#,#wa#
–  Two)prior)probabili5es)pr(x1))and)pr(y1))
Quantum'Model'Predic1ons'
Three'key'predic1ons:'
'
1.  Order'effects'
–  2D'and'4D'models'
2.  Reciprocity'(Inverse'fallacy)'
–  2D'model'
3.  Memoryless'effects'
–  2D'model'
Experiment 1: Order Effects
Order Effects
•  The$sequen)al$processing$of$events$in$quantum$
probability$theory$naturally$gives$rise$to$order$
effects.$
Pr(E|X, Y)
≠
Pr(E|Y, X)
Experiment 1
•  Causal&reasoning&about&a&novel&biological&
category,&Lake&Victoria&Shrimp&(Rehder,&2003)&
•  Three&binary&features:&
High or low ACh neurotransmitter
Accelerated or decelerated sleep
cycle
Normal or high body weight
Feature RelaGonships
Two causal relaGonships: High%ACh%!%High%Body%Weight%
Low%ACh%!%Normal%Body%Weight%
Accelerated Sleep Cycle ! High Body
Weight
Decelerated Sleep Cycle ! Normal Body
Weight
Sequential Judgments
Pr(high'body'weight)'
Pr(high body weight | high
ACh)
Pr(high body weight | high
ACh,
decelerated sleep
cycle)
Results (N = 94)
p(e = 1| x = 1, y = 1) vs p(e=1| y = 1, x = 1)
80
x = 1, y = 1
y = 1, x = 1
60
40
20
0
Prior
100
Probability of Effect
Probability of Effect
100
p(e=1| x = 1, y = 0) vs p(e=1| y = 0, x = 1)
80
60
40
20
0
After First Cause After Both Causes
p(e=1| x = 0, y = 1) vs p(e=1| y = 1, x = 0)
80
x = 0, y = 1
y = 1, x = 0
60
40
20
0
Prior
After First Cause After Both Causes
Prior
After First Cause After Both Causes
p(e=1| x = 0, y = 0) vs p(e=1| y = 0, x = 0)
100
Probability of Effect
Probability of Effect
100
x = 1, y = 0
y = 0, x = 1
80
x = 0, y = 0
y = 0, x = 0
60
40
20
0
Prior
After First Cause After Both Causes
Modeling Experiment 1
Fit$each$model$to$40$mean$judgments$(20$mean$judgments$per$condi8on)$
Experiment 1: 2−Dimensional Model
Experiment 1: 8−Dimensional Model (Unrestricted)
1
1
0.9
0.8
0.8
0.7
0.7
0.6
0.6
Model
Model
0.9
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
BIC = -185.88
0.1
0
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
BIC = -171.70
0.1
1
0
0.1
0.2
0.3
0.4
Data
0.9
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.4
0.3
0.3
0
BIC = -180.90
0
0.1
0.2
0.3
0.4
0.5
Data
0.6
0.7
0.8
0.7
0.8
0.9
1
0.5
0.4
0.1
0.6
Experiment 1: 8−Dimensional Model (Causal Power)
1
Model
Model
Experiment 1: 4−Dimensional Model
1
0.2
0.5
Data
0.9
1
BIC = -174.97
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Data
0.6
0.7
0.8
0.9
1
Experiment 2: Invariances
of the 2D model
Two Invariances
Reciprocity:+
Pr(z1+|+x1)+=+Pr(x1+|+z1)++
+
Memoryless+effect:+
Pr(x1+|+y1)+=+Pr(x1+|+z1,+y1)+
+
+
Y"
X"
Z"
Results Exp. 2
•  Within&subjects&design&(N&=&58)&
•  S6muli,&instruc6ons,&and&learning&stage&
iden6cal&to&Experiment&1&
Probabili(es+
Reciprocity+
Memory+less+
Bayes+Factor+
Pr(x1+|+z1)+
Pr(z1+|+x1)+
6.41+
Pr(y1+|+z1)+
Pr(z1+|+y1)+
7.43+
Pr(x1+|+z1)+
Pr(x1+|z1,+y1)+
2.13+
Pr(x1+|+z0)+
Pr(x1+|z1,+y0)+
2.96+
Pr(y1+|+z1)+
Pr(y1+|z1,+x1)+
1.37+
Pr(y1+|+z0)+
Pr(y1+|z1,+x0)+
6.88+
Conclusions
The 2-­‐D model suggests that only one piece of informaGon is processed at a Gme A mental representaGon without joint events might be a simple and more efficient way to evaluate informaGon Related findings: -­‐“singularity principle” of hypotheGcal thinking (Evans, 2006) -­‐ “structurally local” causal reasoning (Fernbach & Sloman, 2009) Thank You
What’s coming next…
11:30-12:00 pm Quantum dynamics Part 1 (Peter Kvam)
1:00-2:00 pm Quantum dynamics Part 2 (Peter Kvam)
2:00-2:30 pm Advanced tools for building quantum models part 1 (James Yearsley)
2:30-3:00 pm coffee
3:00-3:45 pm Advanced tools for building quantum models part 2 (James Yearsley)
3:45-4:00 pm Wrap up and discussion (James, Jennifer, and Peter)