Ldb Convergenze Parallele_trueblood_02

Modeling Human Judgments with Quantum Probability

Theory

Jennifer S. TruebloodUniversity 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 versus Vectors

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

B̄

F̄

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|B̄i|Bi = .31|F i+ .95|F̄ i|B̄i = .95|F i � .31|F̄ i

| i = 0.46|F̄ i � .89|F i

p(F ) = (�.89)2 = 0.79

p(B) = (.16)2 = 0.026p(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 = feministp(B) = ||PB | i||2

= ||PB · I · | i||2= ||PB(PF + PF̄ )| i||2= ||PBPF | i+ PBPF̄ | i||2= ||PBPF | i||2 + ||PBPF̄ | i||2 + IntB

B

B̄

F̄

F| i






p(F \B) = p(F )p(B|F )= ||PBPF | i||2

B

B̄

F̄

F| i

Feminist is considered first because it is more representative of Linda






p(F \B) = p(F )p(B|F )= ||PBPF | i||2

p(F \B) > p(B) =) IntB < �||PBPF̄ | i||2

B

B̄

F̄

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 FallacyLinda 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 · (PBPF̄ )] < �p(F̄ )p(B|F̄ )

• Disjunction fallacy requires

2 ·Re[(PF̄PB )T · (PF̄PB̄ )] < �p(B̄)p(F̄ |B̄)

• 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||2sim(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

Com

mun

ist

Not c ommunist

C aribbean

Not C a ribbea n

Russia

J ama ic

aCub

a

Sim(R, J) / ||PJ | Ri||2 = cos

2(✓RC + ✓CJ) = 0.33

Sim(C, J) / ||PJ | Ci||2 = cos

2✓CJ = 0.79

Sim(R,C) / ||PC | Ri||2 = cos

2✓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




• Incompatible events

• Two events cannot be realized simultaneously

• Events are processed sequentially


Compatibility







} ClassicProbability


Compatibility







} }QuantumProbability

ClassicProbability


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 the Voting Basis:

Id = {U |Di, U |Ri, U |Ii}

V = {|Di, |Ri, |Ii} Id = {|Li, |Ci, |Mi}


What if Voting 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







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







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

• Eight total sequential judgments (2 cases x 2 orders x 2 strengths)


ExamplePeople 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.




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.





•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.




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.




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 Cases0

0.2

0.4

0.6

0.8

1SD versus SP

Prob

abilit

y of

Gui

lt

SP,SDSD,SP


0.2

0.4

0.6

0.8

1SD versus WP

Prob

abilit

y of

Gui

lt

WP,SDSD,WP


0.2

0.4

0.6

0.8

1WD versus SP

Prob

abilit

y of

Gui

lt

SP,WDWD,SP


0.2

0.4

0.6

0.8

1WD versus WP

Prob

abilit

y of

Gui

lt

WP,WDWD,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

664

!h1ê+

!h1ê�

!h2ê+

!h2ê�

3

775 =)

2

664

↵h1ê+

↵h1ê�

↵h2ê+

↵h2ê�

3

775 =)

2

664

↵h1ê+

0↵h2ê+

0

3

775

Neutral Prosecution

Prosecution

Upn Positive Evidence


State Revision • Suppose the prosecution presents evidence (e+) favoring guilt

2

664

!h1ê+

!h1ê�

!h2ê+

!h2ê�

3

775 =)

2

664

↵h1ê+

↵h1ê�

↵h2ê+

↵h2ê�

3

775 =)

2

664

↵h1ê+

0↵h2ê+

0

3

775

Neutral Prosecution

Prosecution

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

Prosecution

Negative Evidence

2

664

↵h1ê+

0↵h2ê+

0

3

775 =)

2

664

�h1ê+

�h1ê�

�h2ê+

�h2ê�

3

775 =)

2

664

0�h1ê�

0�h2ê�

3

775

Defense Defense

Udp


State Revision • Now, suppose the defense presents evidence (e-) favoring innocence

Prosecution

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

664

↵h1ê+

0↵h2ê+

0

3

775 =)

2

664

�h1ê+

�h1ê�

�h2ê+

�h2ê�

3

775 =)

2

664

0�h1ê�

0�h2ê�

3

775

Defense Defense

Udp


Example Fits


0.2

0.4

0.6

0.8

1Quantum Model: SD versus SP

Prob

abilit

y of

Gui

lt

SP,SD (data)SD,SP (data)SP,SD (QI)SD,SP (QI)


0.2

0.4

0.6

0.8

1Quantum Model: SD versus WP

Prob

abilit

y of

Gui

lt

WP,SD (data)SD,WP (data)WP,SD (QI)SD,WP (QI)


0.2

0.4

0.6

0.8

1Averaging Model: SD versus SP

Prob

abilit

y of

Gui

lt

SP,SD (data)SD,SP (data)SP,SD (Avg)SD,SP (Avg)


0.2

0.4

0.6

0.8

1Averaging Model: SD versus WP

Prob

abilit

y of

Gui

lt

WP,SD (data)SD,WP (data)WP,SD (Avg)SD,WP (Avg)


Example Fits


0.2

0.4

0.6

0.8

1Quantum Model: SD versus SP

Prob

abilit

y of

Gui

lt

SP,SD (data)SD,SP (data)SP,SD (QI)SD,SP (QI)


0.2

0.4

0.6

0.8

1Quantum Model: SD versus WP

Prob

abilit

y of

Gui

lt

WP,SD (data)SD,WP (data)WP,SD (QI)SD,WP (QI)


0.2

0.4

0.6

0.8

1Averaging Model: SD versus SP

Prob

abilit

y of

Gui

lt

SP,SD (data)SD,SP (data)SP,SD (Avg)SD,SP (Avg)


0.2

0.4

0.6

0.8

1Averaging Model: SD versus WP

Prob

abilit

y of

Gui

lt

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.


•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 li\le 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






2

4

6

8

10

12

14

16

18

20Belief−Adjustment Model

Con

fiden

ce in

Gui

lt

P,D (data)D,P (data)P,D (B−A)D,P (B−A)


2

4

6

8

10

12

14

16

18

20Quantum Model

Con

fiden

ce in

Gui

lt

P,D (data)D,P (data)P,D (QI)D,P (QI)

N = 164Thursday, September 5, 13





2

4

6

8

10

12

14

16

18

20Belief−Adjustment Model

Con

fiden

ce in

Gui

lt

P,D (data)D,P (data)P,D (B−A)D,P (B−A)


2

4

6

8

10

12

14

16

18

20Quantum Model

Con

fiden

ce in

Gui

lt

P,D (data)D,P (data)P,D (QI)D,P (QI)

N = 164Thursday, September 5, 13

Thank You

• What’s coming next...

• Quantum Dynamics

• Disjunction Effect and Violations of Savage's Sure Thing Principle

• Comparing Quantum and Markov Models with the Prisoner’s Dilemma Game


Small Business & Entrepreneurship

Ldb Convergenze Parallele_trueblood_02